Bell's derivation; socks and Jaynes

[harrylin]

Hello,

For this little discussion I base myself on Bell's paper on Bertlmann's socks:
http://cdsweb.cern.ch/record/142461

Although I have participated in a number of discussions about Bell's theorem, I always had the uneasy feeling not to fully understand the definitions of symbols and the notation - in particular how to account for lambda in probability calculations.

So, although I intend to discuss here the validity (or not) of Jayne's criticism of Bell's equation no.11, I'll start very much more basic. Using Bell's example of socks, I think that we could write for example:

P1(pink) = 0.5

Here P1(pink) stands for the probability to observe a pink sock on the left foot on an arbitrary day. An experimental estimation of it is found by taking the total from many observations, divided by the number of observations.

As the colour depends on Bertlmann's mood, we can then account for that mood as an unknown variable "lambda" (here I will just put X, for unknown). However, any local realistic theory that proposes such an unknown variable as explanation, still must predict the same observed result. Therefore, I suppose that if we include X as causal factor, we must still write:

P1(pink|X) = 0.5

Thus far correct?

 

[IsometricPion]

P1(pink) = 0.5

Here P1(pink) stands for the probability to observe a pink sock on the left foot on an arbitrary day. An experimental estimation of it is found by taking the total from many observations, divided by the number of observations.

As the colour depends on Bertlmann's mood, we can then account for that mood as an unknown variable "lambda" (here I will just put X, for unknown). However, any local realistic theory that proposes such an unknown variable as explanation, still must predict the same observed result. Therefore, I suppose that if we include X as causal factor, we must still write:

P1(pink|X) = 0.5

Thus far correct?

I don't think this is quite correct. Rather, if X correlates with Bertlmann wearing a pink sock then P1(pink|X)=(P1(pink)/P(X))>0.5. Instead, $\int{}P_1(pink|X)P(X)dX=P_1(pink)=0.5$ (obviously if X is a causal factor it must correlate with P1(pink)). I think what Bell is saying in equation 11 is that if one knew λ (in addition to the local conditions) there would be no residual correlations between the distributions of the measurements (after accounting for its effects).

 

[harrylin]

I don't think this is quite correct. Rather, if X correlates with Bertlmann wearing a pink sock then P1(pink|X)=(P1(pink)/P(X))>0.5. Instead, $\int{}P_1(pink|X)P(X)dX=P_1(pink)=0.5$ (obviously if X is a causal factor it must correlate with P1(pink)). [..]).

Thanks for that clarification! I had not looked at it that way. However, X is like EPR's hidden function: Bertlmann's unknown and unpredictable mood determines what socks he will wear. X stands for the physical model, which is here an invisible random function (indeed, it happens in his head) that delivers one of {pink, not pink}. Obviously the chance to observe a Bertlmann pair of socks on Bertlmann's feet is simply 1. Then we must have, for the case that half of the time a pink sock is observed on the left foot:
P1(pink|X)=P1(pink)/P(X) = P1(pink)/1 =0.5

It's exactly the same as for a fair coin: P(head | fair coin) = 0.5.

I can imagine that someone would like to split the probability estimation up into unknown "knowns": then we can separate it into the cases that Bertlmann decides to put a pink sock on his left leg, and the cases that he decides to put another colour on his left leg. However, what we are interested in the result over many times, and then we are necessarily back at where we were here above. Thus, I don't see any use for that.

 

[IsometricPion]

X stands for the physical model, which is here an invisible random function (indeed, it happens in his head) that delivers one of {pink, not pink}. Obviously the chance to observe a Bertlmann pair of socks on Bertlmann's feet is simply 1. Then we must have, for the case that half of the time a pink sock is observed on the left foot:
P1(pink|X)=P1(pink)/P(X) = P1(pink)/1 =0.5

It's exactly the same as for a fair coin: P(head | fair coin) = 0.5.

I misinterpreted what you meant by X. I took it to mean a variable taking on values from the set of moods, some subset of which would correlate with Bertlmann wearing a pink sock instead of the mood model itself. In the latter case, I certainly agree with your results.

