# Bell Theorem and probabilty theory

1. Jan 7, 2009

### mn4j

This is not true. If the coins are designed in such a way that they are biased to produce those results at 120 degrees apart, it would not be at odds with probability.

The problem with bells theorem which many people still fail to realize to their own undoing is that it is only valid within the assumptions it is predicated on. But there are several problems with bells assumptions, or more specifically his understanding of the meaning of probability. For example

According to Bell, in a local causal theory, if x has no causal effect on Y, P(Y|xZ) = P(Y|Z) (J.S. Bell, "Speakable and unspeakable in quantum mechanics", 1989)

However this is false, as explained in (ET Jaynes, "Clearing up mysteries, the original goal", 1989), P(Y|xZ) = P(x|YZ) in a local causal theory.

This is easily verified: Consider an urn with two balls, one red and one white (Z). A blind monkey draws the balls, the first with the right hand and second with the left hand. Certainly the second draw can not have a causal effect on the first draw? Set Y to be "First draw is a red ball" and x to be Second draw is a red ball.

Exp 1. Monkey shows you the ball in his left hand (second draw) and it is white. What is p(Y|xZ). The correct answer is 1. However, according to Bell and believers, since the second draw does not have a causal effect on the first, p(Y|xZ) should be the same as P(Y|Z) = 1/2 ??? This is patently false.

Bells problem is that he did not realize the difference between "logical independence" and "physical independence" with the consequence that whatever notion of locality he was representing in his equations is not equivalent to Einstein's.

2. Jan 7, 2009

### DrChinese

Re: Trouble with QM's theory on Bell's discovery

This is plain wrong, and on a lot of levels. Besides, you are basically hijacking the OP's thread to push a minority personal opinion which has been previously discussed ad nauseum here. Start your own thread on "Where Bell Went Wrong" (and here's a reference as a freebee) and see how far your argument lasts. These kind of arguments are a dime a dozen.

For the OP: You should try my example with the 3 coins. Simply try your manipulations, but then randomly compare 2 of the 3. You will see that the correlated result is never less than 1/3. The quantum prediction is 1/4, which matches experiments which are done on pretty much a daily basis.

Last edited: Jan 7, 2009
3. Jan 7, 2009

### JesseM

Re: Trouble with QM's theory on Bell's discovery

Yes, mn4j's analogy shows a misunderstanding of Bell's inequality. Here's a more accurate analogy I came up with a while ago:
And you can modify this example to show some different Bell inequalities, see post #8 of this thread if you're interested.

4. Jan 7, 2009

### mn4j

Re: Trouble with QM's theory on Bell's discovery

Jaynes, E. T., 1989, "Clearing up Mysteries - The Original Goal ," in Maximum-Entropy and Bayesian Methods, J. Skilling (ed.), Kluwer, Dordrecht, p. 1 (http://bayes.wustl.edu/etj/articles/cmystery.pdf)

Specifically, it treats the Bell inequalities from p.10. It also shows how the QM expectation value is consistent with probability theory and explains very clearly the mistakes Bell made that caused the conflict.

If you have access to Bell's original paper (NOT the multitude of third-party proofs on the web) you can read and follow alongside and see exactly where Bell introduced the conflict (equation 2) in Bell's original paper.

5. Jan 7, 2009

### JesseM

Re: Trouble with QM's theory on Bell's discovery

It's not consistent with probability theory applied to a local hidden variables scenario. If you disagree, please look at the example I gave and explain how any local hidden variables scenario (i.e. a scenario where each lotto card has a preexisting fruit under each box, and scratching merely reveals the fruit that was already underneath) could be consistent with the statistics given.

