Bell's derivation; socks and Jaynes

[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.

 

[DrChinese]

*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.

It is safe to say that Bell used a definition of realism that Einstein would have appreciated. Specifically, Einstein stated that there IS a reality independent of the act of observation. In addition, EPR defines something called elements of reality and clearly Bell was trying to model that. And nicely he did!

So I would happily say that Jaynes and others may have different definitions of realism, and under their definitions, local realism is quite possibly not ruled out.

 

[lugita15]

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.

I think you misunderstood me. I was asking a rhetorical question. Clearly, if the correlation depends on something other than local variables, it is definitionally not a local theory, end of story. That was the point I was trying to make.

 

[PeterDonis]

In addition, EPR defines something called elements of reality and clearly Bell was trying to model that. And nicely he did!

I agree, Bell did a good job of capturing what EPR were getting at.

 

[DrChinese]

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.

If Bell's is wrong (which can be stated many different ways, and has already in this thread: What is a better definition of realism?

If p(x,y,z)>=0 for any x, y, z doesn't work, I think we have a bigger problem. I keep asking this, and so far, not a single local realist will give me a satisfactory *alternative* definition. All the while rejecting Bell's. And Einstein's!

 

[PeterDonis]

I think you misunderstood me. I was asking a rhetorical question. Clearly, if the correlation depends on something other than local variables, it is definitionally not a local theory, end of story. That was the point I was trying to make.

It's not that simple. First of all, remember that Jaynes views probabilities as expressing our knowledge about reality, not reality itself. When he writes conditional probabilities that condition on "other than local variables", he's not saying there's any "action at a distance" that occurs physically; he's merely saying that, *logically*, knowledge of those "other than local variables" can in principle change your posterior probability estimates.

Second, however, Jaynes hiimself points out that, actually, the probabilities P(A|ab\lambda) and P(B|ab\lambda) (i.e., the ones that apparently depend on *both* sets of measurement settings, but *not* on either measurement result--each of these appears as one of two factors in the two versions of the "factorized" equation that I took from Jaynes' equation 15) can actually be simplified, because it's easy to show that knowledge only of the *direction* of the "a" measurement, for example, gives no additional information about the probabilities of possible results of the "b" measurement. So the two conditional probabilities above can actually be simplified to P(A|a\lambda) and P(B|b\lambda)--meaning that the probabilities that condition only on the measurement settings (not on the results) *are* "local" in the sense you are using the term.

The additional information that *does* change the posterior probabilities is knowledge of the *results* of the measurements, A and B. But the observer at the "a" measurement doesn't know the result of the "b" measurement until it reaches him via a light signal, and vice versa. So the actual correlations that are observed could, in principle, be explained entirely by information traveling at light speed or less; there is nothing in the probability functions themselves, once simplified as above, that rules that out.

 

[PeterDonis]

If p(x,y,z)>=0 for any x, y, z doesn't work, I think we have a bigger problem.

I'm not sure what you're driving at. Do you see something in a viewpoint like Jaynes' that appears to violate this condition?

 

[PeterDonis]

What is a better definition of realism?

I wasn't questioning Bell's definition of "realism", only of "local realism", and only the "local" part.

As far as alternative definitions, I don't have any ready-packaged one, but I do have an observation: in Quantum Field Theory, the definition of "causality" is that field operators have to commute at spacelike separations. There is nothing in there about lack of correlations or what variables correlations can depend on; the only requirement is that, if two measurements are spacelike separated, the results can't depend on which one occurs first. The QM probabilities in EPR experiments certainly meet that requirement. Would something along these lines count as an alternative definition of "local realism"?

 

[IsometricPion]

I wasn't questioning Bell's definition of "realism", only of "local realism", and only the "local" part.

There is nothing in there about lack of correlations or what variables correlations can depend on; the only requirement is that, if two measurements are spacelike separated, the results can't depend on which one occurs first.

I assumed local meant that each outcome was only a function of properties at the measuring device (i.e., the settings of the device and the values of any hidden variables evaluated at that event in space-time). Note that without the sufficiency gauranteed by realism one cannot factor the joint probability as I did above (the last equality in the last line is no longer valid in general).

I assumed realism to mean that a thing's state exists independent of measurement. I took this to imply that the measurement is determined completely by the other properties in the system (i.e., A=A(B,a,b,λ); B=B(A,a,b,λ) and since this must hold for the states of both objects, A=A(a,b,λ); B=B(a,b,λ)).

I made no reference to correlations anywhere except when calculating the joint probability at the end. Do you mean (in "questioning... only the 'local' part") that the analysis is no longer local when one is examining correlations between events separated by a spacelike interval? If so, I would say that locality does not apply to analyses, only to interactions between things modeled by the theory in question (though I could be wrong). As an aside, Wikipedia- Principle of locality indicates that QFT obeys locality.

 

[lugita15]

As an aside, Wikipedia- Principle of locality indicates that QFT obeys locality.

