Bell Theorem and probabilty theory

[JesseM]

Again I am waiting for you to show where you got this equation. What rule or theory of mathematics or logic permits you to write this equation the way you did. Bell did not give a citation and neither have you. Jaynes equations are the result of the well known and established product rule of probability theory. What is the equation you or Bell is using?

The equation just comes from thinking about the statistics of this particular problem, but I'll spell it out a bit. Do you agree that if X and Y are statistically independent, then P(XY)=P(X)*P(Y)? If so, would you also agree that even if X and Y are not independent in the sample space as a whole, if they are independent in the subset of trials where Z occurred, then we can write P(XY|Z)=P(X|Z)*P(Y|Z)? Clearly this sort of thing applies if you consider statements about the hidden state of the first ball and urn after the first drawing, like H1 and H2. If you already know H1, "first ball has hidden color red, urn contains two balls with hidden colors red and white", then knowing A (that the first ball was observed to have the color red when it was revealed) gives you no further knowledge about the likelihood that the second ball will be red (B), and likewise knowing B gives you no further knowledge about the likelihood that the first ball will be observed to be red (in fact knowing H1 leads you to predict A with probability 1).

So, it's reasonable that P(AB|H1) = P(A|H1)*P(B|H1) in this problem (not as a general statement about probabilities in arbitrary problems), and likewise for any other hidden state that might exist immediately after the first ball is picked. It's also reasonable that if every point in the sample space has some H_i associated with it (i.e. on every trial there is some fact of the matter of what the hidden color of the first ball was after it was picked, and what hidden colors remained in the urn), and all the different H_i's are mutually exclusive, then we can write P(AB) = \sum_i P(AB|H_i)*P(H_i). Do you disagree with that? If not, it's easy to combine these equations to get P(AB) = \sum_i P(A|H_i)P(B|H_i)P(H_i).

So if there is some point of disagreement here, please be specific about where it is. Also, if you disagree, please play around with the above equation a little and see if you can find any urn examples where the equation would give the wrong numbers for the probabilities.

 

[mn4j]

Do you agree that if X and Y are statistically independent, then P(XY)=P(X)*P(Y)?

Of course. Working from the product rule
P(XY|Z) = P(X|Z)*P(Y|Z)
the reason P(XY|Z) = P(X|Z)*P(Y|Z) in the case of logical independence is because P(Y|XZ) = P(Y|Z). This means that knowledge of X gives us no additional information about Y, so the equation reduces to P(XY|Z) = P(X|Z)*P(Y|Z).

If so, would you also agree that even if X and Y are not independent in the sample space as a whole, if they are independent in the subset of trials where Z occurred

This makes no sense. If you write P(XY|Z) = P(X|Z)*P(Y|Z), you are saying that based on all the information Z, X is logically independent of Y. The reason this is false in this particular case is precisely because there is logical dependence.

To see this you only need to ask the question, does knowing the spin at A tell me anything about the spin at B? If it does, then they are logically dependent and you can NOT reduce P(Y|XZ) to P(Y|Z). Now remember we are still talking about a specific experiment NOT repeated events. The fact that the experiment can be repeated is inconsequential.

The definitive premise in any hidden variable theorem is that the particles left the source with enough information to determine the outcome at the stations (together with the settings at the station). This means for two entangled particles, any information you gain about one, MUST tell you something about the other. This was the basis of the EPR paradox which Bell was trying to refute. So logical dependence is a MUST in the specific case Bell is treating. Therefore there is no justification for why he reduced P(Y|XZ) to P(Y|Z). Do you understand this?

 

[JesseM]

This makes no sense. If you write P(XY|Z) = P(X|Z)*P(Y|Z), you are saying that based on all the information Z, X is logically independent of Y.

I'm not sure what you mean by the phrase "based on all the information Z, X is logically independent of Y". The equation P(XY|Z)=P(X|Z)*P(Y|Z) certainly doesn't mean X is logically independent of Y in the series of trials as a whole, it just means that in the subset of trials where Z occurred, the frequency of X is independent of Y and vice versa. Another way of putting this is that the statistical dependence of X and Y is wholly accounted for by Z, meaning that if you know Z, then knowing X gives you no additional information about Y, and likewise if you know Z, knowing Y gives you no additional information about X.

Consider the urn example again. If I've already learned that H1 is true (that the first ball selected had the hidden color red prior to being observed) and I want to bet on whether or not B is true (that the second ball selected will be observed to be red), do you agree that knowing A (that the first ball selected was observed to be red) tells me absolutely nothing new that should cause me to modify my bet when all I knew was H1?

Of course this example is somewhat trivial since H1 implies A with probability 1. But you could consider a more complicated example too. Suppose that instead of colored balls, the urn contains three plastic balls with dimmed lights inside, and that pressing a button on top of each ball causes either the internal red light to turn on, or the internal green light. The balls are externally indistinguishable but their hidden internal mechanisms are somewhat different, so to distinguish them we'll call the three balls D, E and F. Each ball has a random element that decides which light will go on when the button is pressed, but because of the different hidden mechanisms the probabilities are different in each case--D has a 70% chance of lighting up red, E has a 90% chance of lighting up red, and F has a 25% chance of lighting up red. Two balls are taken from the urn, one is given to Alice and the second is given to Bob, then they go into separate rooms and press the buttons and note the results.

Now, since the balls are taken from the same urn without replacement, there is some statistical dependence between the results seen by Alice and the results seen by Bob--for example, if Alice sees her ball light up green, that makes it more likely that the she got ball F, which in turn makes it less likely that Bob got F and thus more likely that Bob will see his light up red. Now consider the "hidden variables" condition H1: "ball D was given to Alice, ball E was given to Bob". Do you agree that if we look at the subset of trials where H1 is true, then in this subset, knowing that Alice's ball lit up red gives us no additional information about the likelihood that Bob's ball will light up red? In other words, P(Bob's ball lights up red | ball D given to Alice, ball E given to Bob) = P(Bob's ball lights up red | ball D given to Alice, ball E given to Bob AND Alice's lit up red) = P(Bob's ball lights up red | ball D given to Alice, ball E given to Bob AND Alice's lit up green) = 90%. The fact that Bob got ball E is enough to tell us there's a 90% chance his will light up red, knowing the color Alice saw doesn't further affect our probability calculation at all. In the same way, knowing that Alice got ball D tells us there's a 70% chance hers will light up red, knowing the result Bob got doesn't change the probability further either. So, P(Alice saw red, Bob saw red|H1) = P(Alice saw red|H1)*P(Bob saw red|H1) = 0.9*0.7=0.63. It's only if we don't know the hidden truth about which ball each one got that knowing what color one of them saw can influence our estimate of the probability the other one saw red. Do you disagree with any of this? If you do, please tell me what you would give for the probability P(Alice saw red, Bob saw red|H1) along with the probabilities P(Alice saw red|H1) and P(Bob saw red|H1).

The definitive premise in any hidden variable theorem is that the particles left the source with enough information to determine the outcome at the stations (together with the settings at the station). This means for two entangled particles, any information you gain about one, MUST tell you something about the other.

Of course. But if God came and told you the hidden state \lambda associated with each particle, then in that case knowing the information gained by one experimenter tells you nothing additional about the result the other experimenter will see, just like if God told Alice that she had gotten ball D, then knowing Bob had seen his ball light up red would tell her nothing additional about the probability her own would light up red. That's why for every possible hidden state H (H1, H2, etc.), it is true that P(Alice sees red, Bob sees red|H) = P(Alice sees red|H)*P(Bob sees red|H). Similarly, in a hidden variables theory of QM that's why if you include the complete set of hidden variables \lambda in the conditional, in that case you can write P(AB|\lambda) = P(A|\lambda)*P(B|\lambda)--you're only looking at the subset of trials where \lambda took on some particular value. But I know you'll probably find reason to dispute this, which is why I want to first deal with the simple example of the light-up balls, and see if in that case you dispute that P(Alice sees red, Bob sees red|H) = P(Alice sees red|H)*P(Bob sees red|H).

 

[mn4j]

I'm not sure what you mean by the phrase "based on all the information Z, X is logically independent of Y". The equation P(XY|Z)=P(X|Z)*P(Y|Z) certainly doesn't mean X is logically independent of Y in the series of trials as a whole

You keep mixing up consideration of an individual experiment with consideration of multiple experiments that is why you are confused. We are not yet talking about multiple experiments, that is where the intergral comes in. We are talking about A SINGLE experiment!

Secondly, if P(XY|Z)=P(X|Z)*P(Y|Z) does not mean X is logically independent of Y, then where did you get the equation from. I have already shown you how you can derive P(XY|Z)=P(X|Z)*P(Y|Z) from the product rule P(XY|Z)=P(X|YZ)*P(Y|Z) WHEN AND ONLY WHEN X is logically independent of Y. You have not shown how you can arrive at that equation without assuming logical independence.

This is very basic probability theory. Unless you understand this, we can not even begin to discuss about hidden variables and multiple experiments.

 

[JesseM]

You keep mixing up consideration of an individual experiment with consideration of multiple experiments that is why you are confused. We are not yet talking about multiple experiments, that is where the intergral comes in. We are talking about A SINGLE experiment!

As I already explained, the probability of X on a single experiment is always guaranteed to be equal to the fraction of trials on which X occurred in a large number (approaching infinity) of repeats of the experiment with the same conditions--it doesn't matter if a large number of trials actually happens, all that matters is that this would be true hypothetically the experiment were repeated in this way. Do you disagree with this? Please give me a simple yes or no answer.

Secondly, if P(XY|Z)=P(X|Z)*P(Y|Z) does not mean X is logically independent of Y, then where did you get the equation from.

It comes from the fact that if you only look at the trials where Z occurred and throw out all the trials where it did not, then in that set of trials X is independent of Y. It would help if you addressed my simple numerical example:

Three balls D, E, F; D has 0.7 chance of lighting up red when button pushed, E has 0.9 chance, F has 0.25 chance.

H1 is the condition that "ball D was given to Alice, ball E was given to Bob".

Do you agree that in this case, P(Alice saw red, Bob saw red|H1) = 0.63, exactly the same as P(Alice saw red|H1)*P(Bob saw red|H1)? If not, what probability would you assign to P(Alice saw red, Bob saw red|H1)? Please give me a specific answer to this question.

I have already shown you how you can derive P(XY|Z)=P(X|Z)*P(Y|Z) from the product rule P(XY|Z)=P(X|YZ)*P(Y|Z) WHEN AND ONLY WHEN X is logically independent of Y. You have not shown how you can arrive at that equation without assuming logical independence.

Of course I have. As I already explained, I am simply assuming that X is logically independent of Y in the subset of trials where Z occurs, even though it they are not independent in the set of all trials. Another way of putting this is that if we know Z occurred, then knowing X tells us nothing further about the probability of Y, even though in the absence of knowledge of Z knowledge of X would have given us some new information about the probability of Y. You apparently are having problems understanding this explanation, but I can't help you unless you are willing to actually think about some specific examples like the one I ask above. Can you please just do me the courtesy of answering my simple questions about these examples?

Here's another example--we have a 6-sided die and a 12-sided die, Alice rolls one and Bob rolls the other. If there's a 50/50 chance the 12-sided die will be given to either one of them, and we aren't told who got which die, then the results they see aren't logically independent--for example, if Alice gets a 10 that allows us to infer Bob will get some number between 1 and 6, whereas before we knew what Alice rolled we would have considered it possible Bob could get any number from 1-12. However, in the subset of cases where Alice gets the 12-sided die and Bob gets the 6-sided die (or if you prefer, in a single case where we know Alice got the 12-sided die and Bob got the 6-sided die), Bob has a probability 1/6 of getting any number 1-6, and a probability 0 of getting a number 7-12, and knowing what Alice rolled has absolutely no further effect on these probabilities. Do you disagree? Again, please think about this specific example and answer yes or no.

 

[mn4j]

As I already explained, the probability of X on a single experiment is always guaranteed to be equal to the fraction of trials on which X occurred in a large number (approaching infinity) of repeats of the experiment with the same conditions

Yes. This makes sense provided your sampling is unbiased. However, this point is irrelevant here. We are not arguing about how Bell calculated P(X|Z), we are arguing about how he calculated P(XY|Z) to be P(X|Z)*P(Y|Z) for a single experiment.

It comes from the fact that if you only look at the trials where Z occurred and throw out all the trials where it did not, then in that set of trials X is independent of Y.

Again, when you write P(XY|Z)=P(X|Z)*P(Y|Z), Z is not considered as an outcome, but as a premise. That means Z is assumed to be true. So it is just confusion to be talking about trials in which Z occurred or did not occur. That is why I suspect you understand the meanings of those equations differently than they mean in Probability theory.

P(AB|Z) means: Given that Z is true, what is the probability that both A and B are True.
Let's look at the Bell's specific case.
Z: Two spin 1/2 particles denoted by 1 and 2 were jointly in a pure singlet state in the past, with 1 moving toward station A, and 2 moving toward station B , but they remain jointly in a pure singlet state, in which their spins are perfectly anti-correlated.

A: spin of particle 1 is found in the up direction at station A
B: spin of particle 2 is found in the up direction at station B

Therefore P(AB|Z) means: What is the probability that both particle 1 and particle 2 are found in the up direction given Z.

The second term in P(AB|Z)=P(A|BZ)*P(B|Z), ie P(A|BZ), means that if we know for sure that B is true, ie that that spin at station 2 is found to be 'up', then that INFORMATION should influence the probability we assign to A. Do you disagree with this?