Edit: A couple of papers that may be relevant to this discussion: Jaynes' view of EPR, a critque of Jaynes' view

 

[harrylin]

@IsometricPion: thanks for the links! I suspect that our own discussion here, which is based on http://cdsweb.cern.ch/record/142461, will show that the Arxiv paper misses the point; we'll see!

instead of running to eq.11, I will first work out the example that Bell gave in his introduction, as he did not do so himself.
Note that in Bell's paper the pictures come after the text. I'll start with a partial re-take.

Elaborating on Bell's example of Bertlmann's socks, we could write for example:

P1(pink) = 0.5

Here P1(pink) stands for the probability to observe a pink sock on the left foot on an arbitrary day. An experimental estimation of it is found by taking the total from many observations, divided by the number of observations.

As the colour depends on Bertlmann's mood, we can account for that mood as an unknown function "lambda" (here I will just put X, for unknown). However, any "classical" theory that proposes such a physical model, still must predict the same observed result. Therefore, if we include X as invisible cause for the outcome, we must still write:

P1(pink|X) = 0.5
(Compare: P(head | fair coin) = 0.5)

Similarly we can write for the right leg:

P2(pink|X) = 0.5

Bell remarks:

Which colour he will have on a given foot on a given day is quite unpredictable. But when you see that the first sock is pink you can already be sure that the second sock will not be pink. Observation of the first, and experience of Bertlmann, gives immediate information about the second.

The fact that "pink" on the left foot implies "not pink" on the right foot implies a strong correlation between results. We can acknowledge that correlation as follows, with for convenience a slight change of notation:

P(L,R|X) =/= P1(L|X) P2(R|X)

Here L stands for "pink on left leg", and R stands for "pink on right leg".

Ok so far?

 

[alsor]

pink = 1, not pink = -1

then:
P(LR) = 0, because L and R are always different;

formally:
P(LR) = P(L|R)*P(R); P(R) = P(L) = 1/2;
but: P(L|R) = 0 <> P(L); (both socks have never the same colour)corr = 0 - 1 = -1, full anti-correlation.And using Bell reasoning:
P(LR) = P(L)*P(R) = 1/2 * 1/2 = 1/4;

corr = 2*1/4 - 2*1/4 = 0.

Two random socks, and completely independent, of course.

 

[harrylin]

@ alsor: it appears that in this matter we both agree with Jaynes.
However, again you ran far ahead of me and I'm not sure if everyone who, so far, didn't "see" this point of Jaynes etc., could follow you. So, I'll continue my slow pace to make sure that everyone who watches this topic can follow me and that we all agree on the basic facts as well as notation. I'll catch up with you later.

 

[IsometricPion]

And using Bell reasoning:
P(LR) = P(L)*P(R) = 1/2 * 1/2 = 1/4;

corr = 2*1/4 - 2*1/4 = 0.

Two random socks, and completely independent, of course.

This misrepresents Bell's model of local hidden variables. Equation 11 of his paper assumes one knows the values of the hidden variables, in this case Bartlmann's mood. So, P(L|bartlmann feels like wearing a pink sock on his right foot)=0 (since beyond his mood one also knows that he does not wear the same color socks) and P(R|bartlmann feels like wearing a pink sock on his right foot)=1. Thus P(L|mood=right, pink)*P(R|mood=right, pink)=1*0=0.

If instead one does not know his mood (or anything about it other than it can take on one of two sets of values), P(L)=P(L|RP)P(RP)+P(L|R¬P)P(R¬P)=0*0.5+1*0.5=0.5, by exchangeability. The problem Jaynes sees with Bell's reasoning is not his statistical or mathematical procedure/ability, rather he thinks Bell is to restrictive in what he (Bell) consideres to be valid variables for the probability distributions for a theory upholding local realism.

 

[harrylin]

[..]The problem Jaynes sees with Bell's reasoning is not his statistical or mathematical procedure/ability, rather he thinks Bell is to restrictive in what he (Bell) consideres to be valid variables for the probability distributions for a theory upholding local realism.

Thanks for the correction; however, although indeed Bell doesn't make a blunder of that proportion, Jaynes certainly points out a subtle error in Bell's equation; according to Jaynes it is not correct.
Anyway, we're not there yet: the problem with the illustration of Bertlmann's socks is that it by far doesn't catch the complexity of the problem at hand. If the observations would always be perfectly anti-correlated, there wouldn't be a riddle.