The article you posted is confused on many points--for example, they claim that "as his words show above, Bell took it for granted that a conditional probability P(X|Y) expresses a physical causal influence, exerted by Y on X". But if you look at Bell's words that they're quoting, this is a misinterpretation. What Bell said was "It would be very remarkable if b proved to be a causal factor for A, or a for B; i.e. if $$P(A | a \lambda)$$ depended on b or P(B | b \lambda) depended on a." Nowhere is Bell saying that in general P(X|Y) being different from P(X) implies that Y was causally influencing X, he's just saying that in this particular case the only sensible way that $$P(A | a \lambda)$$ could depend on b would be if b exerted an influence on the hidden variables $$\lambda$$. And the reason for that has to do with the specific meaning of the terms--here a represents one experimenter's choice of what variable to measure, and b represents the other experimenter's choice of what variable to measure. It is assumed that these choices happen at a spacelike separation from one another so there can be no direct causal influence from one to another in a local hidden variables theory, and it is assumed that the choices are spontaneous and random, so that there were no factors in their common past light cone which preconditioned both the experimenter's choices and the hidden variables $$\lambda$$ for each particle in such a way that they could be correlated. In physics there are two ways that events at different locations can be correlated, either by a direct causal influence from one to another, or by some factor in their common past light cone which influenced both; the monkey/urn example is simply an example of the latter, since both the probability of a given ball being found in one hand and the probability of a given ball being found in another are determined by the selection procedure from the urn, with the combination of balls in the urn at the time the second ball was selected having been influenced by the event of the first ball being selected. In the case of the experimenter's choices, Bell was simply passing over the possibility of some factor in the common past light cone of both measurements somehow predetermined both their choices of what to measure and the hidden variables in such a way as to produce a correlation, because he was assuming the choices were really free; once this possibility is eliminated, the only remaining possibility for a dependence between $$P(A | a \lambda)$$ and b would be if the experimenter's choice of measurement b somehow exerted a faster-than-light influence on the hidden variables $$\lambda$$ of the other particle. In later writings Bell and other physicists have explicitly recognized this loophole of the experimenter's choices being predetermined by factors in their common past light cone, see for example near the end of this article from the Stanford Encyclopedia of Philosophy where they write:
In terms of my lotto card analogy, it would be as if the machine printing the cards could know ahead of time whether me and my partner would both scratch the same box on our respective cards or different boxes (because there was some causal factor in the past light cone of the scratching events which determined in advance which boxes we would scratch, and the machine could examine this causal factor to know ahead of time what the choices would be), and the machine would vary the statistics of what fruits were behind each box depending on what our future choices were going to be on each trial.

6. Jan 7, 2009

### mn4j

Re: Trouble with QM's theory on Bell's discovery

You have not understood it. Bell is thinking that events at A should not influence events at B. Physically that is correct. However he imposes that physical condition on the probabilities. So his equations are wrong. Logically, the probability of an event at A can influence the probability of an event at B even if there is no physical dependence. That is precisely the reason why he makes the fatal error. You only need to look at the example of the monkey pulling two balls to see this. Physically, the second ball can have no influence on the first. But if you impose this condition on the probability, you end up with a probability of 1/2 for the first ball even after you have seen the second ball which is wrong. If you know that the second ball is white, the probability that the first one picked was red is 1 not 1/2.

Following the correct probability rules, QM is in line with probability theory.

7. Jan 7, 2009

### JesseM

Re: Trouble with QM's theory on Bell's discovery

He is also saying the experimenter's choices of what variable to measure a and b (i.e. the spin axes they are measuring on a given trial), which are distinct from the outcomes they get A and B, should not influence the outcomes at the other experimenter's detector, or even be correlated with the outcomes at the other experimenter's detector, because their choices are freely-made.
And what physical condition would those be? Are you talking about the one in the quote "It would be very remarkable if b proved to be a causal factor for A, or a for B; i.e. if $$P(A | a \lambda)$$ depended on b or $$P(B | b \lambda)$$ depended on a"? If not, please tell me what specific probability condition you're talking about. If you are talking about that quote--which seems to be the one the authors are focusing on with their urn analogy--then my analysis is correct, Bell's statement depended on his specific understanding of what a meant physically (a choice of what variable to measure by the experimenters), he wasn't saying that in general P(X|Y) can only depend on Z if Z exerts a causal influence on X.
In a local realist universe, the only way this can work is if their are causal factors in the past of both A and B that influenced both and caused them to be correlated--this is exactly what's going on in the urn example. Do you think there's a third option for how A and B can be correlated that's not based on A and B exerting a causal influence on one another, and is also not based on some set of factors C in the overlap between the past light cones of A and B which causally influenced both? If so, please explain it.
No, but the past event of the monkey pulling a ball from the urn is in the common past of both events (the event of our looking to see what color the monkey chose on his first pick, and the event of our looking to see what color the monkey chose on his second), and this shared past is what explains the correlation, obviously. Did you even bother to read my explanation???
Nope, it's not compatible with both the condition of local hidden variables and the condition that the experimenter's choices of what variable to measure were free ones which were not causally determined by some past factors which also determined the hidden variables associated with the particles (the 'no-conspiracy' assumption discussed in the quote from the Stanford Encyclopedia). The authors of the paper you mention have only rediscovered the conspiracy loophole, although they don't seem to have realized that the only way their observations about probabilities are relevant to Bell's theorem is if there was a conspiracy in the initial conditions which caused the experimenter's choices to be causally correlated with the hidden variables.

Last edited: Jan 7, 2009
8. Jan 7, 2009

### mn4j

Re: Trouble with QM's theory on Bell's discovery

The rules of probability demand that you calculate probabilities in specific ways. Bell used equation (12) when he should have used equation (13) and equation (14) when he should have used equation (15) [numbers according to Jaynes article]. The reason he did this was because he thought there should be no physical causal relationship. However, lack of a physical causal relationship does not mean lack of logical causality. This is explained in more detail in Jaynes' book " https://www.amazon.com/Probability-Theory-Logic-Science-Vol/dp/0521592712

The point is that by knowing the outcome of the second pick, the calculated probability for the first pick changes from what it was if the second outcome was unknown. The calculation of probabilites is completely after the fact of the experiment. This tells you that there is a logical link between the two in the way the calculation should be done even though we know that the second ball can not cause the first one to be different.