Now to the answer of your question:

Z: Three balls D, E, F; D has 0.7 chance of lighting up red when button pushed, E has 0.9 chance, F has 0.25 chance. D given to Alice and E given to Bob, and button is pressed.
A: Alice sees red
B: Bob sees red

P(AB|Z) = P(A|BZ)P(B|Z) according to the product rule.

P(A|BZ) = P(A|Z) since knowing that Bob saw red, does not tell me anything about what Alice might see. In other words,
A and B are logically independent.

Therefore P(AB|Z) = P(A|Z)P(B|Z) and the answer is 0.7 * 0.9 = 0.63

As you see, I am using the product rule consistently here.

 

[mn4j]

I decided to to post the question as a separate post and changed it a bit so that the result is clearer.

Now answer this one.

Z: Two red LEDs D, E were made on a circuit so that when D was observed to be lit, the probability of E lighting up was 0.2 and when E was lit the probability of D lighting up was 0.2. Also, the circuit was designed such that at least one LED was lit on each button press with no other bias imposed on the LED other than the correlation above. D given to Alice and E given to Bob, and button is pressed.
A: Alice sees red
B: Bob sees red

What probability will you assign to P(AB|Z).

 

[JesseM]

As I already explained, the probability of X on a single experiment is always guaranteed to be equal to the fraction of trials on which X occurred in a large number (approaching infinity) of repeats of the experiment with the same conditions

Yes. This makes sense provided your sampling is unbiased. However, this point is irrelevant here. We are not arguing about how Bell calculated P(X|Z), we are arguing about how he calculated P(XY|Z) to be P(X|Z)*P(Y|Z) for a single experiment.

Yes, but P(XY|Z) can easily be made into a statement about a large number of trials too--just do the experiment many times with the same conditions, then look at the subset of trials where Z did in fact occur, and look at the fraction of these in which X and Y occurred as well. Do you disagree that this fraction should always be the same as P(XY|Z)?

If you don't disagree, then again, my point is just that you can have a situation where X is independent of Y in this subset, even though it is not independent of Y in the complete set of all trials.

Again, when you write P(XY|Z)=P(X|Z)*P(Y|Z), Z is not considered as an outcome, but as a premise.

That's not exactly true, we're still dealing with the same larger sample space that includes events in which Z did not occur. It's just that when we write P(XY|Z), we want to look only at the subset of points in the sample space where Z did occur, and then look at how frequently X and Y occurred in this subset. Of course this is logically equivalent to just defining a new sample space that includes only events where Z occurred, and looking at how frequently X and Y occur in the complete sample space (i.e. P(XY) in this smaller sample space which includes only events where Z occurs is equal to P(XY|Z) in the larger sample space which includes events where Z does not occur).

P(AB|Z) means: Given that Z is true, what is the probability that both A and B are True.
Let's look at the Bell's specific case.
Z: Two spin 1/2 particles denoted by 1 and 2 were jointly in a pure singlet state in the past, with 1 moving toward station A, and 2 moving toward station B , but they remain jointly in a pure singlet state, in which their spins are perfectly anti-correlated.

A: spin of particle 1 is found in the up direction at station A
B: spin of particle 2 is found in the up direction at station B

Therefore P(AB|Z) means: What is the probability that both particle 1 and particle 2 are found in the up direction given Z.

The second term in P(AB|Z)=P(A|BZ)*P(B|Z), ie P(A|BZ), means that if we know for sure that B is true, ie that that spin at station 2 is found to be 'up', then that INFORMATION should influence the probability we assign to A. Do you disagree with this?

I agree that this is true in your equation with the meaning you've assigned to the terms, but I disagree that it'd necessarily still be true if we change the meaning of Z so that it now represents some complete state of the hidden variables. With Z given this meaning, P(A|BZ)=P(A|Z), and in those situations P(AB|Z)=P(A|Z)*P(B|Z), meaning that if you know Z and use that to calculate the probability of A, then knowing B gives you no further information that would cause you to alter your calculation of the probability of B. Note that this was true in my thought-experiment with the light-up balls, if A and B represented statements about the colors seen by Alice and Bob when they pushed their buttons, and Z represented a statement about which of the three balls D, E, F were given to Alice and Bob.

Now to the answer of your question:

Z: Three balls D, E, F; D has 0.7 chance of lighting up red when button pushed, E has 0.9 chance, F has 0.25 chance. D given to Alice and E given to Bob, and button is pressed.
A: Alice sees red
B: Bob sees red

P(AB|Z) = P(A|BZ)P(B|Z) according to the product rule.

P(A|BZ) = P(A|Z) since knowing that Bob saw red, does not tell me anything about what Alice might see.

Yes, that's the key. So you agree that in this example, P(AB|Z)=P(A|Z)P(B|Z).

In other words,
A and B are logically independent.

Yes, But A and B are not logically independent in the sample space for this problem as a whole, which includes events where Z is not true (for example, it also includes events where F is given to Alice and D is given to Bob). In other words, P(AB) is not equal to P(A)P(B) in this sample space, despite the fact that P(AB|Z)=P(A|Z)P(B|Z). Do you agree or disagree?

Now you should answer this one:
Z: Two red LEDs D, E were made to be anti-correllated so that when D was observed to lit, the probability of E lighting up was 0.25 and E was lit the probability of D lighting up was 0.6. D given to Alice and E given to Bob, and button is pressed.
A: Alice sees red
B: Bob sees red

What probability will you assign to P(AB|Z).

This problem seems to be not very well-defined, I don't think enough information is given to calculate P(A|Z) and P(B|Z) as a whole. All we can say is that P(A|BZ)=0.6, and P(B|AZ)=0.25. From these facts alone how do you calculate P(AB|Z)?

Also, this example is not really in the spirit of a "local hidden variables" explanation because it's not clear from how it's described whether the correlation involves actual communication between the two LEDs ('spooky action at a distance'), or whether the correlation is ensured by internal mechanisms (hidden variables) in each LED which on each trial were given to them by the "source" that is sending them out to Alice and Bob, such that if you know the mechanism associated with a particular LED on a given trial that allows you to calculate the probability it'll light up, and knowing about what happened to the second LED gives you no further information about the probability the first one will light up, just like with my example of the light-up balls above where you apparently agreed this was true (you agreed that P(A|BZ)=P(A|Z), where Z gave you the information about the internal mechanism about the ball given to Alice). The latter is how it's supposed to work in a local hidden variables theory.

We could assume a very simple local hidden-variables explanation in your example--suppose the "source" assigning LEDs to Alice and Bob has only two types, a type L (for light) that always lights up when the button is pushed, and a type D (for dark) that always fails to light up. Then on x trials, the source sends an L to Alice and a D to Bob; on y trials, the source sends an L to both; and on z trials, the source sends a D to Alice and an L to Bob (assume the source never sends a D to both, otherwise the problem won't be well-defined). Assume x, y, and z are all fractions, so that x + y + z = 1. Then we can write P(Bob's lights up|Alice's lit up) = P(Bob's lights up AND Alice's lit up)/P(Alice's lit up) = y/(x + y) = 0.25, which gives us x=3y. Likewise, P(Alice's lights up|Bob's lit up)=P(Bob's lights up AND Alice's lit up)/P(Bob's lit up)=y/(z + y) = 0.6, which gives z=(2/3)y. So, we can substitute x=3y and z=(2/3)y into x + y + z = 1 to get (14/3)y = 1 which implies y=3/14. So, x=9/14 and z=2/14. So, now we can define a new condition Z' which implies your Z but contains more detail:

Z': The source has a collection of L LEDs that always light up, and D LEDs that never do, and there is a probability of 9/14 it gives Alice an L and Bob a D, and probability of 3/14 it gives both Alice and Bob an L, and a probability of 2/14 it gives Alice a D and Bob an L.

A: Alice's lights up
B: Bob's lights up

In this case P(A|BZ')=0.6 and P(B|AZ')=0.25 as before. But here we can also show that P(AB|Z')=3/14, and that P(A|Z')=12/14 while P(B|Z')=5/14. Here, note that it is not true that P(A|BZ')=P(A|Z'), so it isn't true that P(AB|Z')=P(A|Z')P(B|Z'). However, this is just because Z' does not give us the complete information about the hidden variables on a given trial. Consider Z1, Z2 and Z3 that do give us this information:

Z1: The source has a collection of L LEDs that always light up, and D LEDs that never do, and it gives Alice an L and Bob a D
Z2: The source has a collection of L LEDs that always light up, and D LEDs that never do, and it gives both Alice and Bob an L
Z3: The source has a collection of L LEDs that always light up, and D LEDs that never do, and it gives Alice a D and Bob an L.

In this case we can say that in our sample space, P(Z1)=9/14, P(Z2)=3/14, and P(Z3)=2/14. In this case it is true that P(AB) = \sum_i P(AB|Z_i)P(Z_i), and it's also true that P(AB|Z1)=P(A|Z1)P(B|Z1) (because P(A|BZ1) = P(A|Z1), and likewise that P(AB|Z2)=P(A|Z2)P(B|Z2) and that P(AB|Z3)=P(A|Z3)P(B|Z3). So, we can rewrite the above summation as P(AB) = \sum_i P(A|Z_i)P(B|Z_i)P(Z_i). Do you disagree with any of this?

 

[mn4j]

Here's another example--we have a 6-sided die and a 12-sided die, Alice rolls one and Bob rolls the other. If there's a 50/50 chance the 12-sided die will be given to either one of them, and we aren't told who got which die, then the results they see aren't logically independent--for example, if Alice gets a 10 that allows us to infer Bob will get some number between 1 and 6, whereas before we knew what Alice rolled we would have considered it possible Bob could get any number from 1-12. However, in the subset of cases where Alice gets the 12-sided die and Bob gets the 6-sided die (or if you prefer, in a single case where we know Alice got the 12-sided die and Bob got the 6-sided die), Bob has a probability 1/6 of getting any number 1-6, and a probability 0 of getting a number 7-12, and knowing what Alice rolled has absolutely no further effect on these probabilities. Do you disagree? Again, please think about this specific example and answer yes or no.

The examples you are giving are deliberately cases in which there is no logical dependence. Go back to the example Bell is considering gave and explain to me how there can be no logical dependence in the subset he is considering.

Or think about this one instead.

Alice has a die with six sides, with an additional hidden biasing property which you don't know about. If you knew everything about the die, it won't be a hidden variable now would it? Bell was not God.

What is the probability that of the die showing a 1 on the first throw. What about the second throw. Does knowing the result of the first throw tell you anything about the probability of the second throw? If you think not, what about the 100th throw? Does knowing the result of the 99 previous throws tell you anything about what the result of the 99th throw might be?

Let's even eliminate the hidden variable completely and consider a unbiased die. Does knowing the result of the first throw tell you anything about the probability of the second throw? If not, then what about the 10th throw? If I told you that the first 9 times, Alice got a all 1's (ones), and ask you to tell me the probability that Alice will get a 1 on the 10th throw. Do you believe your answer will still be 1/6. Common sense tells you that the more 1's Alice get's in a row, the less likely it becomes for her to continue making a 1.

 

[mn4j]

If you don't disagree, then again, my point is just that you can have a situation where X is independent of Y in this subset, even though it is not independent of Y in the complete set of all trials.

That's not exactly true, we're still dealing with the same larger sample space that includes events in which Z did not occur. It's just that when we write P(XY|Z), we want to look only at the subset of points in the sample space where Z did occur, and then look at how frequently X and Y occurred in this subset. Of course this is logically equivalent to just defining a new sample space that includes only events where Z occurred, and looking at how frequently X and Y occur in the complete sample space (i.e. P(XY) in this smaller sample space which includes only events where Z occurs is equal to P(XY|Z) in the larger sample space which includes events where Z does not occur).

It all boils down to the point that you do not understand the meaning of logical independence. Look at this

http://math-cs.cns.uni.edu/~campbell/stat/prob4.html#prod

This problem seems to be not very well-defined, I don't think enough information is given to calculate P(A|Z) and P(B|Z) as a whole. All we can say is that P(A|BZ)=0.6, and P(B|AZ)=0.25. From these facts alone how do you calculate P(AB|Z)?

You apparently did not see the correct question in the following post. Could you answer that one instead? Sorry about the mixup.

Let me answer it also.

P(AB|Z) = P(A|BZ)P(B|Z) according to the product rule.

P(A|BZ) = 0.2
P(B|Z) = P(A|Z) = 0.5 since there is no bias between A and B, they are both equally likely. I suspected you will have a problem with this one because it appears, you do not understand probabilities as meaning more than frequencies.

therefore P(AB|Z) = 0.2 * 0.5 = 0.1

 

[JesseM]

Here's another example--we have a 6-sided die and a 12-sided die, Alice rolls one and Bob rolls the other. If there's a 50/50 chance the 12-sided die will be given to either one of them, and we aren't told who got which die, then the results they see aren't logically independent--for example, if Alice gets a 10 that allows us to infer Bob will get some number between 1 and 6, whereas before we knew what Alice rolled we would have considered it possible Bob could get any number from 1-12. However, in the subset of cases where Alice gets the 12-sided die and Bob gets the 6-sided die (or if you prefer, in a single case where we know Alice got the 12-sided die and Bob got the 6-sided die), Bob has a probability 1/6 of getting any number 1-6, and a probability 0 of getting a number 7-12, and knowing what Alice rolled has absolutely no further effect on these probabilities. Do you disagree? Again, please think about this specific example and answer yes or no.

The examples you are giving are deliberately cases in which there is no logical dependence.

But of course there is a logical dependence in the sample space as a whole! Remember I said there was a 50/50 chance the 12-sided die would be given to either one, so the sample space includes both events where Alice got the 12-sided die and events where Bob got it. If we don't know who got it (i.e. we don't know the hidden variables which determine the probability distribution of each one's rolls), and then we find out Alice got a 10, this will cause us to revise the probabilities that Bob will get a given number--do you disagree?