Now, I'm afraid that his next illustration of Lille and Lyon matches it even less well; thus, for this discussion I have been trying to come up with a variant of Bertlmann's socks that addresses the fact that the local conditions affect the observed correlation, but I didn't come up with a good looking one (I thought of observation of white or yellow socks in daylight/artificial light, as well as mud on his socks, but I'm not satisfied). Any better suggestion? If not, we should perhaps move on to the introduction of eq.11.

 

[ThomasT]

Although I have participated in a number of discussions about Bell's theorem, I always had the uneasy feeling not to fully understand the definitions of symbols and the notation - in particular how to account for lambda in probability calculations.

Metaphors are unnecessary and sometimes confusing, imho. Why not just refer to Bell's original formulation of a local realistic QM expectation value. Where does lambda appear and what does it refer to?

 

[harrylin]

Metaphors are unnecessary and sometimes confusing, imho. Why not just refer to Bell's original formulation of a local realistic QM expectation value. Where does lambda appear and what does it refer to?

While it may appear that he defines it very precisely, different people interpret it slightly differently in the literature. Moreover, I wasn't in the clear about notation. However, this discussion is already making it quite clear (I just needed a memory refresh!); we're now moving on to Bell vs Jaynes.

 

[jtbell]

we're now moving on to Bell vs Jaynes.

I'm pretty much a spectator in these discussions, but I'd like to point out that there was a long thread here about three years ago, about Jaynes's objections to Bell:

https://www.physicsforums.com/showthread.php?t=283519

This was before you joined PF, so you may not have seen this. It may or may not fit in with the direction you were planning to go.

It was split off from another thread, by the way, which is why it appears to start in the middle of a discussion.

 

[harrylin]

I'm pretty much a spectator in these discussions, but I'd like to point out that there was a long thread here about three years ago, about Jaynes's objections to Bell:

https://www.physicsforums.com/showthread.php?t=283519

This was before you joined PF, so you may not have seen this. It may or may not fit in with the direction you were planning to go.

It was split off from another thread, by the way, which is why it appears to start in the middle of a discussion.

Thank you! Indeed I had not seen that one... BTW I was also very much a spectator of another current thread in which I saw the suggestion to start this topic. Now I'll first check out the old thread.

 

[harrylin]

Thank you! Indeed I had not seen that one... BTW I was also very much a spectator of another current thread in which I saw the suggestion to start this topic. Now I'll first check out the old thread.

Ok, I'm afraid that I will need some time to work through that old thread; and I'm very busy this week.

Still, I started reading it and I notice some disagreement about what Bell claimed to prove. There is no use getting into arguments about the meaning of "local realism" and philosophy. What the "local realist" Einstein insisted on, and what Bell claimed to be incompatible with QM, was "no spooky action at a distance". Or, as Bell put it in his first paper:

that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past.

Those who deviate from that issue are shooting at straw men.

Bell puts it this way in his Bertlmann's socks paper:

What is held sacred is the principle of "local causality" - or "no action at a distance". [...] What [Einstein] could not accept was that an intervention at one place could influence, immediately, affairs at the other.

The focus of this discussion is Bell's attempt to prove that Einstein's "no action at a distance" principle is incompatible with QM, in the light of Jayne's first criticism.

 

[harrylin]

I'm pretty much a spectator in these discussions, but I'd like to point out that there was a long thread here about three years ago, about Jaynes's objections to Bell:

https://www.physicsforums.com/showthread.php?t=283519

This was before you joined PF, so you may not have seen this. It may or may not fit in with the direction you were planning to go.

It was split off from another thread, by the way, which is why it appears to start in the middle of a discussion.

I now had a better look at it, and I think that in particular posts #26 and #31 are important. Anyway I'll give a short summary of how I now see it.

If Jaynes' criticism focuses on Bell's equation no.11 in his "socks" paper, it was perhaps due to a misunderstanding about what Bell meant (his comments were based on an earlier paper).

P(AB|a,b,x) = P(A|a,x) P(B|b,x) (Bel 11)

Here x stands for Bell's lambda, which corresponds to the circumstances that lead to a single pair correlation (in contrast to my earlier X, which causes the overall correlation for many pairs).