9. Jan 7, 2009

### JesseM

Re: Trouble with QM's theory on Bell's discovery

Apparently you missed (or misunderstood) the part immediately after (13) where the authors say:
In other words, they're saying that if you grant the assumption that "knowledge of the experimenter's free choices (a,b) would give us no information about $$\lambda$$", which is exactly the same "no-conspiracy" condition I have been discussing, then the left hand of (14) reduces to the right hand, and their equation (13) reduces to the equation (12) that Bell used. It's only if you drop this this assumption that you're left with the "fundamentally correct" equations (13) and (15) in which no assumption about (a,b)'s relation to other events has been made.
No, the explanation for the assumption is not just that there is no direct causal link between an experimenter's choice of what to measure at one detector b and the outcome at the opposite detector A. What is also being assumed is that in a local realist universe, there should be no causal factor in the common past of these two events that would lead to a correlation between b and A, because the experimenter's choices are assumed to be "free". Again, in a local realist universe there are two ways of explaining a correlation between two events X and Y; #1 is that either X is exerting a direct causal influence on Y or vice versa, but #2 is that some event(s) Z in their common past exerted a causal influence on both in such a way as to create the correlation. For example, if I put a black ball in one box and a white ball in another, and send one box to Alice and another to Bob, then if Alice opens her box and sees a white ball she'll know Bob must have gotten the black ball; this isn't because the event of Alice opening her box exerted a causal influence on the event of Bob opening his box or vice versa, but it is because both events were determined by the past events of my putting a black ball in one box and a white ball in the other. In fact this is exactly the idea behind the $$\lambda$$ in a local hidden variables theory, it represents some factors which both particles share because these factors were determined at the past event of both particles being created at the same location.
Please name an example of "logical causality" between two events A and B that does not reduce to either a direct causal link between A and B or some event C in the common past of both A and B that conditioned both of them and caused the correlation.
Yes, and the explanation for that is a common causal factor in the past of both picks. In a local realist universe, when you have consistent correlated outcomes of pairs of experiments, the explanation is always either 1) one outcome exerting a direct causal influence on the other, or 2) a causal factor in the past which influenced both later outcomes. Do you agree or disagree with this?

Last edited: Jan 7, 2009
10. Jan 7, 2009

### mn4j

Re: Trouble with QM's theory on Bell's discovery

You fail, just like Bell, to appreciate the difference between ontological correlations, which are results of real experimental data, and logical correlations which are epistemological in nature. Bell was not analysing real experimental results, he was constructing a mental picture of what we might see if we measured it. This means he was bound by the rules of probability to do the calculations differently than he did.

You asked for an example of "logical causality". I have given one already. The one of the urn. Knowledge of the second ball being red, causes the probability of the first ball to change from 1/2 to 1 even though there is no physical causation happening. To say that the result is only because of a common physical cause in the past does not work because the calculation of probabilities is completely after the monkey had picked the balls, yet calculating the probability of the first ball before and after the revealing of the second ball you get a different value. There is no way any new knowledge of the second ball can physically cause a change in the first ball is there?

Another example:
Consider the the following proposition
- "A implies B", logically, this implies "not B implies not A".
Now pick any A and B of your choice for which there is a physical causality such that A causes B. It would still hold logically that "not B implies not A", yet it would make no sense to talk of "not B" causing "not A" physically.

The point therefore is that lack of physical causality is not a good enough excuse to assume lack of logical causality in your equations as bell did. You must still use the correct rules of probability to calculate.

11. Jan 7, 2009

### JesseM

Re: Trouble with QM's theory on Bell's discovery

But Bell was assuming a realist picture in which any hidden variables that determine results of measurements already have well-defined values before measurement. You can imagine God continuously measuring $$\lambda$$ as the particles propogate, until after each one is measured by the experimenters, if it helps.
I asked for an example of logical causality where the measured correlation could not also be physically explained either in terms of one measurement influencing the other directly or both measurements being influenced by events in the past. The correlation between the colors of the two balls is completely based on the fact that they were both selected from the same urn.
But if you take the God's eye view in which all objective facts exist before measurement, then what's happening here is the "hidden variable" of the second ball being red determines that when I look at it, the event "I see a red ball" is guaranteed to happen with probability 1. Meanwhile, the fact that the first ball picked was white, and was picked from an urn containing one red ball and one white ball, is what determines there'd be a probability 1 that the "hidden variable" associated with the second ball turned out to be red. In other words, P(I see a red ball when I look at second pick | I saw a white ball when I looked at the first pick) = 1 can be broken down into:

P(first ball picked had hidden state 'white' after being picked | I saw a white ball when I looked at the first pick) = 1
P(second ball picked had hidden state 'red' after being picked | first ball picked had hidden state 'white' after being picked) = 1
P(I see a red ball when I looked at second pick | second ball picked had hidden state 'red'' after being picked) = 1

In a realist universe, you should always be able to take such a God's eye view where the true state of all the unknown factors is part of the probability calculation (although these true states are represented as variables since we don't necessarily know them), and probability is interpreted in frequentist terms (in terms of the statistics of repeated trials) rather than Bayesian terms, and this is exactly what Bell was doing in his proof.

Meanwhile, I don't see the authors of the paper you link to arguing that we can only use epistemological Bayesian probabilities rather than objective frequentist probabilities, that appears to be your own original argument which has nothing to do with their own. The fact that they include $$\lambda$$ in their probability equations, despite the fact that $$\lambda$$ represents hidden variables whose value we can never actually know, shows this.
Why not? "Causality" just means that one physical fact determines another physical fact according to the laws of physics, and an absence of a certain physical event is still a physical fact about the universe, there's no reason it can't be said to "cause" some other fact.

12. Jan 7, 2009

### DrChinese

Re: Bell Theorem

ZapperZ, thanks for splitting out the thread.

mnj4,

It is the realism argument that is most important to Bell, and it really doesn't matter your point about conditional probabilities. This is of little importance to Bell.

The mathematical description of realism is:

1 >= P(A, B, C) >= 0

The reason is that A, B and C individually are elements of reality, because they can be predicted in advance in a Bell test. The issue is not whether the measurement somehow distorts the results, it is whether these elements of reality (EPR) exist simultaneously independently of the ACT of observation.

If you believe they do, you are supporting realism. And NO physical theory of local realism (i.e. hidden variables) can reproduce the predictions of QM. If that statement is not true, you can provide a counterexample! But as pointed out earlier, this has befuddled everyone who has tried (so far).

Your urn example makes no sense. There are NO conditionals before any measurement occurs. So if we ask what percentage of humans are men, are Democrats, and/or are college educated, I can make predictions all day long and my answers will always satisfy the realism criterion above. It doesn't matter whether the attributes are causally related somehow or not. And yet when QM is applied, you cannot get around the negative probabilities for some of the subensembles if Malus is followed. You MUST know this already, how can you not?

Last edited: Jan 7, 2009
13. Jan 7, 2009

### mn4j

Re: Trouble with QM's theory on Bell's discovery

Exactly! Therefore any local realist theorem MUST also consider that measurements at A and B MUST be logically correlated! Bell sets out trying to determine the probability of observing certain events at A based on measurements at B. But in doing so, in he fails to incorporate logical dependence. This is the crucial point. And this is why you get the wrong answer for the urn if you calculate using bell's equation! Because there is no concept of logical dependence in it even though there must be! And the reason bell makes this mistake is because he does not clearly separate logical dependence from physical causation.

I am not arguing this at all. I have not even started talking about $$\lambda$$ because there are other assumptions Bell makes about $$\lambda$$ that are unfounded. The simplest explanation of what I am saying is that if A and B are determined by a local-realist theorem of hidden variables, then the probabilities of events at A MUST be logically dependent on those at B, even if there is no direct physical causation from A to B and vice versa. Therefore, failure to include logical dependence in his treatment is unfounded.

Last edited: Jan 7, 2009
14. Jan 7, 2009

### DrChinese

Re: Trouble with QM's theory on Bell's discovery

You have it backwards, as usual. Bell is pointing out what actually should have been obvious in retrospect: that subensembles cannot respect Malus as does QM.

1. If you had a group of 100 trials of Alice and Bob at settings of 0 and 45 degrees respectively, then we would expect a coincidence rate of 50%. Now, I ask does this statement somehow violate your definition of proper probability? I certainly hope not...

2. Now, imagine that we have the 4 permutations of the above and choose to subdivide it into another group, the results (which by the realistic definition must exist and be well-defined) of measurements (let's call this Carrie) at 22.5 degrees - i.e. midway between the 0 and 45 degree mark. Surely you already know that the correlation rate between Alice and Carrie must equal the correlation rate between Bob and Carrie. Is that too difficult? Does this statement somehow violate your definition of proper probability? I certainly hope not...

And finally, you must also already know that there is no set of trials possible in which 1. is true and also 2. is true. Oh, and that also matches the prediction of QM for 22.5 degrees (for photons) which of course is about 85%.

Now, none of this is a problem if you drop the requirement of realism. If you said that Bell's argument was not well-phrased, I would probably agree with you. But there is nothing wrong with the conclusion it leads to.