It seems that part of the problem here is that you are fairly confused about the difference between the sample space for a problem and the conditions in a statement about conditional probability in a problem. They are not the same thing! For a simple example, consider the following:

Sample space--pairs of flips of a fair coin, possible outcomes HH, HT, TH, and HH, each with probability 1/4.

Event A: first flip was H.
Event B: Second flip was H.
Event C: Both flips gave the same result.

For this sample space, all of the following are true:
P(A) = P(B) = P(C) = 1/2
P(B|A) = 1/2
P(B|C) = 1/2
P(B|AC) = 1

Do you disagree?

Now, consider the more complex example of the balls with the hidden internal mechanisms. Here, we have:

Sample space: Three balls D, E, F; D has 0.7 chance of lighting up red when button pushed, E has 0.9 chance, F has 0.25 chance. The source randomly chooses which balls to give to Alice and Bob, so P(Alice gets D, Bob gets E)=P(Alice gets D, Bob gets F)=P(Alice gets E, Bob gets D)=P(Alice gets E, Bob gets F)=P(Alice gets F, Bob gets D)=P(Alice gets F, Bob gets E)=1/6.

H1 is the event "Alice gets D, Bob gets E".
H2 is the event "Alice gets D, Bob gets F".
H3 is the event "Alice gets E, Bob gets D".
H4 is the event "Alice gets E, Bob gets F".
H5 is the event "Alice gets F, Bob gets D".
H6 is the event "Alice gets F, Bob gets E".
A is the event that Alice saw red when she pushed the button on her ball.
B is the event that Bob saw red when he pushed the button on his ball.

Do you agree that in this sample space as a whole, A is not logically independent of B, so that if we don't know which hidden-variable condition H1-H6 obtained on a given trial, then knowing that B occurred should cause us to revise our estimate of the probability of A, and P(AB) is not equal to P(A)*P(B) in this sample space? On the other hand, do you also agree that if we pick a particular hidden-variable condition H1-H6, then it will be true that P(AB|H1)=P(A|H1)*P(B|H1), and so on for conditions H2, H3, H4, H5 and H6? Do you agree that in this sample space, P(AB) = \sum_i P(AB|H_i )P(H_i ), so if we combine this with the fact that P(AB|Hi)=P(A|Hi)*P(B|Hi), we get the equation P(AB) = \sum_i P(A|H_i )P(B|H_i )P(H_i ) which is directly analogous to the hidden-variable integral in Bell's paper, or equation (12) in the Jaynes paper?

 

[JesseM]

It all boils down to the point that you do not understand the meaning of logical independence. Look at this

http://math-cs.cns.uni.edu/~campbell/stat/prob4.html#prod

Of course I understand it. Do you agree that it may be possible to find a sample space where it is true that P(AB) is not equal to P(A)P(B), meaning they are not logically independent, yet it's also true that P(AB|Z)=P(A|Z)P(B|Z) in this sample space? This would mean that if we consider a different sample space where Z is included as part of the conditions of the sample space (i.e. the subset of trials in the original sample space where Z occurred), then in this new sample space A is logically independent of B.

You apparently did not see the correct question in the following post. Could you answer that one instead? Sorry about the mixup.

Sure, your new problem was this:

Z: Two red LEDs D, E were made on a circuit so that when D was observed to be lit, the probability of E lighting up was 0.2 and when E was lit the probability of D lighting up was 0.2. Also, the circuit was designed such that at least one LED was lit on each button press with no other bias imposed on the LED other than the correlation above. D given to Alice and E given to Bob, and button is pressed.
A: Alice sees red
B: Bob sees red

Your own answer contains a leap that can't really be justified in any rigorous terms:

Let me answer it also.

P(AB|Z) = P(A|BZ)P(B|Z) according to the product rule.

P(A|BZ) = 0.2
P(B|Z) = P(A|Z) = 0.5 since there is no bias between A and B, they are both equally likely.

Your original statement of the condition Z did not say "there is no bias between A and B", it simply didn't comment on the overall likelihood that A and B occurred. If you want to include it as an additional part of Z that A and B are equally likely overall, then that's fine, but the problem as stated didn't contain enough information to give a well-defined answer.

I suspected you will have a problem with this one because it appears, you do not understand probabilities as meaning more than frequencies.

I recognize that in Bayesian reasoning observers can have "prior" probability distributions which simply represent their beliefs about the likelihood different events will occur; however, in problems that don't contain enough information to be interpreted in frequentist terms, this is highly subjective, different people can have different prior probability distributions and there are no well-defined rules for deciding what the "correct" prior is. So I should say that in any situation where enough information is given so that everyone will be forced to assign events the same probabilities, in those situations you should always be able to re-interpret probabilities in frequentist terms. All the examples we have been dealing with up until now, like the light-up balls or the urn examples, are sufficiently well-defined so that there should be no subjective elements in assigning probabilities. And Bell's proof assumes that there is a complete set of local physical facts which determine probabilities, such that if an omniscient observer had access to all possible physical facts, he could assign probabilities to events like P(AB) without any subjective element either. The variable \lambda is supposed to represent the complete set of hidden facts on any given trial--we don't know the value of \lambda on a given trial, but that's similar to the fact that we don't know which of the six hidden conditions H1-H6 obtained on a given trial in the example with the light-up balls in my previous post, we nevertheless can agree that from the perspective of an omniscient observer who does know, it must be true that P(AB) = \sum_i P(A|H_i )P(B|H_i )P(H_i ) (and of course, we do know whether A and B obtained on each trial). This equation is obviously very similar to (12) in Jaynes' paper, and the reasoning behind both is analogous, if you don't understand my summation equation above you won't be able to understand the reasoning behind (12).

 

[mn4j]

On the other hand, do you also agree that if we pick a particular hidden-variable condition H1-H6, then it will be true that P(AB|H1)=P(A|H1)*P(B|H1), and so on for conditions H2, H3, H4, H5 and H6?

No I do not. Simply knowing the values of the hidden variables is not enough to reduce P(A|BH1) to P(A|H1). You MUST also know that A is logically independent of B in the specific event. That is why you should focus on only one event and show me how it is that knowledge of the hidden variable reduces P(A|BH1) to P(A|H1).

 

[JesseM]

No I do not. Simply knowing the values of the hidden variables is not enough to reduce P(A|BH1) to P(A|H1).

But in post #36 you already agreed with something equivalent:

Now to the answer of your question:

Z: Three balls D, E, F; D has 0.7 chance of lighting up red when button pushed, E has 0.9 chance, F has 0.25 chance. D given to Alice and E given to Bob, and button is pressed.
A: Alice sees red
B: Bob sees red

P(AB|Z) = P(A|BZ)P(B|Z) according to the product rule.

P(A|BZ) = P(A|Z) since knowing that Bob saw red, does not tell me anything about what Alice might see.

The only difference is that in my statement of the problem I have more adequately distinguished what was the sample space and what was a specific hidden-variables condition which only obtained in some events in the sample space (see my questions about the coinflip in post #41--I'd like to know whether you agree or disagree there, and whether in general you understand the difference between the sample space and conditions in a conditional probability equation). Specifically I wrote:

Sample space: Three balls D, E, F; D has 0.7 chance of lighting up red when button pushed, E has 0.9 chance, F has 0.25 chance. The source randomly chooses which balls to give to Alice and Bob, so P(Alice gets D, Bob gets E)=P(Alice gets D, Bob gets F)=P(Alice gets E, Bob gets D)=P(Alice gets E, Bob gets F)=P(Alice gets F, Bob gets D)=P(Alice gets F, Bob gets E)=1/6.

H1 is the event "Alice gets D, Bob gets E".

Do you agree that in any event (i.e. single trial) with this sample space where H1 occurred, the proposition you labeled Z will also be true?

You MUST also know that A is logically independent of B in the specific event. That is why you should focus on only one event and show me how it is that knowledge of the hidden variable reduces P(A|BH1) to P(A|H1).

"Logical independence" refers to the sample space, asking whether A is logically independent of B in a specific event doesn't make much sense. Do you agree that any statement about probabilities must involve a well-defined sample space?

 

[mn4j]

But in post #36 you already agreed with something equivalent:

Not correct! In your statement of the problem, there was no dependence between D E and F. That is why I could reduce P(A|BZ) to P(A|Z), not because I knew values of the probabilities P(E), P(D), P(F). Do not confuse the value of a specific hidden variable with the probability of having that particular value in an event.

asking whether A is logically independent of B in a specific event doesn't make much sense.

False. My very first post on this issue gave you an example. The case of the monkey picking two balls.
1 event in that case is the the picking of the two balls and yet in that event, I showed you that the probability of picking the first ball, is logically dependent on the outcome of the second and vice versa. That example was not taken from thin air. It was chosen precisely because it mirrors the EPR experiment.

1 event in the EPR case is a measurement at 2 stations A and B for two electrons with a common past.

 

[JesseM]

Not correct! In your statement of the problem, there was no dependence between D E and F.

What does "dependence between D E and F" mean? D E and F aren't events, they are just names of balls. Also, when you say "in my statement of the problem", do you mean the original statement you were responding to when you agreed P(A|BZ) = P(A|Z), or do you mean the more recent statement where I distinguished between the sample space and the events H1-H6?

That is why I could reduce P(A|BZ) to P(A|Z), not because I knew values of the probabilities P(E), P(D), P(F).

Like I said, not events, so these don't make sense as stated, but I would guess that by P(E) you probably mean the probability that ball E will light up when pressed, and similarly for D and F?

In other words, it is impossible to tell just from the probabilities P(E), P(D), P(F) whether E, D, F are dependent on each other or not.

I see, you're worried about whether there's a logical dependence between the event of ball E lighting up and the event of ball D lighting up, and so on for other pairs? Well, this is supposed to be a local hidden variables explanation where the probability a given ball lights up is wholly determined by its internal mechanism (internal hidden variables), so you can assume there is no such logical independence. I'll restate the problem as follows:

Sample space: Three balls D, E, F; D has 0.7 chance of lighting up red when button pushed, E has 0.9 chance, F has 0.25 chance. The probability a given ball will light up is wholly determined by its internal mechanism, so there is no logical dependence between the event of ball D lighting up when the button is pushed and any other outside events, and likewise for balls E and F. The source randomly chooses which balls to give to Alice and Bob, so P(Alice gets D, Bob gets E)=P(Alice gets D, Bob gets F)=P(Alice gets E, Bob gets D)=P(Alice gets E, Bob gets F)=P(Alice gets F, Bob gets D)=P(Alice gets F, Bob gets E)=1/6.

H1 is the event "Alice gets D, Bob gets E".
H2 is the event "Alice gets D, Bob gets F".
H3 is the event "Alice gets E, Bob gets D".
H4 is the event "Alice gets E, Bob gets F".
H5 is the event "Alice gets F, Bob gets D".
H6 is the event "Alice gets F, Bob gets E".
A is the event that Alice saw red when she pushed the button on her ball.
B is the event that Bob saw red when he pushed the button on his ball.

With this revised description of the sample space and various possible events, do you now agree that P(AB|H1) = P(A|H1)P(B|H1)? If so, do you also agree that A and B are not logically independent in the sample space as a whole, i.e. P(AB) is not equal to P(A)P(B)?

asking whether A is logically independent of B in a specific event doesn't make much sense.

False. My very first post on this issue gave you an example. The case of the monkey picking two balls.
1 event in that case is the the picking of the two balls and yet in that event, I showed you that the probability of picking the first ball, is logically dependent on the outcome of the second and vice versa. That example was not taken from thin air.

Yes, but the logical dependence here is based on your knowledge of the sample space--the reason one event is logically dependent on the other is because if we consider the sample space of all possible outcomes for both the first and second pick, we can assign probabilities to each point based on the way the balls are picked from the urn, and we find that if we know we picked a point in that sample space where the first ball was red, that tells us something more about the probability that we picked a point in the sample space where the second ball was also red. If you didn't know anything about the sample space or the probability of different points in the sample space--for example, if you had no idea whether on the second pick the monkey would get a ball, an egg, or a squirrel, or if you did know the urn contained only balls but didn't know what color balls were in the urn originally, or didn't know whether new balls were being placed in the urn between picks to replace the one chosen, or didn't know whether the monkey picks balls randomly from the urn or whether he has a particular taste for picking balls of a particular color--in that case, knowing the result of the first pick wouldn't tell you anything definite about the probability of getting a given color on the second pick.

It was chosen precisely because it mirrors the EPR experiment.

My light-up ball experiment mirrors the EPR experiment in this way too--P(AB) is not equal to P(A)P(B), so if you find out B and you don't already know anything about the hidden variables, this gives you new information about P(A).

 

[mn4j]

What does "dependence between D E and F" mean? D E and F aren't events, they are just names of balls.

P(D) is the probability for observing the event that "D lights up when the button is pressed" etc.

Also, when you say "in my statement of the problem", do you mean the original statement you were responding to when you agreed P(A|BZ) = P(A|Z), or do you mean the more recent statement where I distinguished between the sample space and the events H1-H6?

I mean (see midway on post #35):

Three balls D, E, F; D has 0.7 chance of lighting up red when button pushed, E has 0.9 chance, F has 0.25 chance. ball D is given to Alice, ball E is given to Bob and they both press the button. What is the probability P(Alice saw red, Bob saw red).​

Like I said, not events, so these don't make sense as stated, but I would guess that by P(E) you probably mean the probability that ball E will light up when pressed, and similarly for D and F?

I thought that was obvious.