According to Jaynes it should be instead, for example:

P(AB|a,b,x) = P(A|B,a,b,x) P(B|a,b,x)

Perhaps Jaynes thought that Bell meant:

P(AB|a,b,X) = P(A|a,X) P(B|b,X)

in which case Jaynes claimed that:

P(AB|a,b,X) = P(A|B,a,b,X) P(B|a,b,X)

This is really tricky.

However, he really was disagreeing with the integral equation.
According to him, it should not be:

P(AB|a,b) = ∫ P(A|a,x) P(B|b,x) p(x) dx

but:

P(AB|a,b) = ∫ P(AB|a,b,x) P(x|a,b) dx

and thus:

P(AB|a,b) = ∫ P(AB|a,b,x) p(x) dx = ∫ P(A|B,a,b,x) P(B|a,b,x) p(x) dx

Is my summary of the disagreement correct?

What is the significance of little p(x) instead of P(x)?

 

[DrChinese]

According to Jaynes it should be instead, for example:

P(AB|a,b,x) = P(A|B,a,b,x) P(B|a,b,x)

http://bayes.wustl.edu/etj/articles/cmystery.pdf

As I read it, this is one of Jaynes's arguments. However, I think it is attacking a straw man. The essence of Bell's argument does not require the factorization so much as a definition of what realism is.

For a SINGLE photon, not a pair: does it have a well-defined polarization at 0, 120, and 240 degrees independent of the act of observation? Once you answer this in the affirmative, as any local realist must, the Bell conclusion (a contradiction between the assumption and QM's predictions) follows quickly. If you answer as no, then you already deny local realism so it is moot.

So I really don't see the significance here of Jaynes' argument. The only people that take it seriously are local realists looking for support for their position. The vast majority of scientists see it for what it is, something of a technicality with no serious implications for the Theorem whatsoever.

In other words, it would be helpful to see an example that somehow related specifically to photon polarization rather than urns (which does not seem to be much of an analogy).

 

[lugita15]

http://bayes.wustl.edu/etj/articles/cmystery.pdf

Speaking of this paper, does anyone know what Jaynes is talking about in the end of page 14 and going on to page 15, concerning "time-alternation theories"? He seems to be endorsing a local realist model which makes predictions contrary to QM, and he claims that experiments peformed by "H. Walther and coworkers on single atom masers are already showing some resemblance to the technology that would be required" to test such a theory. Does anyone one know whether such a test has been peformed in the decades since he wrote his paper?

 

[IsometricPion]

Is my summary of the disagreement correct?

What is the significance of little p(x) instead of P(x)?

I think it is clear that λ in Bell's paper corresponds to x here (rather than X). While I am less certain of Jaynes' meaning I think it is probably x as well (since it appears on the left side of the | indicating that it is a variable in some of his equations). Jaynes is pretty consistent, so I would expect everything he denotes by λ to refer to the same thing (i.e., all his λ's should correspond with x's rather than X's).

Jaynes refers to probabilities essentially as logical statements of uncertain truth value. His P(y|Y) correspond to logical statements where Y is the predicate and y is the antecedent the truth value of which one is uncertain (the amount of (rational) belief one has that y has a value between u and v is P(u≤y≤v|Y)=∫_u^v P(y|Y)dy). He refers to any probability not of this form as p(y), since one cannot ascribe a logical statement to such a probability without more information regarding its context. Since the context here is clear and consistently applied, I think it is just a matter of formalism (i.e., there is no substantial difference). (Jaynes defines what he means by these symbols in Appendix B of Probability Theory: The Logic of Science.)

Jaynes states what he thinks are Bell's hidden assumptions:

(1)...Bell took it for granted that a conditional probability P(X Y) [sic P(X|Y)] expresses a physical causal influence, exerted by Y on X. ...

(2)The class of Bell theories does not include all local hidden variable theories...

He goes on to mention a type of local hidden variable theory he does not think Bell's theorm covers, though I do not yet understand his argument as to why it isn't covered.

I think a key point to this discussion is how to define local realism in terms of the functional dependence of probability distributions of outcomes of the EPR (thought) experiment. Once this is agreed upon (i.e., all the variables and symbols we are using are well-defined) the rest should just be a matter of mathematics (about which I think we all should be able to agree).