15. Jan 7, 2009

### JesseM

Re: Trouble with QM's theory on Bell's discovery

But the only type of "dependence" when you do frequentist calculations is statistical dependence in the objective facts of the matter over many trials, and this statistical dependence must always be explainable in terms of physical causation. For example, when I broke down the urn/ball probabilities like this:

P(first ball picked had hidden state 'white' after being picked | I saw a white ball when I looked at the first pick) = 1
P(second ball picked had hidden state 'red' after being picked | first ball picked had hidden state 'white' after being picked) = 1
P(I see a red ball when I looked at second pick | second ball picked had hidden state 'red'' after being picked) = 1

...none of these were subjective probabilities. For example, "P(first ball picked had hidden state 'white' after being picked | I saw a white ball when I looked at the first pick)" can be interpreted in frequentist terms as meaning "if you look at a large number of trials, and then look at only the subset of trials where the event of me seeing a white ball when I looked at the first pick occurred, in what fraction of this subset was it also true that the the ball had the hidden state white at the moment it was picked"? And the answer of course, is 1, the reason being that on any trial where the ball had hidden state white after it was picked, this caused me to predictably see a white ball when I looked at the color.

So you see, when you're dealing with a frequentist notion of probabilities in a realist universe, any statistical correlations must involve physical causation, either one fact causing the other or both facts being caused by some common cause in their past. Do you disagree? If so, please name an example of a situation where we can interpret probabilities in a purely frequentist manner yet there is a statistical correlation that can not be explained in one of these two causal ways.
What "Bell's equation" are you talking about? Please write it down, and explain what the variables are supposed to represent in terms of the urn example that you think leads the equation to be wrong.
Again, when you interpret probabilities in frequentist/realist terms, all statistical correlations (which I guess is what you mean by 'logical dependence') must be explained in terms of physical causes, though the explanation may involve a common cause in the past of two events rather than either one directly influencing the other. And Bell did not assume there'd be no statistical correlation between A and B--the whole point of including $$\lambda$$ was to show there could be such a correlation in a local realist universe, as long as it was explained by the source creating both particles with correlated hidden variables (a common cause in the past), just like my example of sending balls to Alice and Bob and always making sure one was sent a black ball and the other was sent a white one, so their measurements results would always be opposite (here I play the role of the 'source' which determines the hidden variables of each box that determine the correlations between their observations when they open their respective boxes).

The place that Bell assumed "no statistical correlation" was between the choice made by one experimenter of what spin axis to perform a measurement on, and the hidden variables of the other particle being measured by the other experimenter far away (as well as the other experimenter's choice of what to measure). In a local realist universe, for this assumption to be wrong, there would have to be some common cause(s) in the past of both an experimenter's choices and the particles' hidden variables which predetermined what they would be, and ensured a statistical correlation (so the source would act like it 'knows in advance' what choice the two experimenters would make, and adjusts the probabilities of emitting particles with different hidden variables accordingly).

16. Jan 7, 2009

### mn4j

Re: Trouble with QM's theory on Bell's discovery

This make no sense. If you had the objective facts, you will not need induction. It will be deductive and you will never have a probability other than 0 or 1. Did you read the article? The Bernouli urn example is treated in the article. The reason inductive reasoning is used, where probabilities can have values between 0 and 1, is because we don't have all the objective facts. Which means we are trying to make a prediction of what we might see if you make the measurement. Bell did not base his calculations on ANY objective facts. The first experiments trying to test his theorem were done years after. By objective facts I assume you mean real experimental data.

Take the example of the urn I gave earlier.

1. You know that there are two balls in the urn, one is red and one is white.
2. You know that the monkey picked one ball first and then another ball second.
3. You are asked to infer what the probability is of the first ball being red before seeing the result of the second ball. Then the second ball is shown to you and you are asked to again infer the probability that the first ball is red.

We both accept that "the second ball is red" has no physically causative effect on the state of the first ball, because it was picked after the second ball. At most, they have a single event in their past which caused them both. Yet, in calculating the probabilities in (3) above, you will not arrive at the correct result if you do not use the right equations which include logical dependence. Note that the urn example is the simplest example of a hidden variable theory. In this case, you are saying that once the balls have left the urn, the outcome of the experiment is determined and there is no superluminal communication between the monkey's right hand and left hand.

Bell's equation written for this situation is essentially,

P(AB|Z) = P(A|Z)P(B|Z) ( see Bell's equation (2) which is the same as eq 12 in Jaynes ) Remember the question is "What is the probability that both balls are red"?

Z: The premise that the urn contains two balls, one white and one red.
A: First ball is red
B: Second ball is red

Calculating based on that equation, the probability that both of those balls is red results in 0.5 * 0.5 or 0.25! Which is wrong! If you don't believe me, do the experiment, 1000000 times, you will never observe a red ball in both hands.