My light-up ball experiment mirrors the EPR experiment in this way too

No! It doesn't because in the EPR case the two electrons are from the same source and are anti-correlated and you do not know prior to measurement whether spin up will go left or right. If you put the two balls in a box and at each event Alice and Bob each pull a ball at random without replacing and press on the button, then it will be closer.

Do you agree that any statement about probabilities must involve a well-defined sample space?

Yes. But this does not mean the sample space should not be modified in calculating the answer. That is precisely the meaning of the product rule.

When you write P(AB|Z) = P(A|BZ)P(B|Z),
Everything right of the "|" is the hypothesis space in which you are calculating the particular term. The hypothesis space for calculating P(AB) is Z. P(A|BZ) means that knowing B changes the hypothesis space for calculating P(A) by adding information to it. This is just another way of saying "we now assume that, not only is Z true but B is also True".

In the EPR case, since you do not know which electron came to which station, if you now know that station B measured the spin up electron, that information changes the hypothesis space such that station A can only measure the spin down electron therefore P(A) = P(A|BZ).

 

[mn4j]

My light-up ball experiment mirrors the EPR experiment in this way too--P(AB) is not equal to P(A)P(B), so if you find out B and you don't already know anything about the hidden variables, this gives you new information about P(A).

What do you mean by find out B. What are the hidden variables in this case that could give me any information about A?

Sample space: Three balls D, E, F; D has 0.7 chance of lighting up red when button pushed, E has 0.9 chance, F has 0.25 chance. The probability a given ball will light up is wholly determined by its internal mechanism, so there is no logical dependence between the event of ball D lighting up when the button is pushed and any other outside events, and likewise for balls E and F. The source randomly chooses which balls to give to Alice and Bob, so P(Alice gets D, Bob gets E)=P(Alice gets D, Bob gets F)=P(Alice gets E, Bob gets D)=P(Alice gets E, Bob gets F)=P(Alice gets F, Bob gets D)=P(Alice gets F, Bob gets E)=1/6.

H1 is the event "Alice gets D, Bob gets E".
H2 is the event "Alice gets D, Bob gets F".
H3 is the event "Alice gets E, Bob gets D".
H4 is the event "Alice gets E, Bob gets F".
H5 is the event "Alice gets F, Bob gets D".
H6 is the event "Alice gets F, Bob gets E".
A is the event that Alice saw red when she pushed the button on her ball.
B is the event that Bob saw red when he pushed the button on his ball.
With this revised description of the sample space and various possible events, do you now agree that P(AB|H1) = P(A|H1)P(B|H1)?

I do not agree. The statement I highlighted in bold above shows you clearly what I have been saying from the beginning. Logical dependence is different from physical dependence. In that sentence you are implying that lack of physical dependence implies lack of logical dependence. Although physical dependence implies logical dependence, lack of physical dependence does not imply lack of logical dependence. Do you disagree?

You can not impose a logical independence condition at will in your hypothesis space. It is part of the mechanism by which you reason out the problem. What you have done is to break the calculator before asking me to use it to calculate a problem. The problem therefore becomes ill-formed because if I know that Bob got F, it DOES tell me something about the probability that Alice got D. You can't force me to not consider that information. By imposing independence in the premise. It doesn't work like that.

This whole argument boils down to the meaning of logical dependence. And I conclude that you either do not understand it, or you understand but refuse to accept it.

 

[JesseM]

I thought that was obvious.

OK, I guessed that was probably what you meant, which is why I presented you with the revised description of the sample space and asked you some questions about it--can you please look over the revised description and answer those questions?

Sample space: Three balls D, E, F; D has 0.7 chance of lighting up red when button pushed, E has 0.9 chance, F has 0.25 chance. The probability a given ball will light up is wholly determined by its internal mechanism, so there is no logical dependence between the event of ball D lighting up when the button is pushed and any other outside events, and likewise for balls E and F. The source randomly chooses which balls to give to Alice and Bob, so P(Alice gets D, Bob gets E)=P(Alice gets D, Bob gets F)=P(Alice gets E, Bob gets D)=P(Alice gets E, Bob gets F)=P(Alice gets F, Bob gets D)=P(Alice gets F, Bob gets E)=1/6.

H1 is the event "Alice gets D, Bob gets E".
H2 is the event "Alice gets D, Bob gets F".
H3 is the event "Alice gets E, Bob gets D".
H4 is the event "Alice gets E, Bob gets F".
H5 is the event "Alice gets F, Bob gets D".
H6 is the event "Alice gets F, Bob gets E".
A is the event that Alice saw red when she pushed the button on her ball.
B is the event that Bob saw red when he pushed the button on his ball.

With this revised description of the sample space and various possible events, do you now agree that P(AB|H1) = P(A|H1)P(B|H1)? If so, do you also agree that A and B are not logically independent in the sample space as a whole, i.e. P(AB) is not equal to P(A)P(B)?

My light-up ball experiment mirrors the EPR experiment in this way too

No! It doesn't because in the EPR case the two electrons are from the same source and are anti-correlated and you do not know prior to measurement whether spin up will go left or right. If you put the two balls in a box and at each event Alice and Bob each pull a ball at random without replacing and press on the button, then it will be closer.

I didn't mean to imply the light-up ball experiment was analogous to the EPR experiment in every respect, but it's analogous to an attempt at a local hidden-variables explanation for the EPR experiment in the specific respect that there is a statistical correlation between the results seen by Alice and the results seen by Bob (i.e. P(AB) is not equal to P(A)P(B)), but this correlation is wholly explained by the hidden internal mechanism associated with each ball being measured. If you want an example that's even more closely analogous to a local hidden-variable explanation for the EPR experiment, look at my lotto card example in post #3 in this thread.

Yes. But this does not mean the sample space should not be modified in calculating the answer. That is precisely the meaning of the product rule.

No, this shows your confusion between the sample space of a problem and a statement of conditional probability. When you write P(AB|Z) that does not imply a modification to the original sample space, it just means you are looking at the subset of points in the sample space where Z occurred, then looking at how frequently AB occurred in this subset. Read page 81 here, for example.

In the EPR case, since you do not know which electron came to which station, if you now know that station A measured the spin up electron, that information changes the hypothesis space such that station B can only measure the spin down electron.

See above for disagreement about your use of "hypothesis space", although I agree that knowing station A measured spin-up changes your estimate of the probability that B will measure spin-up or spin-down, assuming you don't know what hidden variables (if any) are associated with each electron (if you did know the full hidden variables associated with B, then in a local hidden variables theory that would tell you whether it would be spin-up or spin-down, so you'd gain no additional information from knowing what station A measured). Similarly, in the light-up ball case where you don't know which case H1-H6 applied (i.e. you don't know which of balls D, E, and F were given to Alice and Bob), if you know whether Bob's ball lit up, that will change your estimate of the probability that Alice's ball will light up: P(A|B) is not equal to P(A).

 

[mn4j]

I recognize that in Bayesian reasoning observers can have "prior" probability distributions which simply represent their beliefs about the likelihood different events will occur; however, in problems that don't contain enough information to be interpreted in frequentist terms, this is highly subjective, different people can have different prior probability distributions and there are no well-defined rules for deciding what the "correct" prior is.

This is not correct. There are well-defined rules. Read up on the "principle of indifference" or "maximum entropy".

When physicists say "any two configurations of atoms with the same total energy are equally probable at equilibrium" you don't expect that someone has actually counted the atoms to calculate their probability do you? They are using the principle of indifference.

Still I would have liked to see how you calculate on this question:

Z: Two red LEDs D, E were made on a circuit so that when D was observed to be lit, the probability of E lighting up was 0.2 and when E was lit the probability of D lighting up was 0.2. Also, the circuit was designed such that at least one LED was lit on each button press with no other bias imposed on the LED other than the correlation above. D given to Alice and E given to Bob, and button is pressed.
A: Alice sees red
B: Bob sees red

What probability will you assign to P(AB|Z).

And please point out the error you claim exists in my answer to it:

P(AB|Z) = P(A|BZ)P(B|Z) according to the product rule.

P(A|BZ) = 0.2
P(B|Z) = P(A|Z) = 0.5 since there is no bias between A and B, they are both equally likely. I suspected you will have a problem with this one because it appears, you do not understand probabilities as meaning more than frequencies.

therefore P(AB|Z) = 0.2 * 0.5 = 0.1

 

[JesseM]

P(AB) is not equal to P(A)P(B), so if you find out B and you don't already know anything about the hidden variables, this gives you new information about P(A).

What do you mean by find out B. What are the hidden variables in this case that could give me any information about A?

Again, the assumption is that the two balls given to Alice and Bob were taken from the set D, E, and F, with the assumptions already given about the probabilities associated with each ball. But I'm talking about a situation where you know the balls were selected this way, but you don't know which specific ball was given to Alice and which specific ball was given to Bob. In this case, if you find out that Bob's ball lit up red, this will change your estimate of the probability that Alice's ball will light up red: P(A) is different from P(A|B).

I do not agree. The statement I highlighted in bold above shows you clearly what I have been saying from the beginning. Logical dependence is different from physical dependence. In that sentence you are implying that lack of physical dependence implies lack of logical dependence. Although physical dependence implies logical dependence, lack of physical dependence does not imply lack of logical dependence. Do you disagree?

No, I don't disagree with that statement in general. But we are talking about the specific case where the outcome of a given ball's button being pressed is assumed to be fully determined physically by the internal mechanisms in that ball, which are not in communication with the outside world. Are you saying that even when we already know which physical mechanisms are inside the balls given to Alice and Bob, you still think that knowing whether Bob's ball lit up or not would cause us revise our estimate of the probability that Alice's ball lit up? Suppose instead Alice and Bob were flipping coins, and we know that each coin's probability of coming up heads or tails is physically determined by its physical structure, and that both coins are fair coins that have a 50/50 chance of landing heads or tails. Would knowing that Bob's coin landed heads somehow cause you to revise your estimate of the probability that Alice's came up heads?

In a universe with local realist laws, the results of a physical experiment on any system are assumed to be determined by some set of variables specific to the region of spacetime where the experiment was performed. There can be a statistical correlation (logical dependence) between outcomes A and B of experiments performed at different locations in spacetime with a spacelike separation, but the only possible explanation for this correlation is that the variables associated with each system being measured were already correlated before the experiment was done, that the systems had "inherited" correlated internal variables from some event or events in the overlap of their past light cones. Do you disagree? If so, try to think of a counterexample that we can be sure is possible in a local realist universe (no explicitly quantum examples).

If you don't disagree, then the point is that if the only reason for the correlation between A and B is that the local variables \lambda associated with system #1 are correlated with the local variables associated with system #2, then if you could somehow know the full set of variables \lambda associated with system #1, knowing the outcome B when system #2 is measured would tell you nothing additional about the likelihood of getting A when system #1 is measured. In other words, while P(A|B) may be different than P(B), P(A|B\lambda ) = P(A | \lambda ). If you disagree with this, then I think you just haven't thought through carefully enough what "local realist" means.

You can not impose a logical independence condition at will in your hypothesis space.

No, but you can if it makes sense given the physical assumptions of the problem. For example, while I can't impose the condition P(AB)=P(A)P(B) in general, if I know I'm dealing with a situation where Bob and Alice are both given different fair coins and asked to flip them, and A=Alice got heads and B=Bob got heads, then of course in this case it makes sense to say P(AB)=P(A)P(B). Similarly, if I know I'm dealing with a situation where Alice got a ball that has a 90% chance of lighting up when the button was pressed due to some internal mechanisms that aren't in communication with anything outside the ball, and where Bob got a ball that has a 70% chance of lighting up due to similar internal mechanisms, then if A=Alice's ball lit up and B=Bob's ball lit up, I can of course say that P(AB)=P(A)P(B) here.

It is part of the mechanism by which you reason out the problem. What you have done is to break the calculator before asking me to use it to calculate a problem. The problem therefore becomes ill-formed because if I know that Bob got F, it DOES tell me something about the probability that Alice got D.

Huh? I never asked you to calculate the probability that Alice got D given that Bob got F. (of course I'd agree that if you don't know what Alice got, knowing Bob got F increases the probability Alice got D!) The only two problems I gave were: #1, the one where you don't know which ball either of them got, and you have to calculate the probability that both balls lit up; and #2, the one where you already know which balls both of them got, and you have to calculate the probability both balls lit up given that knowledge.

 

[JesseM]

This is not correct. There are well-defined rules. Read up on the "principle of indifference" or "maximum entropy".

They are well-defined, but you have been misled by Jaynes if you think they are universally agreed upon by all people who study statistics--see this page on the internal division between frequentists and bayesians among statisticians. There is no objective reason why adopting the principle of indifference is more correct than adopting some other rule--indeed, it is possible to think of well-defined physical situations where, if you don't give enough information about the situation for the frequentist probabilities to be clear, then if one person fills in the blanks using the principle of indifference while second person uses a different prior distribution, the second person may be closer to the "correct" frequentist probabilities. For example, if I tell two people "where I live, on some days it rains and on others it doesn't, what's the probability it rains on a given day", then if person 1 uses the principle of indifference here he'll say the answer is 0.5, while if the other person blindly guesses the answer is 0.2, it may in fact be correct that the ratio of rainy days to sunny days over a long period in my area is closer to 0.2 than 0.5.

In any case, I hope you can agree that in some cases enough information about a problem is given so that there is no need to make use of the principle of indifference--these are precisely the problems where enough information is given so we can see exactly what the frequencies of different events would be over a large number of trials. And in any discussion of the proof of Bell's theorem, it should be understood that the proof takes the point of view of some omniscient being who knows the value of all physical variables that are relevant to determining the outcome of experiments, even if some of these variables would be "hidden" to human experimenters--based on this hypothetical point of view, it shows that it's possible to prove that in a local realist universe there'd be certain probability relations between the types of events which can be seen by human experimenters, and that these probability relations are in fact violated by quantum mechanics.

