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## Trying to Understand Bell's reasoning

 Quote by JesseM That's because your version of H is too vague and doesn't actually specify whether the red card was the one that was picked to send to Alice and the white card was the one that was picked to send to Bob, or vice versa. If you completely specified the hidden properties of the envelope that was sent to Bob--namely "the person who picked the cards from the box put the red card in Bob's envelope, and the envelope continued to have that hidden card on its journey to Bob"--then in that case it would be true that P(B|H)=P(B|AH).
 Quote by billschnieder I am not sure I agree with this. In probability theory, when we write P(A|H), we are assuming that we know H but not A. If we knew A already (a certainty), there is no point calculating a probability is there?
I think I explained clearly in the text above that I was calculating the probability of B, not A, given either just H or given both H and A. If you want to calculate the probability of A rather than B, then you can easily modify the paragraph above:

That's because your version of H is too vague and doesn't actually specify whether the red card was the one that was picked to send to Alice and the white card was the one that was picked to send to Bob, or vice versa. If you completely specified the hidden properties of the envelope that was sent to Alice--namely "the person who picked the cards from the box put the red card in Alice's envelope, and the envelope continued to have that hidden card on its journey to Alice"--then in that case it would be true that P(A|H)=P(A|BH).

 Quote by billschnieder In this case when Bell obtains cos(theta), his equation will only be valid for two angles when cos(theta)= 0 or 1! How then can this equation apply to other angles? Therefore I don't think that is the reasoning here.
Huh? The argument is about what probabilities would be calculated by an ideal observer if they had access to the hidden variables H (which are assumed to have well-defined values at all times in a local realist theory), not just what probabilities are calculated by normal observers who don't know the values of the hidden variables. How could it be otherwise, when H explicitly appears in the conditional probability equations?
 Quote by billschnieder Furthermore, the variables are hidden from the perspective of the wise men, it is not God trying to calculate the probabilities but the wise men, because they do not have all the information. We are only looking from God's perspective to verify that the equation the wise men choose to use corresponds to the factual situation and in the example I gave it does not appear to.
No, you're completely confused, the argument is about taking a God's-eye-view and saying that no matter how we imagine God would see the hidden variables, in a local realist theory God would necessarily end up making predictions about the statistics of different correlations that are different from what we humans actually observer in QM.
 Quote by billschnieder But in my example, with the information known by God, this is the case, every box only contains two cards, one red and one white and in each iteration of the experiment one of the cards is sent to Bob and the other to Alice. The equation P(AB|H) = P(A|H)P(A|BH) always works but P(AB|H) = P(A|H)P(A|H) works only in very limited case in which H is no longer hidden and calculating probabilities is pointless.
It's not "pointless" if you can use this hypothetical God's-eye-perspective (where nothing is hidden) to show that if the hidden variables are such that Alice and Bob always get the same result when they perform the same measurement, that must imply certain things about the statistics they see when they perform different measurements--and that these statistical predictions are falsified in real quantum mechanics! This is a reductio ad absurdum argument showing that the original assumption that QM can be explained using a local realist theory must have been false.

Perhaps you could take a look at the scratch lotto analogy I came up with a while ago and see if it makes sense to you (note that it's explicitly based on considering how the 'hidden fruits' might be distributed if they were known by a hypothetical observer for whom they aren't 'hidden'):

Suppose we have a machine that generates pairs of scratch lotto cards, each of which has three boxes that, when scratched, can reveal either a cherry or a lemon. We give one card to Alice and one to Bob, and each scratches only one of the three boxes. When we repeat this many times, we find that whenever they both pick the same box to scratch, they always get the same result--if Bob scratches box A and finds a cherry, and Alice scratches box A on her card, she's guaranteed to find a cherry too.

Classically, we might explain this by supposing that there is definitely either a cherry or a lemon in each box, even though we don't reveal it until we scratch it, and that the machine prints pairs of cards in such a way that the "hidden" fruit in a given box of one card always matches the hidden fruit in the same box of the other card. If we represent cherries as + and lemons as -, so that a B+ card would represent one where box B's hidden fruit is a cherry, then the classical assumption is that each card's +'s and -'s are the same as the other--if the first card was created with hidden fruits A+,B+,C-, then the other card must also have been created with the hidden fruits A+,B+,C-.

The problem is that if this were true, it would force you to the conclusion that on those trials where Alice and Bob picked different boxes to scratch, they should find the same fruit on at least 1/3 of the trials. For example, if we imagine Bob and Alice's cards each have the hidden fruits A+,B-,C+, then we can look at each possible way that Alice and Bob can randomly choose different boxes to scratch, and what the results would be:

Bob picks A, Alice picks B: opposite results (Bob gets a cherry, Alice gets a lemon)

Bob picks A, Alice picks C: same results (Bob gets a cherry, Alice gets a cherry)

Bob picks B, Alice picks A: opposite results (Bob gets a lemon, Alice gets a cherry)

Bob picks B, Alice picks C: opposite results (Bob gets a lemon, Alice gets a cherry)

Bob picks C, Alice picks A: same results (Bob gets a cherry, Alice gets a cherry)

Bob picks C, Alice picks picks B: opposite results (Bob gets a cherry, Alice gets a lemon)