However, the correct equation should have been,
P(AB|Z) = P(A|B)P(B|Z) = P(B|A)P(A|Z) (Equation 15 in Jaynes).
which results in 0 * 0.5 = 0, the correct answer.

As you see, even though we accept that there is no physical causality from the second draw to the first draw, we still must include logical dependence to calculate the probabilities correctly. This means we must have a P(A|B) or P(B|A) term in our equation.
It is just like proponents of bell now point to experiments with real monkeys and 2-ball urns and say "since bell obtained 0.25 instead of 0 and real experiments obtain 0, it must mean that the experiments disprove local reality of the urn and the monkey."

You see, what is moving faster than light is not any physical influence. It is the same logical influence which caused our probability to suddenly change as soon as we knew the result of the second ball.

You are not reading what I write. I'll use a dramatic example.

Do you agree with the above?

There is a logical dependence between "not Dead" and "not Executed". If a person is not dead, it MUST follow that the person is not Executed. However, you can not say "not Dead" physically causes "not Executed", otherwise nobody will ever be executed. Logical dependence is not the same as physical causation.

He must have. Equation (2) in his article, (12) in Jaynes, means just that. The correct way to include logical dependence is in how you set up the equations, not by introducing additional parameters. Adding numbers to the balls in the urn example above, will not change the results if you do not use the right equations.

Maybe I should ask you a question. If you know the outcome at A, the settings at A and the settings at B will you be able to deduce the outcome at B? Isn't this the premise of any hidden variable theorem, that the outcome is determined by the values they left the source with and the settings at the terminals?
Now explain to me why when you calculate the probabilities, you do not include logical dependence. By this I mean that the term P(A|B) never occurs in any of bell's equations.

Just like knowing the result of the first second draw should influence the way you calculate the probability of the second draw. Shouldn't it? How is it supposed to influence it if you do not have a P(A|B) term??

Look at Bell's original article, DrChinese has it on his website. Everything starts from Eq. (2). Clearly from (2) there are only two options
1- Either bell did not understand how to apply probability rules or
2- He assumed that knowing the results at B should not influence the way we calculate the probability at A.
Either one is devastating to his result.

There is no other way, if you see it , let me know. The equations speak for themselves!

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17. Jan 7, 2009

### DrChinese

Re: Trouble with QM's theory on Bell's discovery

No one is saying that there is anything moving faster than light.

I think you are missing the bigger picture here. It is almost like you found a misspelled word and want to throw out a work for that reason. I have derived Bell's Theorem a variety of ways, and never do I bother with his "P(AB|Z) = P(A|Z)P(B|Z)" which you say is wrong. There are plenty of things in the original that could be presented differently, and something things I personally think are obscure in the extreme. But the ideas ultimately are fine. Like the EPR paper, he makes a few statements - the references to Bohmian Mechanics, for example - which leave a poor aftertaste.

What is generally accepted as the legacy of the paper is the following conclusion: No physical theory of local Hidden Variables can ever reproduce all of the predictions of Quantum Mechanics.

The key here is the realism requirement, which is not explicitly mentioned at all! 1 >= P(A, B, C) >= 0. Einstein would have naturally agreed with this, because he insisted that particle attributes existed independently of the act of observation: "I think that a particle must have a separate reality independent of the measurements. That is: an electron has spin, location and so forth even when it is not being measured. I like to think that the moon is there even if I am not looking at it."

So I would say that the 2 main assumptions of the paper are: a) realism, which is what the inequality is based upon - and NOT separability, as is often mentioned; and b) locality, because he was trying to point out that there might be some unknown force that can communicate between entangled particles or their measuring apparati superluminally.

I would like to make it clear that no dBB-type theory I have ever seen explains any mechanism by which Bell test results occur. They always simply state that they generically reproduce QM predictions. Maybe, maybe not. But the mechanism itself is *always* missing.

On the other hand: If you assume locality, then you can still stay with QM as a non-realistic local theory and be done with it. And there is no problem at all. Look at the bigger picture, and you will see why Bell is widely praised and accepted today: Bell tests are the heart of entanglement measures, and are leading to innovative new ways to explore the quantum world. And lo and behold, there are still no local realistic theories which have survived the rigors of both theory and experiment.

18. Jan 7, 2009

### mn4j

Re: Bell Theorem

If the equations on which his conclusions are based are wrong then the conclusions are baseless. Just because you prove it differently does not mean your equations are correct. For example, where in your eqations do you account for logical dependence? If you think logical dependence is not necessary then what you are modeling is not a local-realist system. No doubt the results contradict real experiments.