 

[lugita15]

He goes on to mention a type of local hidden variable theory he does not think Bell's theorm covers, though I do not yet understand his argument as to why it isn't covered.

Yes, that's what I was asking about in my previous post. He claims that ordinary Bell tests won't be able to test this "time-alternating" model, but some other experiments could test it.

 

[Delta Kilo]

http://bayes.wustl.edu/etj/articles/cmystery.pdf

1. Urn example is a red herring. The cases are not equivalent. There is no correspondence established between A,B,a,b on one hand and R1,R2 on the other. Specifically, λ was lost along the way. Let's try to put it back.

λ is going to be the complete state of the urn before the first draw - that's our hidden random variable. A would be the location of the ball to be drawn first - a freely chosen parameter, mutually independent from λ. a=a(A,λ) is the outcome - deterministic function of A and λ. Now the state of the urn after the first draw is γ=γ(A,λ) - another deterministic function of A and λ. And finally b=b(B,γ) - is yet another deterministic function.

Now b=b(B,γ)=b(B,γ(A,λ))=b(A,B,λ), and b clearly depends on A, that is ∃A,B1,B2,λ: b(A,B1,λ) ≠ b(A,B2,λ). Here b(...) is a deterministic function and A,B1,B2,λ are merely placeholders, arguments of ∃, loop variables is you wish. There is a clear causal link: given the same initial state of the urn, the choice of ball in the first draw causally affects the results of the second draw.

In contrast, in Bell's case we have explicitly denied this causal link as violating locality: ∀A,B1,B2,λ: b(A,B1,λ) = b(A,B2,λ). Note we are not talking here about randomness, conditional probabilities, observer's state of knowledge etc., we are simply trying to establish a link between the value of a deterministic function and its parameter. See the difference?

So the two cases have different physical models behind them and it is not a surprise the results of one are not applicable to another. In fact, the urn would be a great example provided first and second draw can be spacelike separated

2. Regarding time-dependence. In Bell's case λ includes everything that might possibly affect the experiment, except for settings A and B. I think it is safe to assume that absolute value of t does not matter (otherwise we're in for a rough ride). Any relative time delay in the experiment will appear as yet another random factor collectively included in λ and the integral in (12) will include integration over the whole range of it. As long as these delays are independent from choices A and B, it's within Bell's framework.

 

[DrChinese]

1. Urn example is a red herring. The cases are not equivalent. There is no correspondence established between A,B,a,b on one hand and R1,R2 on the other. ...

Thanks for clarifying. I still have a hard time figuring out how the choice of measurement angle (for Alice and Bob) fits in. But you don't need to try to explain that point unless you want to.

 

[PeterDonis]

There is a clear causal link: given the same initial state of the urn, the choice of ball in the first draw causally affects the results of the second draw.

Yes, but that's not Jaynes' point. His point is that the choice of ball in the *second* draw cannot possibly causally affect the results of the *first* draw--yet logically, the two are not independent, so the probabilities don't factorize. (For example, if there is only one red ball in the urn, and we are told that a red ball was drawn on the second draw, the probability of drawing a red ball on the first draw given that data is zero.)

 

[DrChinese]

Yes, but that's not Jaynes' point. His point is that the choice of ball in the *second* draw cannot possibly causally affect the results of the *first* draw--yet logically, the two are not independent, so the probabilities don't factorize. (For example, if there is only one red ball in the urn, and we are told that a red ball was drawn on the second draw, the probability of drawing a red ball on the first draw given that data is zero.)

I get that the Jaynes' critique has to do with the factorizing that Bell pushed, but really how does that change the Bell result in any way?

If you assume realism (simultaneous existence of particle attributes independent of the act of observation) AND locality (that the results of Alice and Bob's observations are causally independent): you easily get the Bell result a number of ways. What does factorizing have to do with this? To me, the factorizing was just a way to express the locality assumption, but certainly not central to the argument.

Although I hardly see how Jaynes' point even applies, the urn example seems so contrived.

 

[PeterDonis]

I get that the Jaynes' critique has to do with the factorizing that Bell pushed, but really how does that change the Bell result in any way?