In this case, you can see that in 1/3 of trials where they pick different boxes, they should get the same results. You'd get the same answer if you assumed any other preexisting state where there are two fruits of one type and one of the other, like A+,B+,C- or A+,B-,C-. On the other hand, if you assume a state where each card has the same fruit behind all three boxes, so either they're both getting A+,B+,C+ or they're both getting A-,B-,C-, then of course even if Alice and Bob pick different boxes to scratch they're guaranteed to get the same fruits with probability 1. So if you imagine that when multiple pairs of cards are generated by the machine, some fraction of pairs are created in inhomogoneous preexisting states like A+,B-,C- while other pairs are created in homogoneous preexisting states like A+,B+,C+, then the probability of getting the same fruits when you scratch different boxes should be somewhere between 1/3 and 1. 1/3 is the lower bound, though--even if 100% of all the pairs were created in inhomogoneous preexisting states, it wouldn't make sense for you to get the same answers in less than 1/3 of trials where you scratch different boxes, provided you assume that each card has such a preexisting state with "hidden fruits" in each box.

But now suppose Alice and Bob look at all the trials where they picked different boxes, and found that they only got the same fruits 1/4 of the time! That would be the violation of Bell's inequality, and something equivalent actually can happen when you measure the spin of entangled photons along one of three different possible axes. So in this example, it seems we can't resolve the mystery by just assuming the machine creates two cards with definite "hidden fruits" behind each box, such that the two cards always have the same fruits in a given box.

And you can modify this example to show some different Bell inequalities, see post #8 of this thread for one example.
 Quote by billschnieder But that is not quite true. The justification for reducing P(A|BH) to P(A|H) is not based on whether B gives [i[additional[/i] information but on whether B gives any information. (see Conditional Independence in Statistical Theory, J.R Statist. Soc B, 1979 41, No. 1, pp. 1-31)
I don't have access to that reference (could you quote it?) but I'm confident it doesn't say what you think it does. In a situation where the probability of A is completely determined by H and the probability of B is also completely determined by H, then it would naturally be true that P(A|BH) would be equal to P(A|H), even if the P(A) was not equal to P(A|B) (i.e. if you don't know H, knowing B does give some information about the probability of A). Do you claim the reference somehow contradicts this?

For example, suppose we have two identical-looking flashlights X and Y that have been altered with internal mechanisms that make it a probabilistic matter whether they will turn on when the switch is pressed. The mechanism in flashlight X makes it so that there is a 70% chance it'll turn on when the switch is pressed; the mechanism in flashlight Y makes it so there's a 40% chance when the switch is pressed. The mechanism's random decisions aren't affected by anything outside the flashlight, so whether or not flashlight X turns on doesn't change the probability that flashlight Y turns on.

Now suppose we do an experiment where Alice is sent one flashlight and Bob is sent the other, by a sender who has a 50% chance of sending X to Alice and Y to Bob, and a 50% chance of sending Y to Alice and X to Bob. Let H1 and H2 represent these two possible sets of "hidden" facts (hidden to Alice and Bob since the flashlights look identical from the outside): H1 represents the event "X to Alice, Y to Bob" and H2 represents the event "Y to Alice, X to Bob". Let A represent the event Alice's flashlight turns on when she presses the switch, B represents the event that Bob's flashlight turns on when she presses the switch.

Here, P(A) = P(A|H1)*P(H1) + P(A|H2)*P(H2) = (0.7)*(0.5) + (0.4)*(0.5) = 0.55
and P(B) = P(B|H1)*P(H1) + P(B|H2)*P(H2) = (0.4)*(0.5) + (0.7)*(0.5) = 0.55

Since P(A|B) = P(A and B)/P(B), we must have P(A|B) = (0.7)*(0.4)/(0.55) = 0.5090909...
So you see that P(A|B) is slightly lower than P(A), which makes sense since if Bob's flashlight lights up, that makes it more likely Bob got flashlight X which had a higher probability of lighting, and more likely A got flashlight Y with a lower probability of lighting.

But despite the fact that B does give some information about the probability of A, it is still true that P(A|B and H1) = P(A|H1) = 0.7, since H1 tells us that Alice got flashlight X, and that alone completely determines the probability that Alice's flashlight lights up when she presses the switch, the fact that Bob's flashlight lit up won't alter our estimate of the probability that Alice's lights up. Likewise, P(A|B and H2) = P(A|H2) = 0.4.