19. Jan 7, 2009

### JesseM

Re: Trouble with QM's theory on Bell's discovery

I didn't say anything about you knowing the objective facts. Again, the frequentist idea is to imagine a God's-eye perspective of all the facts, and knowing the causal relations between the facts, figure out what the statistics would look like for a very large number of trials. Then, if you want to know the probability that you will observe Y when you have already observed X, just look at the subset of these large number of trials where one of the facts is "experimenter observes X", and figure out what fraction of these trials would also include the fact "experimenter observed Y".

If you believe there are objective facts in each trial, even if you don't know them, then it should be possible to map any statement about subjective probabilities into a statement about what this imaginary godlike observer would see in the statistics over many trials--do you disagree? For example, suppose there is an urn with two red balls and one white ball, and the experiment on each trial is to pick two balls in succession (without replacing the first one before picking the second), and noting the color of each one. If I open my hand and see that the first one I picked was red, and then I look at the closed fist containing the other and guess if it'll be red or white, do you agree that I should conclude P(second will be white | first was red) = 1/2? If you agree, then it shouldn't be too hard to understand how this can be mapped directly to a statement about the statistics as seen by the imaginary godlike observer. On each trial, this imaginary observer already knows the color of the ball in my fist before I open it, of course. However, if this observer looks at a near-infinite number of trials of this kind, and then looks at the subset of all these trials where I saw that the first ball was red, do you agree that within this subset, on about half these trials it'll be true that the ball in my other hand was white? (and that by the law of large numbers, as the number of trials goes to infinity the ratio should approach precisely 1/2?)

If you agree with both these statements, then it shouldn't be hard to see how any statement about subjective probabilities in an objective universe should be mappable to a statement about the statistics seen by a hypothetical godlike observer in a large number of trials. If you think there could be any exceptions--objectively true statements of probability which cannot be mapped in this way--then please give an example. It would be pretty earth-shattering if you could, because the frequentist interpretation of probabilities is very mainstream, I'm sure you could find explanations of probability in terms of the statistics over many trials in virtually any introductory statistics textbook.
Exactly, there was an event in their past which predetermined what the color of both the first and second ball would be (the event of the first ball being picked from the urn containing only a white and red ball). Don't you remember that this was exactly my point, that in a realist universe any statistical correlation between events must be explainable either in terms of one event causing the other or in terms of a common cause (or set of causes) in their common past? I asked if you had any counterexamples to this general statement about statistical correlations, the urn example certainly isn't one.
What exactly do you mean by "logical dependence"? The probabilities can of course be calculated in the same frequentist manner as I discussed above--if you imagine a large number of trials of this type, it's certainly true that on the subset of trials where the first ball picked was white, the second ball was always red in 100% of this subset, and likewise in the subset of trials where the first ball picked was red, the second ball was always white in 100% of this subset.
You are distorting Bell's claims again. He does not claim that as some sort of general rule, P(AB|Z) = P(A|Z)P(B|Z) for any arbitrary observations or facts A, B, and Z. Instead, he says that for the specific case where a and b represent the events of some experimenter's choices of what variable to measure on a given trial, we can assume that these choices are really "free" and were not predetermined by some common cause in the past which also determined the state of the hidden variables $$\lambda$$. And thus, in this specific case with a and b having that specific meaning, we can write the equality in equation (14) from the Jaynes paper you referenced:

$$P(AB | ab\lambda ) = P(A | a \lambda) P(B | b \lambda)$$

Jaynes does not disagree that this equation is correct if you make the assumption about a and b not being predetermined by factors that also determined $$\lambda$$, that's why he prefaces that equation by saying "But if we grant that knowledge of the experimenters' free choices (a,b) would give us no information about $$\lambda$$". If you want to question the assumption of "free choice" (which just means choices not determined by factors which also determined the hidden variables produced by the source on a given trial, they might be determined by other complex factors in the experimenter's brains prior to the choice), then go ahead, this is a known loophole in the proof of Bell's theorem. But don't act like Bell was making some very broad statement about probability that would be true regardless of what events/observations the symbols a and b are supposed to represent.
And this is a strawman, since Bell never suggested such a broad equation that was supposed to work regardless of what the symbols represent. Try to think of an experiment where the symbol a represents the free choice of experimenter #1 of what measurement to perform (like which of the three boxes on the lotto card to scratch in my example), and b represents the free choice of experimenter #2 at some distant location (such that no signal moving at the speed of light can cross from the event of experimenter #1 making his choice/measurement to the event of experimenter #2 making his choice/measurement), and A represents the outcome seen by #1 while B represents the outcome #2, and $$\lambda$$ represents some factors in the systems being measured that determine (or influence in a statistical way) what outcome each sees when they perform their measurement. With the symbols having this specific meaning, can you think of an experiment in a local realist universe where the equation