If you assume realism (simultaneous existence of particle attributes independent of the act of observation) AND locality (that the results of Alice and Bob's observations are causally independent): you easily get the Bell result a number of ways. What does factorizing have to do with this? To me, the factorizing was just a way to express the locality assumption, but certainly not central to the argument.

I thought that having the probabilities for the two measurements factorize was a crucial step in deriving the Bell Inequalities; without the factorization the inequalities can't be derived.

Although I hardly see how Jaynes' point even applies, the urn example seems so contrived.

He was just using it as a simple example, easy to visualize, where A is causally independent of B but the joint probability of A and B doesn't factorize into separate probabilities for A and B. (I realize I'm speaking loosely, if I need to tighten it up I'll do so.)

 

[Delta Kilo]

Yes, but that's not Jaynes' point. His point is that the choice of ball in the *second* draw cannot possibly causally affect the results of the *first* draw--yet logically, the two are not independent, so the probabilities don't factorize.

Bell stipulates that conditional probability of outcome a is independent from free parameter B. Jaynes says conditional probability of one outcome R1 is not independent from another outcome R2. See the difference? If we are talking about outcomes, then of course outcomes a and b are not independent in Bell's case, but it is not the point.

In Bell''s case outcome a and experiment parameters A,B are connected (or not connected as the case might be) through deterministic function a(A,λ). If we try to introduce similar notion in Jayne's case, we'll see the function behaves differently, describing different physical model.

 

[PeterDonis]

Bell stipulates that conditional probability of outcome a is independent from free parameter B. Jaynes says conditional probability of one outcome R1 is not independent from another outcome R2. See the difference?

I see the distinction you are making, but I'm not sure you are capturing Jaynes' claim correctly. The "factorization" I am referring to is this:

P(AB|abx) = P(A|ax) P(B|bx)

where a, b are the free parameters (the settings of the two measuring devices in the EPR case; the parameters that determine which ball is picked out of the urn in the urn case); A, B are the outcomes (spin up/down in the given direction in the EPR case; color of ball picked in the urn case), and x represents the hidden variables (and any other prior information which does not vary between the two measurements).

In other words, the factorization claim is a claim about *both* of the things you refer to: it says each measurement outcome is independent of *both* the other outcome *and* the other set of free parameters. Jaynes is basically arguing that *both* of those claims of independence could in principle be false even in a case of causal independence.

The only question I can see is whether the equation I wrote above (which is pretty much the one in Jaynes' paper) correctly captures the corresponding part of Bell's paper. I'll have to dig up my copy of Speakable & Unspeakable in Quantum Mechanics to review Bell's paper to verify that.

 

[harrylin]

I'm now way behind - hope to catch up with the discussion here some time tomorrow!

@DrChinese:
The act of observation affects what is observed: that is a central point of QM and EPR certainly could accept that. Thus, while Jaynes may be attacking a straw man, I'm sure that you do and it distracts from the discussion here. As a reminder of my post #14: The discussion of this thread is about the first equations of Bell's derivation which claims to prove that the following opinion is incompatible with QM (and I now omit a sound bite that apparently bugs you):

"What is held sacred is the principle of [..]"no action at a distance". [..] What [Einstein] could not accept was that an intervention at one place could influence, immediately, affairs at the other".

I agree with you that Jayne's urn example isn't a very good one. However, I also don't appreciate much Bell's example of Lyon and Lille, as it doesn't "catch" the freely chosen detector settings well; and to go to polarisation would defeat the purpose of Bell's attempt. He had been there, it is where he came from - before he motivated his mathematical operations with his example of Lyons and Lille.

@IsometricPion:
Thanks for the clarification of p vs P and the link!

@Delta Kilo: Yes, we agree that the urn isn't a very good example. And neither do I think that Lyon and Lille is a very good example.
But I now have an idea of a better variant of Bertlmann's socks - and I really don't know if it will support Jaynes (which would defeat Bell), or if it will support Bell (which would not exactly defeat jaynes, but ...).
That will have to wait for tomorrow, or Monday.

 

[Delta Kilo]

The "factorization" I am referring to is this:

P(AB|abx) = P(A|ax) P(B|bx)

where a, b are the free parameters (the settings of the two measuring devices in the EPR case; the parameters that determine which ball is picked out of the urn in the urn case); A, B are the outcomes (spin up/down in the given direction in the EPR case; color of ball picked in the urn case), and x represents the hidden variables (and any other prior information which does not vary between the two measurements).