When physicists say "any two configurations of atoms with the same total energy are equally probable at equilibrium" you don't expect that someone has actually counted the atoms to calculate their probability do you? They are using the principle of indifference.

It's a physical fact of our universe that the principle of indifference happens to work well in situations where we want to calculate the future evolution of a system in a certain observed macrostate. On the other hand, since the laws of physics are time-symmetric, if you know the complete physical state of a system at a given time you can calculate backwards to see what its state would have been at earlier times--if you can only see the current macrostate, assuming the principle of indifference with regard to microstates would lead you to predict the system was in a higher entropy state in the past, a prediction would be wrong in most cases. See Loschmidt's paradox.

Anyway, as I said, discussions of the principle indifference are irrelevant in the context of Bell's theorem, because the theorem is explicitly based on imagining that we (or some imaginary omniscient observer) could know the full state of every system, with no information lacking.

Still I would have liked to see how you calculate on this question:

Z: Two red LEDs D, E were made on a circuit so that when D was observed to be lit, the probability of E lighting up was 0.2 and when E was lit the probability of D lighting up was 0.2. Also, the circuit was designed such that at least one LED was lit on each button press with no other bias imposed on the LED other than the correlation above. D given to Alice and E given to Bob, and button is pressed.
A: Alice sees red
B: Bob sees red

What probability will you assign to P(AB|Z).

And please point out the error you claim exists in my answer to it:

P(AB|Z) = P(A|BZ)P(B|Z) according to the product rule.

P(A|BZ) = 0.2
P(B|Z) = P(A|Z) = 0.5 since there is no bias between A and B, they are both equally likely. I suspected you will have a problem with this one because it appears, you do not understand probabilities as meaning more than frequencies.

therefore P(AB|Z) = 0.2 * 0.5 = 0.1

Actually I realize I was mistaken when I said the problem didn't give enough information, I was thrown off by your "since there is no bias between A and B" comment. In fact it is possible to deduce the likelihood of A and B in a frequentist picture here. Just let x be the fraction of trials where only Alice's LED lit up, y be the fraction where they both lit up, and z be the fraction where only Bob's lit up. Then the fraction of trials where Bob's lit up is (y + z), in which case the fact that Alice's LED has only an 0.2 probability of lighting up as well when Bob's is lit up means y/(y + z) = 0.2. Likewise, the fraction of trials where Alice's lit up is (x + y), in which case y/(x + y) = 0.2. From these equations we can conclude that z=4y and x=4y, so since we know that x + y + z = 1, that gives us y=1/9, which means x=z=4/9 (note that this means P(A|Z) = P(B|Z) = 5/9, not 0.5...you apparently forgot that these possibilities are not mutually exclusive so they don't have to add up to 1!) So, the correct probability for both lighting up is just y, i.e. 1/9. Obviously this is different from the answer you got of 0.1. If you think your answer is right and mine is wrong, please give me your own answers for P(Alice saw red, Bob didn't see red | Z), and P(Alice didn't see red, Bob saw red | Z). Hopefully you'd agree that since at least one LED always lights up red, if we take these probabilities and add them to P(Alice saw red, Bob saw red | Z), i.e. P(AB|Z) which you claim is 0.1, the sum should be equal to 1?

 

[mn4j]

You are correct that my answer of 0.1 was wrong. And yours (1/9) is correct. And the reason why mine is wrong is because in calculating one of the probabilities, I did not consider all the information provided in Z. Even though I knew the information. In other words, Z did not mean the same thing in my calculation of every term.

Since we have come full circle, please answer the following simply with yes or no.

1. When you reduce
P(AB) = \sum_i P(AB|Z_i)P(Z_i)
to
P(AB) = \sum_i P(A|Z_i)P(B|Z_i)P(Z_i)
because of independence in the specific Z_i subset. Do you agree that Z_i MUST mean the same thing in all the terms P(A|Z_i) , P(B|Z_i) and P(Z_i)?

2. Do you agree that if Z_i can be split up into individual pieces of information a_i, b_i, \lambda then
P(AB) = \sum_i P(A|Z_i)P(B|Z_i)P(Z_i)
is equivalent to
P(AB) = \sum_i P(A|a_ib_i \lambda)P(B|a_ib_i \lambda)P(a_ib_i \lambda)
but is NOT equivalent to
P(AB) = \sum_i P(A|a_i)P(B|b_i)P(\lambda)

3. Do you believe that if knowledge of a_i gives us no addition information about B and if knowledge of b_i gives us no addition information about A, and knowledge of a_i and b_i give us no additional information about \lambda, then
P(AB) = \sum_i P(A|a_ib_i\lambda)P(B|a_ib_i\lambda)P(\lambda)
should give us the same result as
P(AB) = \sum_i P(A|a_i\lambda)P(B|b_i\lambda)P(\lambda)

4. Do you agree that in Bell's proof, calculating with P(AB) = \sum_i P(A|a_ib_i\lambda)P(B|a_ib_i\lambda)P(\lambda) gave a result which was in agreement with Quantum mechanics but calculating with P(AB) = \sum_i P(A|a_i\lambda)P(B|b_i\lambda)P(\lambda) gave a result which was not.

5. If you agree with 3 and 4, then can you explain to me how the two statements can both be true.

 

[JesseM]

1. When you reduce
P(AB) = \sum_i P(AB|Z_i)P(Z_i)
to
P(AB) = \sum_i P(A|Z_i)P(B|Z_i)P(Z_i)
because of independence in the specific Z_i subset. Do you agree that Z_i MUST mean the same thing in all the terms P(A|Z_i) , P(B|Z_i) and P(Z_i)?

Yes, for any value of i, Z_i should mean the same thing everywhere. Of course Z_1 means something different than Z_2 and so forth...one might involve the hidden-variable condition that Alice got ball D and Bob got ball E, another might involve the hidden-variable condition that Alice got ball D and Bob got ball F. You have to sum over all possible hidden-variable conditions in the above equation to get the total probability of P(AB). Agreed?

2. Do you agree that if Z_i can be split up into individual pieces of information a_i, b_i, \lambda then
P(AB) = \sum_i P(A|Z_i)P(B|Z_i)P(Z_i)
is equivalent to
P(AB) = \sum_i P(A|a_ib_i \lambda)P(B|a_ib_i \lambda)P(a_ib_i \lambda)
but is NOT equivalent to
P(AB) = \sum_i P(A|a_i)P(B|b_i)P(\lambda)

Yes, I agree that if Z_i can be split up in the way you suggest, the sum is equivalent to the first option but not in general equivalent to the second option. But your equation actually has little relevance to Bell's proof, because you haven't put any subscript on the \lambda, implying you think the hidden variables should be exactly the same on every trial! Of course this was not what Bell assumed, he imagined the hidden variables associated with the particles could be different on different trials, and that these hidden variables would explain why an experimenter measuring on a particular axis sometimes gets spin-up and sometimes gets spin-down. So, you really need to give \lambda a subscript and sum over all possible values of this subscript. Also, of course on every trial where the first experimenter makes measurement a1 we do not assume the other experimenter makes b1, so you need multiple subscripts on those letters too. So if you want to write an equation more in keeping with Bell's proof, it should be something like:
P(AB) = \sum_i \sum_j \sum_k P(A|a_i b_j \lambda_k)P(B |a_i b_j \lambda_k)P(a_i b_j \lambda_k)
Or if you assume we are looking at the subset of trials where experimenter #1 made measurement a and experimenter #2 made measurement b (where a and b now stand for two specific measurements rather than variables) we can write:
P(AB|ab) = \sum_k P(A|ab \lambda_k)P(B |ab \lambda_k)P(ab \lambda_k)

3. Do you believe that if knowledge of a_i gives us no addition information about B and if knowledge of b_i gives us no addition information about A, and knowledge of a_i and b_i give us no additional information about \lambda, then
P(AB) = \sum_i P(A|a_ib_i\lambda)P(B|a_ib_i\lambda)P(\lambda)
should give us the same result as
P(AB) = \sum_i P(A|a_i\lambda)P(B|b_i\lambda)P(\lambda)

Yes, although here the equation more in keeping with Bell's proof would be P(AB) = \sum_i \sum_j \sum_k P(A|a_i \lambda_k)P(B |b_j \lambda_k)P(\lambda_k)

4. Do you agree that in Bell's proof, calculating with P(AB) = \sum_i P(A|a_ib_i\lambda)P(B|a_ib_i\lambda)P(\lambda) gave a result which was in agreement with Quantum mechanics but calculating with P(AB) = \sum_i P(A|a_i\lambda)P(B|b_i\lambda)P(\lambda) gave a result which was not.

As before, the idea of Bell's proof is to do a sum over possible values of \lambda (he did an integral because he imagined it taking a continuous range of possible values), but it'd be easy to modify your equations above in this way. But I don't know what you mean by "calculating with"--of course we have no idea of the actual possible values of \lambda_k and thus no idea of the exact value of terms like P(A|a_i\lambda_k) or P(\lambda_k)! The idea is just that if we imagine there is a "conspiracy" in the initial conditions that determines both \lambda_k on a given trial and determines what choices of measurements a_i and b_j the experimenters will make on the same trial, then it's possible to imagine that they could be correlated in a way that would be consistent with quantum predictions even in a universe with local realist physics (I can give you a specific numerical example if you like). It's only if you specifically assume no correlation between the experimenter's choice of measurement on a given trial and the source's "choice" of hidden variables to assign the particles on a given trial that you can derive the Bell inequalities which are inconsistent with QM.

 

[mn4j]

Let's see here:

(Q1 & Q2): So you agree that in
P(AB) = \sum_i P(A|Z_i)P(B|Z_i)P(Z_i) ,
Z_i MUST mean exactly the same thing in all the terms (see your answer to Q1.), and you also believe that

P(AB) = \sum_i P(A|a_ib_i \lambda_i)P(B|a_ib_i \lambda_i)P(a_ib_i \lambda_i)
can be equivalent to
P(AB) = \sum_i P(A|a_i)P(B|b_i)P(\lambda_i) in the case being considered by Bell (your answer to Q2).

Which implies that in Bell's thinking (and yours) there MUST be no dependence between a_i, b_i and \lambda_i, AND there MUST be no dependence between any pair of variables in the complete set (a_i,b_i and \lambda_i), irrespective of which what i is. For example, there MUST be no dependence between a_1 and a_2 or between b_5 and \lambda_{11} . Do you agree with this?


Now do you agree that time t_i can also be a piece of the information contained in Z_i. If you do, can you also envision that the settings at both stations, a_i, b_i could have time-like correlated components in which case integration over time can not be factorized and the settings a_i will be correlated with b_i without any need for spooky action at a distance? In other words, will this new scenario, not considered by Bell, be a local hidden variable model?

(Q3 & Q4):
Do you agree that unless the above conditions are valid, the correlation obtained by Bell will be consistent with QM?

Let me quote Bell's exact words here:

Thirdly, and Finally, there is no difficulty in reproducing the quantum mechanical correlation (3) if the results A and B in (2) are allowed to depend on \vec{b} and \vec{a} respectively as well as on \vec{a} and \vec{b}. For example, replace \vec{a} in (9) by \vec{a'}, obtained from \vec{a} by rotation towards \vec{b} until
1 - \frac{2}{\pi}\theta' = cos \theta
where \theta' is the angle between \vec{a} and \vec{b}. However, for given values of the hidden variables, the results of measurements with one magnet now depend on the setting of the distant magnet, which is just what we would wish to avoid.

In other words, Bell's proof is only valid for the set of hidden variable theories consistent with his assumptions about independence outlined above. Do you agree? (I know you believe that every possible hidden variable theory is covered by it)

Now let us look at this equation which is just the product rule:
P(AB) = \sum_i P(A|a_ib_i \lambda_i)P(B|a_ib_i \lambda_i)P(a_ib_i\lambda_i)

Consider a hidden variable theory in which time is considered a variable as well so that a_i, b_i and \lambda_i are time dependent variables. Note that time dependence of the settings at the stations does not take away the experimenters free will to change a_i or b_i. For example, the measuring device could be a pendulum and the experimenter has the free will to choose the length of the string. It is also not difficult to imagine that the Stern-Gerlach magnet could be made up of electrons exhibiting harmonic motion, even though the experimenter can freely choose the angle of the magnet. At the same time, it is not difficult to imagine that the electrons leaving the source would have the same harmonic motion, for instance due to the fact that they are governed by the same physical law, without any spooky action at a distance.

In this case, one can refactor the above equation easily to
P(AB) = \sum_i P(A|a_ib_i t_i\lambda_i)P(B|a_ib_it_i \lambda_i)P(a_ib_it_i\lambda_i)

Do you agree that in this case, since all the variables after | are dependent on time and thus on each other, it is not valid to reduce this equation to
P(AB) = \sum_i P(A|a_i)P(B|b_i )P(\lambda_i)
In which case such a hidden variable theory was not considered by Bell. If you do, how can you say Bell's theorem disproves all hidden variable theorems. If you think this is not a hidden variable theory, tell me why.

Consider a different hidden variable theory in which the material of which the magnet is a deterministic learner. By this I mean, on interacting with an electron from the i-th measurement event, the material updates it's state based on the value of hidden variable of the electron and it's own hidden variable value. In other words, there is some memory effect left over from the interaction. Then when the electron from the (i+1)th even arrives, the same process repeats over and over.

Do you now see that in such a case, a_i is not independent from a_{i+1} and therefore Bell's factorisation P(AB) = \sum_i P(A|a_i)P(B|b_i )P(\lambda_i) is no longer valid? Do you agree with this? If you think this is not a hidden variable theorem, explain how?