I'm sure that whatever the reference you gave says, it doesn't imply that this reasoning is incorrect.
 Quote by billschnieder You re saying pretty much the same thing there, that A gives no additional information to H. But that is not the meaning of conditional independence. Conditional independence means that A gives us no information whatsoever about B.
I gave a pretty detailed argument in posts #61 and 62 on that thread, starting with the paragraph towards the end of post #61 that says "Let me try a different tack". If you aren't convinced by my comments so far in this post, perhaps you could identify the specific point in my argument on the other thread where you think I say something incorrect? For example, do you disagree with this part?
 I'd like to define the term "past light cone cross-section" (PLCCS for short), which stands for the idea of taking a spacelike cross-section through the past light cone of some point in spacetime M where a measurement is made; in SR this spacelike cross-section could just be the intersection of the past light cone with a surface of constant t in some inertial reference frame (which would be a 3D sphere containing all the events at that instant which can have a causal influence on M at a later time). Now, let $$\lambda$$ stand for the complete set of values of all local physical variables, hidden or non-hidden, which lie within some particular PLCCS of M. Would you agree that in a local realist universe, if we want to know whether the measurement M yielded result A, and B represents some event at a spacelike separation from M, then although knowing B occurred may change our evaluation of the probability A occurred so that P(A|B) is not equal to P(A), if we know the full set of physical facts $$\lambda$$ about a PLCCS of M, then knowing B can tell us nothing additional about the probability A occurred at M, so that P(A|$$\lambda$$) = P(A|$$\lambda$$ B)?
In case we are dealing with a local realist universe that is not deterministic, I think I should add here that the PLCCS of M is chosen at a time after the last moment of intersection between the past light cones of M and B, so that no events that happen after the PLCCS can have any causal influence on B. Continuing the quote:
 If so, consider two measurements of entangled particles which occur at spacelike-separated points M1 and M2 in spacetime. For each of these points, pick a PLCCS from a time which is prior to the measurements, and which is also prior to the moment that the experimenter chose (randomly) which of the three detector settings under his control to use (as before, this does not imply the experimenter has complete control over all physical variables associated with the detector). Assume also that we have picked the two PLCCS's in such a way that every event in the PLCCS of M1 lies at a spacelike separation from every event in the PLCCS of M2. Use the symbol $$\lambda_1$$ to label the complete set of physical variables in the PLCCS of M1, and the symbol $$\lambda_2$$ to label the complete set of physical variables in the PLCCS of M2. In this case, if we find that whenever the experimenters chose the same setting they always got the same results at M1 and M2, I'd assert that in a local realist universe this must mean the results each of them got on any such trial were already predetermined by $$\lambda_1$$ and $$\lambda_2$$; would you agree? The reasoning here is just that if there were any random factors between the PLCCS and the time of the measurement which were capable of affecting the outcome, then it could no longer be true that the two measurements would be guaranteed to give identical results on every trial.

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 Quote by billschnieder Of course, IF we know the contents of each square and we know the square Alice is going to pick, then we can predict with certainty what Alice will observe without disturbing the card in any way. But the contents of the boxes which are already existing, are elements of reality. However, "Alice scratched box 1" is not an element of reality until Alice actually scratches box 1. "Tomorrow at 2pm Alice will scratch box 1" is also an element of reality if in fact that is the box Alice will scratch even if she has not scratched any box yet. It is not ambiguous at all. This is consistent with the EPR definition of "elements of reality" and clearly, it does not mean that outcomes pre-exist measurement. ... What I have explained is that, it is not reasonable to translate this statement to "Einstein can see the moon, even if he is not looking at it." If "seeing the moon" is an outcome of an experiment, you can not claim that realism means that "seeing the moon" pre-existed the act of actually "seeing". Once I understand Bell's justification for "skinning the cat" the way he did in the original paper, I will move to the others. But for now I am only interested in understand his original paper. Thanks for the links to your website. I will check it out.
OK, so which side of the definition are you taking? Does a particle have definite values for observation outcomes PRIOR to the actual act of measurement, or not? That is the question a realist (such as Einstein) answers as "yes". Others, including Bohr and many/most of the scientific establishment, would answer "no". I am squarely in the "no" camp in case, because I believe there is observer dependence (context) - in case you had not already figured that out. And one of the primary reasons for that belief is Bell.

What is *your* answer? It would be helpful to get a straight answer. If you don't like the question, Bell is not likely to mean much to you - since this is the keystone to the paper.

 Quote by DrChinese Does a particle have definite values for observation outcomes PRIOR to the actual act of measurement, or not? That is the question a realist (such as Einstein) answers as "yes".
Not according to my understanding of EPR. The EPR question is whether it is possible to supplement QM with "elements of reality" such that QM becomes complete and can then predict individual events. That is what I explained in my oppenning post no? Einstein definitely did not say "I can see the moon even if I am not looking at it"! Yet given the elements of reality such as the position of the moon, the sky conditions, the position of a person and his gazing direction, the surface conditions around were the person is standing, Eistein will be able to predict with certainty whether a person in that scenario will see the moon or not, without disturbing any individual. Those are the elements of reality. But it definitely does not mean a person can see the moon without looking at it.

The question is not whether "outcomes pre-exist measurement". As I have explained already, there are many real locally causal situations in which outcomes do not pre-exist the act of measurement. Why are you so bent in insisting that this is the issue. I don't see that question/definition in the EPR paper at all. Maybe I missed it. Could you point me to where EPR says "outcomes must pre-exist measurement"?