Where did you get this equation from? Please point me to this or similar equation in the original Bell's paper. Hint: there isn't one, in fact there is no product of two probabilities anywhere in Bell's paper.

The closest equation in Bell's paper would be equation (2) or (19):
$P(\vec{a},\vec{b})=\int{d\lambda \rho(\lambda)A(\vec{a},\lambda),B(\vec{b},\lambda)}$
Let's play "spot the difference":
1. A and B are deterministic functions of deterministic parameters (λ here being variable of integration rather than random variable).
2. There is only one probability on the right side, ρ(λ), and it is not conditional.
3. The whole thing is under the integral.
In other words it is a completely different equation.

 

[PeterDonis]

Where did you get this equation from?

I got it from Jaynes' paper, equation (14). (I was lazy and didn't use LaTeX, so I wrote "x" instead of lambda. I'll quit doing that here.) I did say in my post that I still wanted to check to see what the corresponding equations in Bell's paper looked like. Now that you have linked to Bell's paper, let's play "spot the correspondence".

You are right that the equation I gave, equation (14) in Jaynes' paper, doesn't really have a corresponding equation in Bell's paper. But Jaynes' equation (14) is not the only equation in his paper that bears on the "factorization" issue. In fact, Jaynes' (14) is really just a "sub-expression" from his equation (12), which looks like this:

$$P(AB|ab) = \int{P(A|a, \lambda) P(B|b, \lambda) p(\lambda) d\lambda}$$

This equation is basically the same as the equation you gave from Bell's paper. Bell's equation is for the expectation value of a given pair of results that are determined by a given pair of measurement settings; Jaynes' equation is for the joint probability of a given pair of results conditional on a given pair of measurement settings. They basically say the same thing.

Jaynes' point is that to arrive at his equation in the first place, Bell has to make an assumption: he has to *assume* that the integrand can be expressed in the factored form given above. In other words, the integrand Bell writes down is not the most general one possible for the given expectation value: that would be (using Bell's notation)

$$P(a, b) = \int{A(B, a, b, \lambda) B(a, b, \lambda) p(\lambda) d\lambda}$$

The question then is whether one accepts Bell's implicit reasoning (he doesn't really go into it much; he seems to think it's obvious) to justify streamlining the integrand as he does. Jaynes does not accept that reasoning, and he gives the urn scenario as an example of why not. I agree that there is one key difference in the urn scenario: the two "measurement events" are not spacelike separated. Jaynes doesn't talk about that at all.

Edit: Bell's notation is actually a bit obscure. He says that A, B stand for "results", but he actually writes them as *functions* of the measurement settings a, b and the hidden variables $\lambda$. He doesn't seem to have a notation for the actual *outcomes* (the values of the functions given specific values for the variables). I've used A, B above to denote the outcomes as well as the functions, since Bell's notation doesn't give any other way to do it. In Jaynes' notation things are clearer; the equivalent to the above would be:

$$P(AB|ab) = \int{P(A|B, a, b, \lambda) P(B|a, b, \lambda) p(\lambda) d\lambda}$$

Edit #2: Corrected the equations above (previously I had A in the second factor in each integrand, which is incorrect). Also, Jaynes notes that there are two possible factorizations; the full way to write the equation just above would be:

$$P(AB|ab) = \int{P(A|B, a, b, \lambda) P(B|a, b, \lambda) p(\lambda) d\lambda} = \int{P(B|A, a, b, \lambda) P(A|a, b, \lambda) p(\lambda) d\lambda}$$

This is basically Jaynes' equation (15) with $\lambda$ integrated out.

 

[harrylin]

Where did you get this equation from? [..]

As I discussed in post #15, it happens to be Bell's equation no.11 in his Bertlmann's socks paper, http://cdsweb.cern.ch/record/142461. I use this later paper as the basis for the discussion of this topic, because it is more elaborate and Bell explains things better.

 

[PeterDonis]

As I discussed in post #15, it happens to be Bell's equation no.11 in his Bertlmann's socks paper, http://cdsweb.cern.ch/record/142461. I use this later paper as the basis for the discussion of this topic, because it is more elaborate and Bell explains things better.