$$P(AB | ab\lambda ) = P(A | a \lambda) P(B | b \lambda)$$

would not work?
Um, have you been ignoring my point all along that in a realist universe, statistical correlations between events are always either due to one event influencing the other or events in their common past which influenced (or predetermined) both? I don't think I was very subtle about the idea that there were two options here. You seem to have simply ignored my second option, which is a little suspicious because it's precisely the one that applies to the case of the second ball drawn from the urn (whose color is predetermined by the event of the first ball being picked from the urn, since the urn only contained two balls to begin with).
And you are not reading what I write, because I already explained that your overly narrow definition of "physical causation" is different from what I mean by the term. Read the end of post #11 again:
Both "the prisoner is not dead" and "the prisoner was not executed" are physical facts which would be known by a hypothetical godlike being that knows every physical fact about every situation, and if this being looked at a large sample of prisoners, he'd find that for everyone to whom "not dead" currently applies, it is also true that "not executed" applies to their past history. So, according to my broad definition of "cause", it is certainly true that "not executed" is a necessary (but not sufficient) cause for the fact of being "not dead".
Wow, you have really missed the most basic idea of the proof. No, of course (12) in Jaynes doesn't mean A and B are independent of $$\lambda$$, where could you possibly have gotten that idea? The equation explicitly includes the terms $$P(A | a \lambda)$$ and $$P(B | b \lambda)$$, that would make no sense if A and B were independent of $$\lambda$$! What equation (12) does show is that Bell was assuming A was independent of b, and B was independent of a. No other independence is implied, if you think it is you really need to work on your ability to read statistics equations.
No, of course it isn't--you've completely left out the hidden variables here! A hidden-variables theory just says that the outcome A seen by experimenter #1 is determined by experimenter #1's choice of measurement a (like the choice of which box to scratch in my lotto card analogy in post #3 on this thread) combined with the hidden variables $$\lambda_1$$ associated with the system experimenter #1 is measuring (like the preexisting hidden fruits behind each box on the lotto card). Likewise, the outcome B seen by experimenter #2 is determined by experimenter #2's choice of measurement b along with the hidden variables $$\lambda_2$$ associated with the system experimenter #2 is measuring. If both experimenters always get the same result on trials where they both choose the same measurement, that must mean that the hidden variables associated with each system must predetermine the same outcome to any possible measurement, as long as you assume the source that's "preparing" the hidden variables on each trial has no foreknowledge of what the experimenters will choose (if it did have such foreknowledge, then it might only predetermine the same outcome to the same measurement on trials where the experimenters were, in fact, going to choose the same measurement to make).
Because the correlation seen between results A and B is assumed to be purely a result of the hidden variables the source associated with each particle--a common cause in the past (again, in a local realist universe all correlations are understood either as direct causal relations or a result of common causes in the past, and A and B are supposed to have a spacelike separation which rules out a direct causal relation in a local realist universe). As long as you include a term for the dependence of both A and B on the hidden variables, there's no need for a separate term for the statistical correlation between A and B. Similarly, if I have an urn containing two reds and one white, and the first I pick is red, I can write the equation P(second ball seen to be white | first ball seen to be red) = 1/2; but if I explicitly include a term for the "hidden variables" associated with what's left in the urn on each pick, I can just rewrite this as:

P(second ball seen to be white | after first pick but prior to examination, first ball had 'hidden state' red and urn had 'hidden state' one red, one white) = 1

and

P(first ball seen to be red | after first pick but prior to examination, first ball had 'hidden state' red and urn had 'hidden state' one red, one white) = 1

which together with

P(after first pick but prior to examination, first ball had 'hidden state' red and urn had 'hidden state' one red, one white) = 1/2

imply the statement P(second ball seen to be be white | first ball seen to be red) = 1/2. And in general we can write the equation:

P(first ball seen to be red, second ball seen to be white) = [SUM OVER ALL POSSIBLE HIDDEN STATES X FOR URN + FIRST BALL AFTER FIRST PICK] P(first ball seen to be red | urn + first ball in hidden state X)*P(second ball seen to be white | urn + first ball in hidden state X)*P(urn + first ball in hidden state X)

...This is directly analogous to equation (12) from the Jaynes paper which you've told me is the same as (2) from Bell's paper.
Nope, see above, in a local realist universe any correlation between A and B should be determined by the hidden state of each particle given to them by the source at the event of their common creation, so there is no need to include P(A | B) as a separate term, just like there's no need in my equation above for the urn as long as you include the hidden state of the urn + first ball after the first pick.

Last edited: Jan 8, 2009
20. Jan 8, 2009

### mn4j

Re: Trouble with QM's theory on Bell's discovery

Read up on probability theory Jesse, what you say makes no sense. If you assume that there is a correlation between the two particles when they left the source, which you must if you are trying to model a local realist theorem, then you MUST also assume that there is logical dependence between the probability of the measurement at A and that at B. Bell did not do that. Remember Bell was trying to model a hidden variable theorem.

Write down the equations you used to arrive at your answers. Then we can talk.