 Quote by billschnieder Not according to my understanding of EPR. The EPR question is whether it is possible to supplement QM with "elements of reality" such that QM becomes complete and can then predict individual events. That is what I explained in my oppenning post no? Einstein definitely did not say "I can see the moon even if I am not looking at it"! Yet given the elements of reality such as the position of the moon, the sky conditions, the position of a person and his gazing direction, the surface conditions around were the person is standing, Eistein will be able to predict with certainty whether a person in that scenario will see the moon or not, without disturbing any individual. Those are the elements of reality. But it definitely does not mean a person can see the moon without looking at it. The question is not whether "outcomes pre-exist measurement". As I have explained already, there are many real locally causal situations in which outcomes do not pre-exist the act of measurement. Why are you so bent in insisting that this is the issue. I don't see that question/definition in the EPR paper at all. Maybe I missed it. Could you point me to where EPR says "outcomes must pre-exist measurement"?
This IS the issue, as DrC has been trying to get you to understand. Read "Quantum mysteries revisited," N. David Mermin, Am. J. Phys. 58, #8, August 1990, pp 731-734. I point you to this paper because you don't have to do any physics to see the problem. He does give you the physics in section III, but the conflict between lhv and QM is illustrated nicely in sections I and II. Let me give you some quotes from that paper (there are three particles traveling from a single source to three detectors, the detectors have two different settings (1 and 2) and there are two different possible outcomes (R and G) for each setting):

"In the absence of connections between the detectors and the source, a particle has no information about how the switch of its detector will be set until it arrives there. Since in each run any detector might turn out to be either the one set to 1 or one of the ones set to 2, to preserve the perfect record of always having an odd number of R flashes in 122, 212, and 221 runs, it would seem to be essential for each particle to be carrying instructions for how its detector should flash for either of the two possible switch settings in might find upon arrival." left hand column, p 732.

"If the instruction sets existed, then 111 runs would always have to produce an odd number of R flashes. But they never do, as I remarked in the third paragraph of this section, ... . Thus, a single 111 run suffices all by itself to give data inconsistent with the otherwise compelling inference of instruction sets." right hand column, p 732.

"Instruction sets require an odd number of R flashes in every 111 run; quantum mechanics prohibits an odd number of R flashes in every 111 run." left hand column, p 733.

 Quote by JesseM No, you're completely confused, the argument is about taking a God's-eye-view and saying that no matter how we imagine God would see the hidden variables, in a local realist theory God would necessarily end up making predictions about the statistics of different correlations that are different from what we humans actually observer in QM.
I disagree. God does not play dice. Make up your mind. Either God has complete information such that everything is certain and there are no "probabilities" or he does not.
 It's not "pointless" if you can use this hypothetical God's-eye-perspective (where nothing is hidden) to show that if the hidden variables are such that Alice and Bob always get the same result when they perform the same measurement, that must imply certain things about the statistics they see when they perform different measurements
Again it appears it is you who is confused, who is calculating the probabilities, God or "they"? Who is "they" by the way. It can't be Alice because she knows nothing about what is happening at Bob, nor can it be Bob. It can not be God either because he already knows everything so there are no "probabilities". So obviously it must be some external person looking at the data, who does not know everything about the cause of the data.

In any case, this is going off my main issue which I explained in my last post as follows:

Note that if we assume that your reasoning is correct, the full equation P(AB|H) = P(A|H)P(A|BH) still works. But in Bell's case, using the full chain rule does not work. So there must be another justification for insisting on P(AB|H) = P(A|H)P(A|H) other than the one you have given.

The justification for reducing P(A|BH) to P(A|H) is not based on whether B gives additional information but on whether B gives any information. (see Conditional Independence in Statistical Theory, J.R Statist. Soc B, 1979 41, No. 1, pp. 1-31) http://people.csail.mit.edu/tdanford...s/dawid-79.pdf

if A and B are equal, numerically P(A|BH) = P(A|H), but you could not say in this situation that since B gives you no additional information to H, therefore A and B are independent. They clearly are dependent. Therefore, although it is correct to reduce P(A|H)P(B|AH) to P(A|H)P(B|H) as a result of conditional independence between A and B, it is not correct to go from the fact that their numerical values are the same, to say that they are conditionally independent. X implies Y does not necessarily mean Y implies X. See the paper above. Link added.

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 Quote by billschnieder ... So it appears to me that Bell's ansatz can not even represent the situation he is attempting to model to start with and the argument therefore fails. What am I missing?
 Quote by DrChinese ... Sometimes, it is easy to over-focus on the details and miss the big picture ...
billschnieder here is the BIG PICTURE that you are missing:

Recognitions:
 Quote by billschnieder I disagree. God does not play dice. Make up your mind. Either God has complete information such that everything is certain and there are no "probabilities" or he does not.
You are definitely confused. Do you even understand what a "local hidden variables" theory is? It says the hidden variables associated with a particle have well-defined values at all times, and they either determine the exact outcomes of each measurement, or they determine the probabilities of different outcomes in a partially random way, with nothing besides the purely local variables associated with a particle (hidden or otherwise) influencing its response to a measurement. In the first case P(A|H) or P(B|H) will always be 1 or 0, so "God does not play dice"; in the second case it may be something else, but in this case there is genuine randomness in nature, so God does play dice (the notion of 'local realism' does not in itself automatically imply determinism). But either way we can show it's impossible to construct a local hidden variables theory that matches the statistics seen by actual human experimenters.