I agree, this paper is a much better basis for discussion. It appears that what Bell wrote as P(a, b) in the earlier paper (the "expectation value"), he writes as E(a, b) in this one (equation 13).

 

[IsometricPion]

Suppose a and λ are sufficient to determine A, while b and λ are sufficient to determine B. I assert this to be true due to the spacelike separation between events A and B (and the fact that the theories under consideration exhibit local realism). Therefore, P(A|B,a,b,λ)=P(A|a,λ) similarly P(B|A,a,b,λ)=P(B|b,λ). This is not to say that there is no correlation between A and B, but any such correlation must occur through λ since it is the only shared part of the conditions sufficient to determine each outcome. Using logic symbols: B→(b^λ) and A→(a^λ), therefore B^b^λ→B^B→B→b^λ similarly A^a^λ→a^λ. So, given this interpretation of local realism (which seems to be consistent with that expressed in Bell's paper) P(AB|a,b,λ)=P(A|B,a,b,λ)P(B|a,b,λ)=P(B|A,a,b,λ)P(A|a,b,λ)=P(A|a,λ)P(B|b,λ).

 

[PeterDonis]

Suppose a and λ are sufficient to determine A, while b and λ are sufficient to determine B. I assert this to be true due to the spacelike separation between events A and B (and the fact that the theories under consideration exhibit local realism).

Yes, agreed, *if* Bell's interpretation of local realism is correct, then any local realistic theory will lead to probabilities that "factorize" in the way you've written them down.

This is not to say that there is no correlation between A and B, but any such correlation must occur through λ since it is the only shared part of the conditions sufficient to determine each outcome.

*If* Bell's version of local realism applies, yes. As I read Jaynes, he is questioning whether Bell's definition of "local realism" is the correct one. Of course, if one is willing to give up either locality or realism if that's what it takes to make sense of the actual QM predictions, Jaynes' question is kind of a moot point.

 

[lugita15]

*If* Bell's version of local realism applies, yes. As I read Jaynes, he is questioning whether Bell's definition of "local realism" is the correct one. Of course, if one is willing to give up either locality or realism if that's what it takes to make sense of the actual QM predictions, Jaynes' question is kind of a moot point.

How can you have a correlation between A and B that does not just occur through λ, and still call a theory local realistic?

 

[PeterDonis]

How can you have a correlation between A and B that does not just occur through λ, and still call a theory local realistic?

A good question, to which I don't have a ready answer. Neither did Jaynes, for that matter. But pointing out that nobody has (yet) thought of a way to answer this question is not the same as *proving* that local realism *requires* the correlations to work the way Bell assumed they did. Bell didn't prove this; he just assumed it. The assumption certainly looks reasonable; it may even be true. But that's not the same as proving it true.

Another point that Jaynes makes is worth mentioning. We don't even understand why quantum measurements work the way they do for spin measurements on *single* particles. I take a stream of electrons all of which have come from the "up" beam of a Stern-Gerlach measuring device. I put them all through a second Stern-Gerlach device oriented left-right. As far as I can tell, all the electrons in the beam are the same going into the second device, yet they split into two beams coming out. Why? What is it that makes half the "up" electrons go left and half go right? Nobody knows.

One response to this, which has been the standard response in QM, is to redefine what counts as a physical explanation. Physics no longer has to explain why particular events happen in particular ways; in QM, it's now sufficient to explain probabilities over ensembles of "similar" events, without even pretending to explain why the individual events themselves turn out the way they do.

Another response, which is Jaynes' response, is to say that our physical knowledge is simply insufficient at this point, and what we ought to be doing, rather than lowering our standards of explanation, is to look harder for underlying mechanisms. Such a search may not yield any results; but Jaynes' claim is basically that since QM was adopted physicists haven't really been trying very hard. Perhaps if we looked harder, we would figure out an underlying mechanism that explained why half the "up" electrons go left and half go right; and once we had that mechanism, we might find that it also explained the EPR correlations in a way that showed how local realism can be true even if the probabilities don't factorize.

I'm not saying Jaynes' response is necessarily right; but it doesn't seem to me that it can just be rejected out of hand either.