As you hopefully can see now, Bell's inequalities is just a mathematical theorem, whose result is a consequence of the specific assumptions imposed, outside of which it will NOT be valid. Bell's theorem has no experimental basis and has NEVER been proven experimentally! All known experiments violate Bell's theorem! Quantum Mechanics violates Bell's theorem! Think about that for a moment.

 

[JesseM]

Let's see here:

(Q1 & Q2): So you agree that in
P(AB) = \sum_i P(A|Z_i)P(B|Z_i)P(Z_i) ,
Z_i MUST mean exactly the same thing in all the terms (see your answer to Q1.), and you also believe that

P(AB) = \sum_i P(A|a_ib_i \lambda_i)P(B|a_ib_i \lambda_i)P(a_ib_i \lambda_i)
can be equivalent to
P(AB) = \sum_i P(A|a_i)P(B|b_i)P(\lambda_i) in the case being considered by Bell (your answer to Q2).

I said in my answer that this equation is not really correct because you use the same subscript for both the measurement choices a_i and b_i and the hidden variable states \lambda_i, implying that each hidden variable state is associated with a unique measurement. In reality, if there's no correlation between the hidden variable states and measurements, then it should be possible to have trials where you have measurement a_1 and hidden variable state \lambda_1, trials where you have measurement a_1 and hidden variable state \lambda_4, trials where you have measurement a_3 and hidden variable state \lambda_1, etc. This is why you have to write it with a multiple sum like I did:

P(AB) = \sum_i \sum_j \sum_k P(A|a_i \lambda_k ) P(B|b_j \lambda_k ) P(\lambda_k)

Also, you made an error when you left the \lambda's out of the P(A|ai) and P(B|bi), they should be included as I did above. I also noticed I made a small error of my own when writing the above equation--it's assumed we set things up so the experimenters make each combination of measurements equally frequently, so P(a1b1)=P(a2b1)=P(a3b1) etc., all have the same probability 1/N where N is the number of possible combinations. So really we should have P(AB) = \sum_i \sum_j \sum_k P(A|a_i \lambda_k ) P(B|b_j \lambda_k ) P(a_i b_j \lambda_k) which implies P(AB) = \sum_i \sum_j \sum_k P(A|a_i \lambda_k ) P(B|b_j \lambda_k ) P(a_i b_j ) P(\lambda_k) based on the assumption of the independence of measurements and hidden variables, which means the proper equation would have the extra constant factor 1/N:

P(AB) = \sum_i \sum_j \sum_k (1/N) P(A|a_i \lambda_k ) P(B|b_j \lambda_k ) P(\lambda_k)

Which implies that in Bell's thinking (and yours) there MUST be no dependence between a_i, b_i and \lambda_i, AND there MUST be no dependence between any pair of variables in the complete set (a_i,b_i and \lambda_i), irrespective of which what i is. For example, there MUST be no dependence between a_1 and a_2 or between b_5 and \lambda_{11} . Do you agree with this?

Yes. Of course with the independence of a-measurements from b-measurements, this is a situation that we arrange just by telling the experimenters to choose randomly on each trial (perhaps each uses a separate random-number generator, or each rolls a separate die or something), it isn't an assumption about the way the laws of physics work.

Now do you agree that time t_i can also be a piece of the information contained in Z_i. If you do, can you also envision that the settings at both stations, a_i, b_i could have time-like correlated components in which case integration over time can not be factorized and the settings a_i will be correlated with b_i without any need for spooky action at a distance? In other words, will this new scenario, not considered by Bell, be a local hidden variable model?

The a's and b's represent choices made by the experimenters, unless you think it would be impossible to set things up so that their choices are uncorrelated with one another, I don't see the relevance here. Remember, Bell's theorem is about picking the optimum experimental situation for ruling out local hidden variables, we don't have to consider arbitrary variations in the experimental settings that we control. We do have to consider variations in the nature of the hidden variables associated with the particles, since we don't control those--so, it would be appropriate to imagine if the hidden variables associated with each particle might vary over time. But remember that according to QM, if both experimenters measure along the same axis they'll always get opposite spins (or the same spins, depending on what particles are used and how they are entangled), even if they measure at different times. And in a local hidden variables theory we are assuming that there is no physical "communication" between the particles once they are separated, any correlation in their measured behavior must be due to local physical variables--which could be time-varying functions--that were "assigned" to each one in a correlated way when the particles were at a common location.

So, if we make the assumption that there's no correlation between the hidden variable functions assigned to each particle when they were in the same location and the experimenters' later choices about how/when to measure them, the only way to explain this perfect correlation when they are measured on the same axis (regardless of when the measurements are made) is if the hidden variables predetermine a single answer each particle will give to being measured on any given axis, and there's no time variation in this answer (though there could be time variation in other aspects of the hidden variables as long as they don't change what answer a given particle would give when measured on a particular axis at different times). Do you agree?

In other words, Bell's proof is only valid for the set of hidden variable theories consistent with his assumptions about independence outlined above. Do you agree? (I know you believe that every possible hidden variable theory is covered by it)

Yes, I agree--the proof covers every possible hidden variable theory that meets the stated conditions--i.e. locality (there are nonlocal hidden variables theories which Bell's theorem doesn't apply to), realism, and the assumption about a lack of "conspiracy" in the

Now let us look at this equation which is just the product rule:
P(AB) = \sum_i P(A|a_ib_i \lambda_i)P(B|a_ib_i \lambda_i)P(a_ib_i\lambda_i)

Consider a hidden variable theory in which time is considered a variable as well so that a_i, b_i and \lambda_i are time dependent variables. Note that time dependence of the settings at the stations does not take away the experimenters free will to change a_i or b_i. For example, the measuring device could be a pendulum and the experimenter has the free will to choose the length of the string. It is also not difficult to imagine that the Stern-Gerlach magnet could be made up of electrons exhibiting harmonic motion, even though the experimenter can freely choose the angle of the magnet. At the same time, it is not difficult to imagine that the electrons leaving the source would have the same harmonic motion, for instance due to the fact that they are governed by the same physical law, without any spooky action at a distance.

I don't understand how it would be relevant if the electrons in the magnet are exhibiting harmonic motion. Even if we assume there are some hidden variables in the electrons making up the magnet that are correlated with the hidden variables being measured, this does not in any way imply a correlation between the hidden variables of the particles being measured and the experimenter's choice of which setting to use. The different "settings" like a1 or a2 don't contain information about all the physical details of the measuring device, they only refer to the single visible aspect of the measurement that's being varied, in this case the detection angle. It may well be that hidden variables associated with the magnet are different on one trial with setting a2 than they are on a different trial with setting a2, it doesn't mean we treat them as different settings. You can include any hidden variables associated with the measuring device in \lambda if you like, it doesn't only have to refer to hidden variables associated with the particles being measured. All that matters is that in local realism, any correlation between physical variables (hidden or otherwise) in the local neighborhood of measurement #1 and physical variables in the local neighborhood of measurement #2 must be explained by common events in the overlap of the past light cones in these two regions, the idea I was talking about in post #51 when I said:

In a universe with local realist laws, the results of a physical experiment on any system are assumed to be determined by some set of variables specific to the region of spacetime where the experiment was performed. There can be a statistical correlation (logical dependence) between outcomes A and B of experiments performed at different locations in spacetime with a spacelike separation, but the only possible explanation for this correlation is that the variables associated with each system being measured were already correlated before the experiment was done, that the systems had "inherited" correlated internal variables from some event or events in the overlap of their past light cones. Do you disagree? If so, try to think of a counterexample that we can be sure is possible in a local realist universe (no explicitly quantum examples).

If you don't disagree, then the point is that if the only reason for the correlation between A and B is that the local variables \lambda associated with system #1 are correlated with the local variables associated with system #2, then if you could somehow know the full set of variables \lambda associated with system #1, knowing the outcome B when system #2 is measured would tell you nothing additional about the likelihood of getting A when system #1 is measured. In other words, while P(A|B) may be different than P(B), P(A|B\lambda ) = P(A | \lambda ). If you disagree with this, then I think you just haven't thought through carefully enough what "local realist" means.

In this case, one can refactor the above equation easily to
P(AB) = \sum_i P(A|a_ib_i t_i\lambda_i)P(B|a_ib_it_i \lambda_i)P(a_ib_it_i\lambda_i)

Do you agree that in this case, since all the variables after | are dependent on time and thus on each other, it is not valid to reduce this equation to
P(AB) = \sum_i P(A|a_i)P(B|b_i )P(\lambda_i)[/tex]

As I said before, "Bell's theorem is about picking the optimum experimental situation for ruling out local hidden variables, we don't have to consider arbitrary variations in the experimental settings that we control." The time of the two measurements is one of those settings that we control. If we want to arrange things so that each experimenter makes their measurement at the same prearranged time on every trial, we're free to do so, in this case when summing over many trials we don't have to sum over variations in time--if we can rule out local hidden variables theories in this experiment, then that means local hidden variables theories can't account for all the physics of our universe, period. And even if the time is varied randomly (each experimenter has a randomized timer that tells them when to choose what axis to measure, for example), we should still be able to arrange things so there's no correlation between the time an experimenter makes the choice and what angle they choose. So in this case it would be valid to reduce your equation above to one where only \lambda is a function of time:

P(AB) = \sum_i \sum_j \sum_k \sum_l (Constant) P(A|a_i \lambda_k (t_l) ) P(B|b_j \lambda_k (t_l) ) P(\lambda_k (t_l) )

And in this case, remember my comments earlier about a time-varying \lambda. As I said, in local realism any correlation between physical variables in the region of the two spacelike-separated measurements--whether the physical variables are hidden variables associated with the particles, hidden variables associated with the measuring devices, or the actual observed choice of measurement settings--must be explained by a common inheritance from the overlap of the past light cone of the measurements. As long as we assume no correlation between the experimenter's choices about what measurement settings to use and the physical variables inherited from this overlap region that explain correlations in hidden variables and outcomes at the two measurement-events, then it must be true that on every trial, the answers for each possible measurement choice were already predetermined in this overlap region, in order to explain how they always get opposite answers when they happen to choose the same measurement setting.

In which case such a hidden variable theory was not considered by Bell. If you do, how can you say Bell's theorem disproves all hidden variable theorems. If you think this is not a hidden variable theory, tell me why.

All we need is a single type of experiment that gives results that contradict local hidden variables theories, and we've shown that such theories can't explain the physics of our universe. If we want to design the experiment so that both measurements are always performed after the same time interval, or so that there is no correlation between the time the measurements are made and the choice of detector angles, we are free to do so. You can consider time variation in the hidden variables themselves since that's out of our control, but I gave an argument above as to why this doesn't make a difference.

Consider a different hidden variable theory in which the material of which the magnet is a deterministic learner. By this I mean, on interacting with an electron from the i-th measurement event, the material updates it's state based on the value of hidden variable of the electron and it's own hidden variable value. In other words, there is some memory effect left over from the interaction. Then when the electron from the (i+1)th even arrives, the same process repeats over and over.

Do you now see that in such a case, a_i is not independent from a_{i+1} and therefore Bell's factorisation P(AB) = \sum_i P(A|a_i)P(B|b_i )P(\lambda_i) is no longer valid? Do you agree with this? If you think this is not a hidden variable theorem, explain how?

Again, a_i only refers to the single visible aspect of the device which we vary, not to other hidden aspects of the device which can be included in \lambda.

To make this more concrete, I think it would really help if you'd address the example of the scratch-off lotto cards I gave in post #3. You are free to imagine that instead of a static fruit printed underneath the scratch-off material on each box, behind the scratch off material is a screen connected to a computer in the card which can vary what fruit will be revealed depending on when an experimenter scratches a box. You can also imagine that the experimenter is using a coin to scratch one of the boxes and reveal the fruit, and that the coin contains all sorts of complicated internal hidden variables that can be in communication with the card it's scratching (including some kind of 'learning material' which remembers which boxes have been scratched in the past and communicates this to the computer in the card). None of this would change the basic fact that if Alice and Bob always get opposite fruits on trials where they pick the same box to scratch, then under a local hidden variables theory where the source creating the cards has no advanced knowledge of what choices they'll make, it should be absolutely impossible for them to get opposite fruits less than 1/3 of the time when they pick different boxes to scratch. Do you disagree?

As you hopefully can see now, Bell's inequalities is just a mathematical theorem, whose result is a consequence of the specific assumptions imposed, outside of which it will NOT be valid. Bell's theorem has no experimental basis and has NEVER been proven experimentally! All known experiments violate Bell's theorem! Quantum Mechanics violates Bell's theorem! Think about that for a moment.

You're getting the terminology confused here--QM violates the Bell inequalities, but Bell's theorem is essentially the statement that "in any universe where the laws of physics obey local realism (along with the no-conspiracy assumption), no experiment should violate the Bell inequalities". So, if Bell's theorem is valid, then experimental violations of Bell inequalities just shows that we do not live in a universe where the laws of physics obey local realism (along with the no-conspiracy assumption).

 

[mn4j]

it's assumed we set things up so the experimenters make each combination of measurements equally frequently, so P(a1b1)=P(a2b1)=P(a3b1) etc., all have the same probability 1/N where N is the number of possible combinations.

Do you agree that to be consistent, you MUST include the possibility that magnets are also governed by local hidden variables so that a_i and b_i represents not only the subset of settings that the experimenter freely chose, but the COMPLETE state of the magnet at the time of the measurement?