If you don't even understand what a local hidden variables theory is, you really need to go back to basics and try to learn something about the ideas behind the proof, rather than rush to critique it before you even have the first idea of what it's saying.
 Quote by JesseM It's not "pointless" if you can use this hypothetical God's-eye-perspective (where nothing is hidden) to show that if the hidden variables are such that Alice and Bob always get the same result when they perform the same measurement, that must imply certain things about the statistics they see when they perform different measurements
 Quote by billschneider Again it appears it is you who is confused, who is calculating the probabilities, God or "they"?
All probability calculations involving hidden variables are from the perspective of "God" (i.e. from the perspective of an idealized observer who knows the values of all hidden variables). But when I said "that must imply certain things about the statistics they see when they perform different experiments", the statistics are those seen by the human experimenters.
 Quote by billschneider Who is "they" by the way. It can't be Alice because she knows nothing about what is happening at Bob, nor can it be Bob.
It's the statistics they find in retrospect once they get together (or send signals) and compare their results, potentially long after they actually perform the measurements at a spacelike separation. For example, for some choice of detector settings they might find that on all trials where they happened to set their detectors at the same angle, they always got the same result (if Alice got spin-up then Bob got spin-up too, and same with spin-down), but on all trials where they happened to set their detectors at different angles, they only got the same result on 1/4 of all these trials. The point of my lotto card example was to show that these statistics are impossible under a local hidden variables theory--if they always get the same result when they scratch the same box on their cards, that implies that on the subset of trials where they scratched different boxes, they should have gotten the same result on 1/3 or more of the trials in that subset. Did you read my lotto card example and consider the math behind it? If we're going to continue to this discussion, you really need to thoughtfully consider the examples and arguments people give you rather than just giving knee-jerk argumentative responses and acting as though you are totally confident that you are right and that every physicist since Bell has missed something obvious that only you were smart enough to discover.
 Quote by billschneider Note that if we assume that your reasoning is correct, the full equation P(AB|H) = P(A|H)P(A|BH) still works. But in Bell's case, using the full chain rule does not work. So there must be another justification for insisting on P(AB|H) = P(A|H)P(A|H) other than the one you have given. The justification for reducing P(A|BH) to P(A|H) is not based on whether B gives additional information but on whether B gives any information. (see Conditional Independence in Statistical Theory, J.R Statist. Soc B, 1979 41, No. 1, pp. 1-31) http://people.csail.mit.edu/tdanford...s/dawid-79.pdf
Where in the paper do you think it says or implies that it's incorrect to say that if B gives no additional information about the probability of A beyond what H gives you, then we can reduce P(A|BH) to P(A|H)? (I'm pretty confident the paper says no such thing and you are misunderstanding it somehow) Can you give a specific quote and page number? And just as you completely ignored my lotto card example, it appears here you completely ignored my flashlight example, and just repeated your original objection almost verbatim. Again, if we're going to have an actual discussion you need to be willing to give thoughtful consideration to examples and arguments, otherwise this will go nowhere.
 Quote by billschneider if A and B are equal, numerically P(A|BH) = P(A|H)
What do you mean "if A and B are equal"? A and B aren't numbers, they're events. Do you mean "if P(A) and P(B) are equal"? But that doesn't make sense either since the conditional probability of A given some other facts can be different than the absolute probability of A, I could easily come up with an example where P(A)=P(B) but numerically P(A|BH) is not equal to P(A|H) (this would have to be an example where B does give some additional information about the probability of A beyond what H alone gives).
 Quote by billschneider but you could not say in this situation that since B gives you no additional information to H, therefore A and B are independent.
I never claimed A and B were independent, so this is irrelevant to my argument. In my flashlight example A and B are clearly statistically dependent (though not causally dependent--do you understand the difference?), since P(A) = 0.55 but P(A|B) = 0.50909..., so B does give you some information about the probability of A. But if you already know H, then B gives you no additional information about the probability of A beyond what you already knew from H, so P(A|H and B) = P(A|H). Do you disagree that this is true in the example I gave?
 Quote by billschneider Therefore, although it is correct to reduce P(A|H)P(B|AH) to P(A|H)P(B|H) as a result of conditional independence between A and B, it is not correct to go from the fact that their numerical values are the same, to say that they are conditionally independent.
Again, no one claimed that A and B are conditionally independent, neither I nor Bell. In my flashlight example they are clearly conditionally dependent--did you read that example and think about it? If not please do so, in particular tell me if you disagree that the following probabilities would be correct in the example:

P(A) = 0.55
P(A|B) = 0.50909...
P(A|H1) = 0.7
P(A|H1 and B) = 0.7
 Don't get too discouraged Bill, there is always an opposition. Think of it as a test of your argument, not a personal attack. Now, I have not studied probability theory or local/nonlocal stuff much, so I am in no position to comment on that, but I can see that there are a lot of posts that aren't addressing his concern. I think there needs to be a well-defined list of axioms laid out for all to use in this argument. It would help a lot. Again, Bill's problem is with [2], not the outcome of Bell's stuff. He agrees that if [2] is true then the other stuff is all true, but the question is all about [2], not about the stuff that follows it. I don't think he is getting lost amongst the trees, I think he is questioning whether one of the trees in the forest really is a tree. Edit: JesseM points out in the next post that what I really meant was [1] everywhere there is a [2] above, which is true.