I've always felt the whole debate about whether quantum mechanics is local to be largely semantics. To my mind, entanglement makes it rather clear that quantum mechanics is nonlocal, but of course many people have the positon that entanglement means that QM possesses "quantum nonlocality" (spooky action at a distance), to be distinguished from "classical nonlocality".

 

[Delta Kilo]

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.

Yes, this is exactly the assumption of local realism: the outcome P(A|a,λ) does not depend on the choice of parameter b and vice versa. Let's see how Bell explains it:

It seems reasonable to expect that if sufficiently many such causal factors can be identified and held fixed, the residual fluctuations will be independent, i.e.,
P(N,M|a,b,λ)=P1(M|a,λ)P2(N|b,λ) (10)

(eq (10) is the same as Bell's eq(11) and Jayne's eq (14) except M and N are renamed to A and B)

The factorization in eq. (10) means P1(...) and P2(...) are independent. Now the reason they are independent is because Bell chose it to be that way, by encapsulating all common factors in parameter λ. The underlying assumption here is that the values M and N are affected by some common factors (represented by λ), local factors a and b, and some residual randomness, independent of either global or local influences. This residual randomness is the reason we have probabilities P1, P2 at all, without it we would have
deterministic functions M(a,λ) and N(b,λ).

Eq (10) is valid for any given a,b,λ. Say, we discovered that the number of cases in both Lyons and Lille is influenced by day of week, so we are only comparing the results on a given day (say on Friday). Then we discovered a correlation with the stock market, so we only compare the results from only those Fridays when stock market was bearish. etc. And we keep doing that that until the residual randomness is independent.

I repeat, P1 and P2 are independent by design. If they turn out not to be independent, it just means we didn't do a good enough job with λ and overlooked some common factor. There is no limit on what λ can contain, except for local factors a and b, in accordance with physical model of local realism.

Now, there is an easy way to get rid of the residual randomness, by lumping it into λ. We can introduce random variables χ and η representing residual randomness of M and N. In case of Lyons and Lille they would represent the health of the population, their susceptibility to heart attack, including random fluctuations. χ(a,λ) might be a random function which tells whether a person in Lyons is going to have a heart attack given local and global factors a and λ. M and M then become deterministic functions M=M(χ,a,λ), N=N(η,b,λ), probabilities P1(M|χ,a,λ) and P2(N|η,b,λ) become {0,1} and eq (10) is automatically satisfied. Then we just redefine λ to include χ,η: λ' = {λ,χ,η}. It does expand the meaning of λ, which now means not just common global factors but any factors at all whether local or global, but excluding a and b. This is basically what was done from the outset in eq (2) in Bell's EPR paper.

Jaynes says that the fundamentally correct equation is
P(AB|abλ) = P(A|Babλ) P(B|abλ) (15)
Well, where did that come from? It's just the axiom of conditional probability P(AB) = P(A|B) P(B) with abλ tucked in. It is of course trivially true, but the locality assumptions and the special role of λ have been thrown out with the bathwater. Basically, while (14) is a physical model of a particular EPR setup with added local realism assumption, (15) is a tautology in a form 2*2*x = 4*x which tells us absolutely nothing.

Now, let's talk about 1st of the two objections:

(1) As his words above show, Bell took it for granted that a conditional probability P(X|Y ) expresses a physical causal influence, exerted by Y on X.

I assume Jaynes refers here to the following quite:

It would be very remarkable if b proved to be a causal factor for A, or a for B; i.e., if P(A|aλ) depended on b or P(B|bλ) depended on a.

Note the subtle difference: Jaynes talks about causal dependence of one outcome random variable on another random variable, while Bell talks about dependence of random variable on free parameter. The difference is, with two random variables they may be dependent and you cannot say whether X causes Y, Y causes X or both X and Y are caused by some third factor. In Bell's case of random variable and free parameter, dependency is clearly one way: the outcome depends on the parameter but not the other way around. The parameter is a given, it does not depend on anything else. This is actually one assumption which is implied and not stated directly. Violation of this assumption represents superdeterminism loophole, which is currently being discussed in another [STRIKE]ward[/STRIKE]thread.

As an illustration of his point, Jaynes gives Bernoulli Urn example. Let's start with eq (16):
P(R1|I)=M/N
I'd say I was introduced here to mimic Bell's λ. But what is the meaning of I exactly?

I = "Our urn contains N balls, identical in every respect except that M of them are red, the remaining N-M white. We have no information about the location of particular balls in the urn. They are drawn out blindfolded without replacement."

So I is not a random variable, nor a parameter. It does not have a set of values you can integrate over. It never changes. Basically it does absolutely nothing. Also note conspicuous absence of local parameter a or its equivalent. And without a, the whole thing misses the point.