I already gave you the example of the measuring device being like a pendulum hidden in a black box where the experimenter freely changes the length of the string but has no other control over the inner working of the box. I also showed you how in fact this is a possible scenario for a local-hidden variable governed Stern-Gerlach magnet where, even though the experimenter can freely choose the angle, they have no control over the harmonic motion of the individual particles making up the magnet. I need a simple yes or no from you whether you think this is possible local-hidden variable description of the behaviour of the Magnet.

Yes. Of course with the independence of a-measurements from b-measurements, this is a situation that we arrange just by telling the experimenters to choose randomly on each trial (perhaps each uses a separate random-number generator, or each rolls a separate die or something), it isn't an assumption about the way the laws of physics work.

Are you aware that any two objects, exhibiting harmonic motion are correlated, by virtue of circular symmetry, irrespective of differences of frequency or phase and such correlation is not necessarily due to spooky action at a distance? If you disagree, consider two harmonic oscilators which obey the following wave equation,

y(t) = A sin(\omega t + \theta)
Pick any two combinations (1,2) of (A, \omega and \theta) and plot y1 vs y2 for the same t for a given time range and see if you change your mind.

The a's and b's represent choices made by the experimenters, unless you think it would be impossible to set things up so that their choices are uncorrelated with one another, I don't see the relevance here. Remember, Bell's theorem is about picking the optimum experimental situation for ruling out local hidden variables, we don't have to consider arbitrary variations in the experimental settings that we control.

Do you believe that the experimenters can control the harmonic behaviour of the atoms and subatomic particles within their magnets? If you don't then you must agree as I said above that a_i and b_i MUST represent not only the subset of settings that the experimenter freely chose, but the COMPLETE state of the magnet at the time of the measurement, including all local-hidden variables of the magnets. For the two oscillations which you plotted above and saw that they correlated, can you explain how it is possible to design an experiment in which such correlation will not be observed, without using any information about the HIDDEN behaviour?

We do have to consider variations in the nature of the hidden variables associated with the particles, since we don't control those--so, it would be appropriate to imagine if the hidden variables associated with each particle might vary over time. But remember that according to QM, if both experimenters measure along the same axis they'll always get opposite spins (or the same spins, depending on what particles are used and how they are entangled), even if they measure at different times.

This is circular reasoning. Bell did not use QM to derive his inequalities. So what QM predicts should happen, is irrelevant to the derivation of Bell's inequalities.

And in a local hidden variables theory we are assuming that there is no physical "communication" between the particles once they are separated, any correlation in their measured behavior must be due to local physical variables--which could be time-varying functions--that were "assigned" to each one in a correlated way when the particles were at a common location.

Bell believed (and apparently you do too), that the only possible way to have any correlation between
a_i and b_i is by psychokinesis (spooky action at a distance). I have just give you above a situation in which there can be correlation between any two harmonic oscillators without psychokinesis and if you are consistent in not only assigning local-hidden variables to the particles but also to the measuring devices, and the local variables can exhibit harmonic time dependent motion, there will be a correlation without any psychokinesis.

So, if we make the assumption that there's no correlation between the hidden variable functions assigned to each particle when they were in the same location and the experimenters' later choices about how/when to measure them, the only way to explain this perfect correlation when they are measured on the same axis (regardless of when the measurements are made) is if the hidden variables predetermine a single answer each particle will give to being measured on any given axis, and there's no time variation in this answer (though there could be time variation in other aspects of the hidden variables as long as they don't change what answer a given particle would give when measured on a particular axis at different times). Do you agree?

No! I disagree, because the assumption of no correlation, excludes other valid local-hidden variable theories explained above, and if indeed this was the assumption Bell made, his theorem is only valid within the confines of the assumption.

Yes, I agree--the proof covers every possible hidden variable theory that meets the stated conditions--i.e. locality (there are nonlocal hidden variables theories which Bell's theorem doesn't apply to), realism, and the assumption about a lack of "conspiracy" in the

I have already shown you above that there are hidden variables which meet the criteria of locality, realism and lack of conspiracy which Bell did not consider. In other words, those are not the only condidions Bell imposed. He also implicity left out time-varying hidden variables of the type I've mentioned.

I don't understand how it would be relevant if the electrons in the magnet are exhibiting harmonic motion.

It is relevant because any two harmonic oscillators are correlated as demonstrated above. Therefore a_i and b_i understood as complete representations of the local state of the measuring stations are not logically independent.

The different "settings" like a1 or a2 don't contain information about all the physical details of the measuring device, they only refer to the single visible aspect of the measurement that's being varied

Why should it matter, if some of these settings are part of the natural dynamics of the measuring device? Why is it inappropriate to also describe the electrons in the devices with local hidden variables in addition to the 'settings'?

You can include any hidden variables associated with the measuring device in \lambda if you like

No. It has to be associated with a_i and b_i not \lambda because \lambda represents the hidden variable shared between the particles and to avoid consipiracy, those variables have to be separate from those of the measuring devices.

it doesn't only have to refer to hidden variables associated with the particles being measured. All that matters is that in local realism, any correlation between physical variables (hidden or otherwise) in the local neighborhood

Wrong. Then it would be a global variable not a local one. Read Bell's article. Global variables don't come in at all. It is very easy to explain spooky action at a distance using global variables!

So in this case it would be valid to reduce your equation above to one where only \lambda is a function of time.


P(AB) = \sum_i \sum_j \sum_k \sum_l (Constant) P(A|a_i \lambda_k (t_l) ) P(B|b_j \lambda_k (t_l) ) P(\lambda_k (t_l) )

No! Give me a good reason why each entity should not get it's own local variables, with the only variables in common being the ones shared by the particles from their source?

..in order to explain how they always get opposite answers when they happen to choose the same measurement setting.

Again, Bell did not use QM to derive the inequalities so this statement is completely out of place. The result in one orientation, says nothing about the mechanism by which the results are obtained!

Again, a_i only refers to the single visible aspect of the device which we vary, not to other hidden aspects of the device which can be included in \lambda.

Give me a good reason why it should not describe the complete state of the measuring device, just like in any real experiment which will ever be performed?

You're getting the terminology confused here--QM violates the Bell inequalities, but Bell's theorem is essentially the statement that "in any universe where the laws of physics obey local realism (along with the no-conspiracy assumption), no experiment should violate the Bell inequalities". So, if Bell's theorem is valid, then experimental violations of Bell inequalities just shows that we do not live in a universe where the laws of physics obey local realism (along with the no-conspiracy assumption).

No. I'm not. Both Bell's theorem and Bell's inequality are only valid within the narrow set of conditions he imposed while deriving Bell's inequalities. Can you point me to a single experiment that confirms Bell's inequalities. If you can't then how can you claim that it has be validated. If Bell's inequalities have never been validated experimentally, how can you claim that Bell's theorem, which is based on the inequalities has been validated.

The argument is like saying:
All real spiders must have 6 legs. Any spiders with more than 6 legs are not real. And then when somebody finds a spider with 8 legs, instead of evaluating the first premise, you instead conclude that the 8 legged spider is not real.

 

[JesseM]

Do you agree that to be consistent, you MUST include the possibility that magnets are also governed by local hidden variables so that a_i and b_i represents not only the subset of settings that the experimenter freely chose, but the COMPLETE state of the magnet at the time of the measurement?

No, as I said the different a's and b's are defined to simply represent the distinct orientations of the spin-measuring device, if you think there are other properties of the measuring devices which vary on different trials and are relevant to determining the measurement outcome, these properties should be included in the \lambda's.

I already gave you the example of the measuring device being like a pendulum hidden in a black box where the experimenter freely changes the length of the string but has no other control over the inner working of the box. I also showed you how in fact this is a possible scenario for a local-hidden variable governed Stern-Gerlach magnet where, even though the experimenter can freely choose the angle, they have no control over the harmonic motion of the individual particles making up the magnet. I need a simple yes or no from you whether you think this is possible local-hidden variable description of the behaviour of the Magnet.

Yes, I already said it was possible, and I already said it should be included in \lambda, the a's and b's are defined to refer just to the single property of the measuring device that the experimenters vary.

Are you aware that any two objects, exhibiting harmonic motion are correlated, by virtue of circular symmetry, irrespective of differences of frequency or phase and such correlation is not necessarily due to spooky action at a distance? If you disagree, consider two harmonic oscilators which obey the following wave equation,

y(t) = A sin(\omega t + \theta)
Pick any two combinations (1,2) of (A, \omega and \theta) and plot y1 vs y2 for the same t for a given time range and see if you change your mind.

Is this equation derived from Newtonian equations where it's assumed that forces are transmitted instantaneously? If so it's not relevant to the question of how things work in a local realist universe with a speed-of-light limit on physical effects. Maybe an equation like that could also apply to something like charged particles being bobbed along by an electromagnetic plane wave, I don't know (though in this case the charged particles would not be influencing one another, they'd both just be passively influenced by electromagnetic waves which must have been generated by other charges in the overlap of their past light cones). It should be obvious that in a relativistic universe, any correlation between events with a spacelike separation must be explainable in terms of other events in the overlap of their past light cones. If you disagree, please give a detailed physical model of a situation in electromagnetism (the only non-quantum relativistic theory of forces I know of) where this would not be true. Or just give a simpler situation compatible with relativity, like two balls being drawn from an urn and shipped off in boxes at sublight speeds to Alice and Bob, where it wouldn't be true.

Do you believe that the experimenters can control the harmonic behaviour of the atoms and subatomic particles within their magnets? If you don't then you must agree as I said above that a_i and b_i MUST represent not only the subset of settings that the experimenter freely chose, but the COMPLETE state of the magnet at the time of the measurement, including all local-hidden variables of the magnets.

Why "must" it? Again, the a's and b's are defined to mean just the settings that the experimenters control. If there are other physical variables associated with the measuring devices, and we choose to define \lambda to include these variables as well as variables associated with the particles being measured, what problem do you see with this? Can't we define symbols to mean whatever we want them to, and isn't it still true that in this case the combination of the a-setting and the \lambda value will determine the probability of the physical outcome A?

For the two oscillations which you plotted above and saw that they correlated, can you explain how it is possible to design an experiment in which such correlation will not be observed, without using any information about the HIDDEN behaviour?

As always, the "information about the hidden behavior" is assumed to be included in the value of \lambda. \lambda can be understood to give the value of all local physical variables in the immediate spacetime region of one measurement which are relevant to determining the outcome of that measurement.

We do have to consider variations in the nature of the hidden variables associated with the particles, since we don't control those--so, it would be appropriate to imagine if the hidden variables associated with each particle might vary over time. But remember that according to QM, if both experimenters measure along the same axis they'll always get opposite spins (or the same spins, depending on what particles are used and how they are entangled), even if they measure at different times.

This is circular reasoning. Bell did not use QM to derive his inequalities. So what QM predicts should happen, is irrelevant to the derivation of Bell's inequalities.

No, but the fact that we always see opposite results on trials where the settings are the same is an observed experimental fact, and a variant of Bell's theorem can be used to show that if we observe this experimental fact and if the experiment is set up in the way Bell describes (with each experimenter making a random choice among three distinct detector angles) and if the universe is a local realist one (with the no-conspiracy assumption), then we should expect to see opposite results at least 1/3 of the time on the subset of trials where the experimenters chose different measurement settings. Since this Bell inequality is violated in real life, that means at least one of the "if" statements must fail to be true as well, and since we can verify directly that the first true were true, it must be the third one about the universe being local realist that's false (see my next post for an elaboration of this logic).

Bell believed (and apparently you do too), that the only possible way to have any correlation between
a_i and b_i is by psychokinesis (spooky action at a distance).

I assume you are still incorrectly defining the a's and b's to refer to all physical aspects of the measuring devices, and that if you used the correct definitions, what you really mean here is that Bell believed any correlation in the values of variables in \lambda associated with one spacetime region and the values of variables in \lambdaassociated with another spacetime region at a spacelike separation from the first would by spooky action at a distance. But of course this isn't true either, the whole point of a hidden variables explanation for correlations in measurement outcomes is that there can be correlations in the values of local hidden variables in different regions with a spacelike separation, as long as these correlations were determined by events in the overlap of the past light cones of the two regions. I've repeated this over and over so there's really no excuse for your continued mischaracterization of the argument.

I have just give you above a situation in which there can be correlation between any two harmonic oscillators without psychokinesis and if you are consistent in not only assigning local-hidden variables to the particles but also to the measuring devices, and the local variables can exhibit harmonic time dependent motion, there will be a correlation without any psychokinesis.

And as I said, in any relativistic model of a harmonic oscillator (which I don't think your equation is, though as I said it might be possible to find a situation in electromagnetism where the equation applies), correlations in the values of physical variables in different regions with a spacelike separation would be explained by physical causes in the overlap of the past light cones of these two regions.

So, if we make the assumption that there's no correlation between the hidden variable functions assigned to each particle when they were in the same location and the experimenters' later choices about how/when to measure them, the only way to explain this perfect correlation when they are measured on the same axis (regardless of when the measurements are made) is if the hidden variables predetermine a single answer each particle will give to being measured on any given axis, and there's no time variation in this answer (though there could be time variation in other aspects of the hidden variables as long as they don't change what answer a given particle would give when measured on a particular axis at different times). Do you agree?

No! I disagree, because the assumption of no correlation, excludes other valid local-hidden variable theories explained above, and if indeed this was the assumption Bell made, his theorem is only valid within the confines of the assumption.

Well, you're simply confused about the physical meaning of a "local realist" universe then. The statement I give above is a general truth about perfect correlations in regions with a spacelike separation in any universe with local realist laws--the only way to explain perfect correlations between events with a spacelike separation is to assume that the events were totally predetermined by other events in the overlap of the past light cones of the two regions. Again, if you disagree, please think up a situation compatible with relativistic physics (no instantaneous Newtonian forces) where this wouldn't be true.