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 Quote by Prologue It would help a lot. Again, Bill's problem is with [2], not the outcome of Bell's stuff. He agrees that if [2] is true then the other stuff is all true, but the question is all about [2], not about the stuff that follows it. I don't think he is getting lost amongst the trees, I think he is questioning whether one of the trees in the forest really is a tree.
He said his problem was with (1), not (2). Specifically he said that he didn't understand why, if H represents the full set of variables that influence the outcome of a given measurement in a local way, then if A is some possible outcome for that measurement, while B is some outcome for a different measurement performed at a spacelike separation from the first, then P(A|H) = P(A|BH). The point I'm making is that this equation is not incompatible with the idea of a statistical correlation between A and B, i.e. the idea that P(A) is different from P(A|B). I gave specific examples like the lotto example and the flashlight example where this would be true. And if the concrete examples don't suffice to show intuitively why the equation should hold in a local hidden variables theory, my argument in post #61 and #62 on the other thread was trying to give a fairly detailed argument as to why P(A|H) = P(A|BH) must be true in a local realist theory, provided we let H represent the complete information about all physical variables (hidden and otherwise) in a past light cone cross-section (PLCCS) of the measurement M1 which might yield result A, with the PLCCS chosen so that no event in it has the measurement M2 which might yield result B in its future light cone (i.e. the cross-section of M1's past light cone is taken at a time after the last time that the past light cones of M1 and M2 intersect).

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 Quote by Prologue Don't get too discouraged Bill, there is always an opposition. Think of it as a test of your argument, not a personal attack. Now, I have not studied probability theory or local/nonlocal stuff much, so I am in no position to comment on that, but I can see that there are a lot of posts that aren't addressing his concern. I think there needs to be a well-defined list of axioms laid out for all to use in this argument. It would help a lot. Again, Bill's problem is with [2], not the outcome of Bell's stuff. He agrees that if [2] is true then the other stuff is all true, but the question is all about [2], not about the stuff that follows it. I don't think he is getting lost amongst the trees, I think he is questioning whether one of the trees in the forest really is a tree. Edit: JesseM points out in the next post that what I really meant was [1] everywhere there is a [2] above, which is true.
A simple review of what I posted previously will show that it is Bill who is missing the train. You don't need to consider separability if you look at the underlying argument. Instead of splitting hairs over semantics, why not address the meat? Just try to answer the following 2 questions: a) do you believe observables have well defined values independent of observation, as Einstein supposed? b) if yes, please present a set of values for angle settings 0, 120 and 240 for some group of photons that you believe is representative.

If you don't get this concept, you are missing the forest. It's cute you pretend that someone is the "loyal opposition" but actually Bill is coming off more as as a craggly contrarian. I am not actually sure craggly is a word, by the way. The meaning of Bell is what is important.

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billschnieder, I think DrChinese, JesseM and RUTA did a good job in trying to explain, but you still seem a little 'skeptical'.

The real problem is that you are making an assumption on completely wrong premises, almost like – "I can prove that the probability for this car making 100 mph doesn’t make sense" – when the car is actually making 200 mph...

And I show you where your assumption goes wrong:
 Quote by billschnieder God observes and smiles because he already knows that the box always contains only two cards one red and one white, which information the wise men do not know, hence "hidden". Therefore the hidden information H = "There are only two cards in the box, one red and one white and the colors of the cards never changes". Obviously, H is completely locally causal. Now lets look at the situation from God's perpective to see if the equation chosen by the wise men is correct.
Your "God" is clearly misinformed. Yes there are "two cards in the box", but then all goes wrong. One card is not red and the other white, both cards are red on one side and white on the other!

This is called spin and superposition, and is absolutely fundamental in QM. Bell of course knew this when writing his ansatz.

 Quote by billschnieder But this is wrong. There are only two possible outcomes in this experiment, (A:red, B:white) or (A:white, B:red) Therefore the probability P(AB|H) should be 0.5!
Run this Java applet:

David Mermin's EPR gedanken experiment animated

...and you see why this is also wrong...

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 Quote by Prologue ... there is always an opposition ...
Correct, and in this case billschnieder is in opposition, and 'we' belong to the 99.99% majority...
 Obviously you are opposing each other so there is no one distinct opposition, you are his opposition, he is yours... I was merely encouraging him to lay it out in detail, I don't have a horse in this race. Most of the arguments of this type (on these forums) end up with people bickering over little details about undefined objects. Define everything from the top, then argue.

 Quote by Prologue Obviously you are opposing each other so there is no one distinct opposition, you are his opposition, he is yours... I was merely encouraging him to lay it out in detail, I don't have a horse in this race. Most of the arguments of this type (on these forums) end up with people bickering over little details about undefined objects. Define everything from the top, then argue.
Disagreements don't change observations. From what I've read, this is an argument that just happens a lot here, over and over. The outcome appears to be the same, and that is that QM violates Bell Inequalities, and it is the best predictive theory on offer. The rest is details and quibbling because there is no other leg to stand on that I'm aware of.

JesseM:

I definitely understand what EPR meant by "elements of reality", and I definitely understand that it DOES NOT mean "I can see the moon when I am not looking at it", which is implied if you say outcomes must pre-exist observation. In any case, this is a rabbit trail and distracts from the main issue which I have already explained.

I am only interested in understand Bell's justification for writing

P(AB|H) = P(A|H) * P(B|H)

P(AB|H) = P(A|H) * P(B|AH)

In all your numerous examples and arguments, the only response relevant to this issue is your claim that, if H is completely specified, A adds no additional information to that already provided by H, therefore P(B|AH) = P(B|H). I don't have to respond to everything else in your rather verbose posts since this is the only point that is relevant to my issue. My response to this point, as I have already pointed out is as follows:

1) The definition of conditional independence is not based on additional information but on any information. In the article I quoted to you, section 2.1 titled "Definitions", page 2, midway down the page, where independence is defined it says:

 X [is independent of] Y if any information received about Y does not alter uncertainty about X;
The same article goes on to show in section 3.1, page 3, where conditional independence is defined that just because P(x|yz) = a(x,z), it does not necessarily show that conditional independence applies. Specifically it says:

 (2a) P(x|y,z)=P(x|z) (2b) P(x|y,z)=a(x,z) read, P(x|y,z) is a function of just x and z. A caution is called for here concerning the use of improper distributions for random variables. It is shown Dawid et al. (1973) that, in such circumstances, it is possible for (2b) to hold and, at the same time, for (2a) to fail. This is referred to as the marginalization paradox.
In any case this is not my main point, so don't focus all your attention here and igore my main point:

2) If P(B|AH) is really equal to P(B|H) as you insinuate, then it shouldn't matter which equation is used. Both should result in the same inequalities right? Do you agree that it should be possible to derive Bell's inequalties from either equation? Now following Bell's logic, try to derive the inequalities from P(AB|H) = P(A|H) * P(B|AH). It can not be done! Can you explain to me why? Just to be clear that you understand this point, let me rephrase it -- P(B|AH) = P(B|H) and P(B|AH) $$\neq$$ P(B|H) can not both be true at the same time right?

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 Quote by billschnieder JesseM: I definitely understand what EPR meant by "elements of reality", and I definitely understand that it DOES NOT mean "I can see the moon when I am not looking at it", which is implied if you say outcomes must pre-exist observation. In any case, this is a rabbit trail and distracts from the main issue which I have already explained.
The definition of "local realism" is not a distraction, it's central to the proof. Nothing I have said implies that "I can see the moon when I am not looking at it", though it does imply that all variables associated with the moon have well-defined variables even when I'm not looking, and therefore we can consider what conclusions could be drawn by a hypothetical omniscient observer who knows the value of all these variables (without assuming anything specific about what the values actually are on any given experimental trial). And all this stuff about variables having well-defined values when I'm not observing them only covers the "realism" aspect of local realism, locality is separate--for example, Bohmian mechanics would be an example of a realist theory that says all physical quantities have well-defined values even when we aren't looking at them, but it's also a non-local theory.
 Quote by billschnieder I am only interested in understand Bell's justification for writing P(AB|H) = P(A|H) * P(B|H) instead of P(AB|H) = P(A|H) * P(B|AH)
Obviously this reduction is fine as long as P(B|H) = P(B|AH). And this is guaranteed to be true in a local realist world where A can't have any causal influence on B, and the only reason for correlations between A and B is some set of conditions H1 and H2 in the past light cones of A and B which influence their probabilities in correlated ways.

Do you understand what a "past light cone" is, and why it's essential to the definition of locality?
 Quote by billschnieder In all your numerous examples and arguments, the only response relevant to this issue is your claim that, if H is completely specified, A adds no additional information to that already provided by H, therefore P(B|AH) = P(B|H). I don't have to respond to everything else in your rather verbose posts since this is the only point that is relevant to my issue.
No, the rest is quite relevant, since I explicitly show various examples where we have two events A and B such that there is a correlation between A and B, but it is completely due to some other set of conditions H in the past and not due to any causal influence between A and B, and this explains why P(B|H) = P(B|AH).

 This forum is meant as a place to discuss [quantum mechanics] and is for the benefit of those who wish to learn about or expand their understanding of said theory. It is not meant as a soapbox for those who wish to argue [quantum mechanics]'s validity, or advertise their own personal theories.
If on the other hand you have some intellectual humility, and are willing to consider that there's a good chance an argument that has been widely accepted by physicists for decades does not have any obvious holes that only you have been able to spot, then you should also consider that if you seem to see such a hole there is probably something basic missing from your understanding of the argument, and listen to the people who are trying to help guide you through the reasoning rather than immediately dismiss whatever they say if you don't spot the relevance right away. Up to you.
 Quote by billschnieder 1) The definition of conditional independence is not based on additional information but on any information.
And what does "the definition of conditional independence" have to do with our discussion? I have already said explicitly that A and B are not conditionally independent, and this was true in my examples as well. A and B are causally independent, which is different.

Are you familiar with the phrase "correlation is not causation"? We might find in some study that two variables A and B, such as sugar consumption and heart disease, are correlated--they are not conditionally independent. It might nevertheless be true that this is not because sugar consumption has any causal influence on heart disease, but rather because high sugar consumption tends to be correlated with some other factor C, like a diet with too much salt, that does have a causal influence on heart disease. In this case we would have a conditional dependence between sugar and heart disease, but no causal influence of sugar consumption on heart disease.