The different "settings" like a1 or a2 don't contain information about all the physical details of the measuring device, they only refer to the single visible aspect of the measurement that's being varied

Why should it matter, if some of these settings are part of the natural dynamics of the measuring device? Why is it inappropriate to also describe the electrons in the devices with local hidden variables in addition to the 'settings'?

Where did you get the idea I said it was inappropriate? I explicitly said you could include these local hidden variables, but they should be included in \lambda, not in the a's and b's.

You can include any hidden variables associated with the measuring device in if you like

No. It has to be associated with a_i and b_i not \lambda because \lambda represents the hidden variable shared between the particles and to avoid consipiracy, those variables have to be separate from those of the measuring devices.

No, there is no rule that \lambda cannot include hidden variables not directly associated with the particles, it can include any physical variables that are local to the spacetime regions of the two measurements. You misunderstand the "no-conspiracy" condition if you think there can't be a correlation between the value of hidden variables associated with the particle and hidden variables associated with the measuring device--it's only a correlation between such hidden variables and the experimenter's free choice of how to set the angle that would be called a "conspiracy".

it doesn't only have to refer to hidden variables associated with the particles being measured. All that matters is that in local realism, any correlation between physical variables (hidden or otherwise) in the local neighborhood

Wrong. Then it would be a global variable not a local one. Read Bell's article. Global variables don't come in at all. It is very easy to explain spooky action at a distance using global variables!

Of course it wouldn't be global, I just said I was talking about variables in the local neighborhood of each measurement. If it makes it more clear, you can use the symbol \lambda to refer to the value of local physical variables in the spacetime region of one experimenter's measurement, and some other symbol like \phi to refer to local physical variables in the spacetime region of the other experimenter's measurement. In this case we can say that if the experimenters always get opposite outcomes when they both pick identical detector angles (call these identical settings a1 and b1), then it must be true that the result for experimenter #1 is fully determined by the combination of a1 and \lambda, while the result for experimenter #2 is fully determined by the combination of b1 and \phi, and that events in the overlap of the past light cones of these two regions cause \lambda and \phi to be correlated in such a way that the predetermined outcome given a1 + \lambda is guaranteed to be the opposite of the predetermined outcome given b1 + \phi.

No! Give me a good reason why each entity should not get it's own local variables, with the only variables in common being the ones shared by the particles from their source?

Each spacetime region can get its own separate local variables as above if you want to write it that way. But once the particle is in the same region as the measuring-device, there's no reason it couldn't have a physical influence on the hidden variables associated with that measuring-device. Of course it would still be true that any correlations in the hidden variables associated with the two measuring-devices in different regions would still be explained by causal influences from the overlap of the past light cones of the regions (in this case, the causal influences would be the hidden variables carried by the two particles which influenced the hidden variables of their respective measuring devices, with the value of each particle's hidden variables having been determined when they both came from the source, an event which was indeed in the overlap of the two past light cones).

Again, Bell did not use QM to derive the inequalities so this statement is completely out of place. The result in one orientation, says nothing about the mechanism by which the results are obtained!

See above, the fact that it's an experimental observation that you get opposite results on the same setting is part of the derivation of the conclusion that in a local realist universe, you should get opposite results at least 1/3 of the time when the experimenters choose different settings. Of course this particular inequality was not actually the one Bell derived in his original paper, though it is a valid Bell inequality--for the inequality he derived in the original paper, see post #8 of this thread which I linked to back in post #3 here. However, this inequality also includes in the derivation the fact that a perfect correlation (or anticorrelation) is seen when the experimenters choose the same detector setting.

Give me a good reason why it should not describe the complete state of the measuring device, just like in any real experiment which will ever be performed?

There is obviously no cosmic force that compels you to assign particular variables a particular physical meaning. Symbols can mean whatever we define them to mean. But by the same token, there is obviously nothing stopping us from using the a's and b's to refer only to the settings chosen by the experimenters, and to include any other physical variables associated with the detectors in the variable representing all the physical hidden variables \lambda (and as I said you could make a minor tweak to the proof to have two different variables for the two distinct spacetime regions if you prefer). Bell's proof definitely depends on the assumption that the a's and b's refer only to the choices made by the experimenters, so if you want to follow Bell's proof you should adopt this convention, which is as good as any other convention.

(continued in next post)

 

[JesseM]

(continued from previous post)

No. I'm not. Both Bell's theorem and Bell's inequality are only valid within the narrow set of conditions he imposed while deriving Bell's inequalities. Can you point me to a single experiment that confirms Bell's inequalities. If you can't then how can you claim that it has be validated. If Bell's inequalities have never been validated experimentally, how can you claim that Bell's theorem, which is based on the inequalities has been validated.

The argument is like saying:
All real spiders must have 6 legs. Any spiders with more than 6 legs are not real. And then when somebody finds a spider with 8 legs, instead of evaluating the first premise, you instead conclude that the 8 legged spider is not real.

You really have the logic totally confused--your analogy has nothing to do with deriving certain conclusions from theoretical assumptions about the laws of physics, saying "in a universe where the laws of physics take form X, under experimental conditions Y we should be guaranteed to see results Z" is a theoretical deduction, nothing like the arbitrary definition "all real spiders must have 6 legs".

Your comment "Can you point me to a single experiment that confirms Bell's inequalities" also shows confusion about the logic of what Bell was trying to do--the whole point is that Bell's inequalities are violated, thus demonstrating that the assumption of local realism (along with the no-conspiracy assumption) must be false! Are you familiar with the idea of "the contrapositive" or "contraposition" in logic? (see here and here). The idea here is that if you can prove that A logically implies B, that is logically equivalent to the statement that if B is false, then logically A must be false as well. As it says at the bottom of the second article from wikipedia, this can be a good way of doing proofs by contradiction:

Because the contrapositive of a statement always has the same truth value (truth or falsity) as the statement itself, it can be a powerful tool for proving mathematical theorems via proof by contradiction, as in the proof of the irrationality of the square root of 2. By the definition of a rational number, the statement can be made that "If \sqrt{2} is rational, then it can be expressed as an irreducible fraction". This statement is true because it is a restatement of a true definition. The contrapositive of this statement is "If \sqrt{2} cannot be expressed as an irreducible fraction, then it is not rational". This contrapositive, like the original statement, is also true. Therefore, if it can be proven that \sqrt{2} cannot be expressed as an irreducible fraction, then it must be the case that \sqrt{2} is not a rational number.

The logic of Bell's theorem, and how when combined with the observed confirmation of quantum predictions it can be used to show that QM is incompatible with local hidden variables, is essentially the same. Here, the A in "A implies B" can be divided into three conditions:

A1: A universe where the laws of physics respect local realism with the no-conspiracy assumption.
A2: An experimental setup where two experimenters make measurements at a spacelike separation, and each is choosing from three possible detector settings which can be labeled a1-a3 for the first experimenter and b1-b3 for the second. The experimenters are making free random choices of which setting to use on each trial.
A3: It is observed to be the case that whenever the experimenters both choose the same setting, they always get opposite results.

Now for B, one version of Bell's theorem proves that these conditions lead logically to the following conclusion:

B: On the subset of trials where experimenters choose different settings, the probability that they get opposite results should be greater than or equal to 1/3.

Now, this experiment can be done, and we can verify that conditions A2 and A3 both apply, yet B is false. So using contraposition, we know some part of A must be false, and since A2 and A3 can be directly verified to be true, the false part must be the theoretical assumption A1.

 

[mn4j]

Yes, I already said it was possible, and I already said it should be included in \lambda, the a's and b's are defined to refer just to the single property of the measuring device that the experimenters vary.

You still have not said why it should not be separate. It would seem that if Bell's proof was robust, it should be able to accommodate hidden variables at the sources in addition to source parameters. It should tell you a lot that the hidden variables must be defined a specific way in order for the proof to work. Since you are the one claiming that Bell's proof eliminates all possible local-hidden variable theorems, the onus is on you to explain why the stations should not be able to get separate local hidden variables. Probably because you already know that Bell's theorem can not be formulated in that context. Therefore, you can not claim with a straight face that all real local hidden variable theorems are out.

Is this equation derived from Newtonian equations where it's assumed that forces are transmitted instantaneously?

You don't recognize a simple wave equation? This has nothing to do with physics, it is mathematics. If you did the plot as I explained you will see that any two sinusoidal waves are correlated irrespective of phase, amplitude and frequency. In other words, there is no way you can design an experiment that will eliminate the correlation if you do not know the specific parameters for each wave.

If so it's not relevant to the question of how things work in a local realist universe with a speed-of-light limit on physical effects.

It is relevant. Especially since we know about wave-particle duality. It should tell you that we do not need psychokinesis to explain correlations between distant objects.

It should be obvious that in a relativistic universe, any correlation between events with a spacelike separation must be explainable in terms of other events in the overlap of their past light cones. If you disagree, please give a detailed physical model of a situation in electromagnetism (the only non-quantum relativistic theory of forces I know of) where this would not be true.

You obviously have not thought it through well enough. Two objects can be correlated because they are governed by the same physical laws, whether or not they share a common past or not. This is obvious. A pendulum clock on opposite sides of the globe made by different local manufacturers are correlated by virtue of the fact that they exhibit harmonic motion. What do you claim is the common event in their past that is the source of their correlations?

Why "must" it? Again, the a's and b's are defined to mean just the settings that the experimenters control. Can't we define symbols to mean whatever we want them to, and isn't it still true that in this case the combination of the a-setting and the \lambda value will determine the probability of the physical outcome A?

It must, because you claim that Bell's theorem eliminates ALL hidden variable theorems. It is telling that the terms were so narrowly defined that other possible local hidden variable theorems do not fit.
No you can't define the terms to mean whatever you want them to. You have to define them so that they include all possible hidden variable theorems. Therefore the conclusion of Bell's theorem is handicapped.

No, but the fact that we always see opposite results on trials where the settings are the same is an observed experimental fact, and a variant of Bell's theorem can be used to show that if we observe this experimental fact and if the experiment is set up in the way Bell describes (with each experimenter making a random choice among three distinct detector angles) and if the universe is a local realist one (with the no-conspiracy assumption), then we should expect to see opposite results at least 1/3 of the time on the subset of trials where the experimenters chose different measurement settings. Since this Bell inequality is violated in real life, that means at least one of the "if" statements must fail to be true as well, and since we can verify directly that the first true were true, it must be the third one about the universe being local realist that's false (see my next post for an elaboration of this logic).

You forgot a very important "if" that is the very topic of this thread, ie

1. "if we observe this experimental fact"
2. "if a local realist universe behaves only as described by Bell's assumptions"
3. "if the universe is a local realist one (with the no-conspiracy assumption)"
4. "if the experiment is set up in the way Bell describes"
5. then we should expect to see opposite results at least 1/3 of the time on the subset of trials where the experimenters chose different measurement settings.

As you can see, violation of 5 can imply that either (2), (3) or (4) or combinations of them are wrong. For some probably religious reason, proponents of Bell's theorem, jump right to (3) and claim that it must be (3) that is wrong.

I have given you already two examples of hidden variable theorems that point to the falsity of (2). In fact, (2) is the proverbial "a spider must have 6 legs". Do you deny that the validity of Bell's theorem rests as much on (2) as on (3) or (4). It remains to be seen whether any experiment has ever been performed which exactly reproduced Bell's assumptions. But that is a different topic.

For other more rigorous proofs why (2) is wrong, see:

• Brans, CH (1988). Bell's theorem does not eliminate fully causal hidden variables. 27, 2 , International Journal of Theoretical Physics, 1988, pp 219-226
  
• Joy Christian, "Can Bell's Prescription for Physical Reality Be Considered Complete?"
  [http://arxiv.org/pdf/0806.3078v1
  
• See, Hess K, and Philipp W (2000). PNAS ͉ December 4, 2000 ͉ vol. 98 ͉ no. 25 pp 14228-14233 for a proof that Bell's theorem can not be derived for time-like correlated parameters, and that such variables produce the QM result.
  
• See also, Hess K, and Philipp W (2003), "Breakdown of Bell's theorem for certain objective local parameter spaces"
  PNAS February 17, 2004 vol. 101 no. 7 1799-1805

Well, you're simply confused about the physical meaning of a "local realist" universe then. The statement I give above is a general truth about perfect correlations in regions with a spacelike separation in any universe with local realist laws--the only way to explain perfect correlations between events with a spacelike separation is to assume that the events were totally predetermined by other events in the overlap of the past light cones of the two regions. Again, if you disagree, please think up a situation compatible with relativistic physics (no instantaneous Newtonian forces) where this wouldn't be true.

I suppose all those people cited above are also confused, as is Jaynes. Yet you have not shown me a single reason why my descriptions of the two scenarios in pos #55 are not valid realist local hidden variable theorems. For some reason you ignored the second scenario completely and did not even bother to say whether a "deterministic learning machine" is local or not.

You can see the following articles, for proof that a local deterministic learning hidden variable model reproduces the quantum result:

• Raedt, KD, et. al.
  A local realist model for correlations of the singlet state
  The European Physical Journal B - Condensed Matter and Complex Systems, Volume 53, Number 2 / September, 2006, pp 139-142
  
• Raedt, HD, et. al.
  Event-Based Computer Simulation Model of Aspect-Type Experiments Strictly Satisfying Einstein's Locality Conditions
  J. Phys. Soc. Jpn. 76 (2007) 104005
  
• Peter Morgan,
  Violation of Bell inequalities through the coincidence-time loophole
  http://arxiv.org/pdf/0801.1776
  
• More about the coincidence time loophole here:
  Larson, JA, Gill, RD, Europhys. Lett. 67, 707 (204)