Similarly, in the lotto card example, there is definitely a conditional dependence between the probability that Alice finds a cherry when she scratches box 1 of her card, and the probability that Bob finds a cherry when he scratches box 1 of his card--in fact, if the first is true, then we know the second is true with probability 1! But this isn't because Alice's scratching box 1 and finding a cherry had any causal influence on Bob's card. Rather it's because of an event in the past light cone of both these other two events, which exerted a causal influence on both--namely the source picking two lotto cards with an identical pattern of "hidden fruits" behind the respective boxes on each card, with the hidden fruits associated with each card staying constant as the cards travel from the source to the locations of Alice and Bob. This is directly analogous to the way a local-hidden variables theory tries to explain why two experimenters always find the same spin (or opposite spin, depending on the type of particle) when they measure each member of a pair of entangled pair along the same axis.
Quote by billschnieder
In the article I quoted to you, section 2.1 titled "Definitions", page 2, midway down the page, where independence is defined it says:
 X [is independent of] Y if any information received about Y does not alter uncertainty about X;
I agree 100%, and have never said anything to suggest I was using a different definition of conditional independence. Again, A and B are not conditionally independent, only causally independent. If you are trying to find the probability of B (which could represent an event like 'Bob measured spin-up when measuring along the 180-degree axis), and you don't know anything besides the fact that it was a randomly-selected trial, then you will calculate some probability P(B). But if you are then asked "I want the probability of B on a trial where A also occurred" (where A could represent 'Alice measured spin-up on the 180-degree axis'), this is "information received about A" which does alter your uncertainty about B (now you are calculating P(B|A), which in a Bell type experiment will be different from P(B)), so B is not independent of A. It is nevertheless true that in a local hidden variables theory, if you had God-like knowledge of all the local hidden variables H associated with B, then learning A would give you no additional information about B, so P(B|H) = P(B|AH). But this would not change the fact that A and B are conditionally dependent, not conditionally independent.
 Quote by billschneider 2) If P(B|AH) is really equal to P(B|H) as you insinuate, then it shouldn't matter which equation is used. Both should result in the same inequalities right?
No, you're not making any sense. The fact that P(B|AH) is equal to P(B|H) is a specific piece of information about the physics of this problem which would not be true for any arbitrary problem where B and H were defined to mean something different physically. P(AB|H) = P(A|H) * P(B|AH) is a statistical identity which would hold mathematically regardless of the physical definitions of what the variables A, B, and H are supposed to mean; P(B|AH) = P(B|H) is an equation that we derive from specific physical considerations of the meanings of the symbols in Bell's proof. It shouldn't surprise you that in a physics proof, proving the conclusion should require making use of the specific physical assumptions of the proof, and that the conclusion can't be proved solely using general statistical identities which are true regardless of the meanings assigned to the variables!
 Quote by billschneider Do you agree that it should be possible to derive Bell's inequalties from either equation?
No, for the reasons above.
 Quote by billschneider Now following Bell's logic, try to derive the inequalities from P(AB|H) = P(A|H) * P(B|AH). It can not be done! Can you explain to me why? Just to be clear that you understand this point, let me rephrase it -- P(B|AH) = P(B|H) and P(B|AH) $$\neq$$ P(B|H) can not both be true at the same time right?
Of course they can't be true at the same time, they would require different physical assumptions about the meaning of A, B, and H. If you don't understand that proofs in physics show that specific physical conclusions follow from specific physical assumptions, and that you can't necessarily prove the same physical conclusions if you start from completely different physical assumptions, then I don't know what else I can say. We can show that E=mc^2 can be proved if we start from some specific physical assumptions like a definition of energy and the fact that c is a constant velocity which is the same in all reference frames; do you think E=mc^2 could still be proved if we used the same mathematical identities but totally changed the physical definitions of E, m, and c? (or didn't make use of any equations which followed specifically from their physical definitions?)

I think that it is simply incorrect to say that Bell was "really" responding to an actual program of Einstein's. It is more to the point to assume that Bell was, to a degree, putting words into Einstein's mouth by saying that Einstein was an advocate of a "more complete" version of QM, whereas Einstein was simply trying to prove that it is utterly fallacious to speak of QM as any kind of physical theory.

The whole idea of Einstein's advocacy of "local hidden variables," in my view, was just an attempt for certain up-and-comers to make names for themselves by way of "one upping" that most famous and venerable of all theoretical physicists.

In other words, since QM is itself just a theory of the necessarily statistical nature of all possible "real world" measurements, and since Einstein upheld that a "complete" physical theory must necessarily provide a spatio-temporal representation of all aspects of the experimental scenario in question (i.e. all measuring devices and things that are to be measured), then it is senseless to say that Bell showed some kind of flaw in the reasoning of EPR.

EPR, I think, was much more of a medidation on the logical foundations of any possible system of thought that can be called a "physical theory," rather than an attempt to show how an already existing theory can somehow be completed.

When the EPR paper finishes...

 While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible.
...I do not see any reason to assume that "such a theory" is necessarily identical with "a completed version of QM."

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