Now if we are to re-introduce a and λ according to Bell's recipe, we would define a as a free local parameter which applies to the first measurement only. Say, a is a location of the ball to be picked during the first draw. Correspondingly b is the location of the ball to be picked on the second draw. λ is a random variable which by definition includes everything else which might possibly affect the outcomes. In this case λ would be exact arrangement of the balls in the urn. Clearly λ and a together completely determine which ball is drawn first: R1 = R1(a,λ). State of the urn after the first draw is γ=γ(a,λ) and second ball R2=R2(b,γ)=R2(a,b,λ). Note that expression for R2 violates Bell locality assumption and so the whole setup is clearly different from Bell's. Anyway, R1 and R2 are fully determined by a, b, and λ and therefore do not depend on anything else:
P(R1|R2abλ)=P(R1|aλ)={1: R1=R1(a,λ), 0: otherwise}
P(R2|R1abλ)=P(R2|abλ)={1: R2=R2(a,b,λ), 0: otherwise}.
Easy to see that factorization P(R1 R2|a,b,λ)= P(R1|aλ)P(R2|abλ) is in fact correct.

R1=R1(a,λ) and R2=R2(a,b,λ) above are deterministic functions, like in EPR paper. We could add some local residual randomness to them to get the equation similar to eq. (10) from Berltmann's Socks paper. For example, a and b would select x-coordinate of the ball to be drawn and y-coordinate would be picked at random. As long as random functions R1 and R2 are independent, the factorization will be valid. Again, this randomness can always be moved from R1 and R2 into λ.

So what is missing in Jaynes paper? Well, the elephant in the room of course, I mean the λ. λ is a key feature of Bell's paper and it is completely absent in Jaynes example. λ by definition encapsulates all randomness and all parameters in the system, except a and b. Once particular values of λ,a,b are fixed, everything else is predetermined. Without λ, the best posteriori estimate of conditional probability P(R1|...) would necessarily include dependency on R2 and vice versa. Once we nail down λ,a,b, all other dependencies disappear.

 

[Delta Kilo]

The additional information that *does* change the posterior probabilities is knowledge of the *results* of the measurements, A and B.

No it does not. The results A and B are already fully defined by a,b, and λ.

 

[PeterDonis]

No it does not. The results A and B are already fully defined by a,b, and λ.

I didn't say that knowledge of the results changes the results. I said that knowledge of one result changes the *posterior probability* that you would compute for the other result.

 

[DrChinese]

I wasn't questioning Bell's definition of "realism", only of "local realism", and only the "local" part.

As far as alternative definitions, I don't have any ready-packaged one, but I do have an observation: in Quantum Field Theory, the definition of "causality" is that field operators have to commute at spacelike separations. There is nothing in there about lack of correlations or what variables correlations can depend on; the only requirement is that, if two measurements are spacelike separated, the results can't depend on which one occurs first. The QM probabilities in EPR experiments certainly meet that requirement. Would something along these lines count as an alternative definition of "local realism"?

Here is my point. Start with ONE photon, not 2. Apply realism to that. That means that there is a well defined value for the result of a polarization measurement at 0, 120 and 240 degrees. So this means that p(0=H,120=H,240=H) or any permutation is >=0. Do you agree with this? If so, yours and mine and Bell's definitions are alike. The problem Bell found starts here. You can see that when you try to put down values for what they would be for any reasonable sample - it won't agree with Malus (and I do mean Malus here).

So what I am saying is that once you set up the realistic scenario you are looking to test, you add an entangled (essentially cloned) photon into help accomplish that. When that photon is tested remotely to the first, you are also require the assumption of observational locality - a setting here does not affect an outcome there, and vice versa. How can a local realist object to this?

So if Jaynes were to agree with this definition of realism, I really don't see what his objection would be to Bell. Again, I am not trying to derail the conversation so much as understand it. If Jaynes is picking on a detail of what Bell wrote, but which has since been readily clarified by hundreds of writers, I just miss the issue entirely.

 

[DrChinese]

... 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.

This helps. Thanks.

 

[Delta Kilo]

I didn't say that knowledge of the results changes the results. I said that knowledge of one result changes the *posterior probability* that you would compute for the other result.

It does it if you don't know λ: P(A|Bab) ≠ P(A|ab)
But for a given a,b,λ it doesn't. P(A|Babλ) = P(A|aλ) = { 1: A=A(a,λ), else 0 }

And that is the crux of the argument. λ is what makes Bell's factorization possible but Jaynes completely ignores it in his paper.

 

[PeterDonis]

It does it if you don't know λ: P(A|Bab) ≠ P(A|ab)
But for a given a,b,λ it doesn't. P(A|Babλ) = P(A|aλ) = { 1: A=A(a,λ), else 0 }

Well of course, if you have a completely deterministic theory (which Bell's "local realistic" theory is), and you have complete knowledge of initial conditions, then you have complete knowledge of outcomes. I was talking about the case (which is the only case of real interest if we're trying to compare a "local realistic" theory in Bell's sense with QM) where we don't know λ, since that's the case Bell and Jaynes are discussing.

(And of course the actual QM probabilities do *not* factorize as above; that is, there is *no* "local realistic", in Bell's sense, set of hidden variables λ that allows perfect prediction of outcomes.)

I'm not disputing that "λ makes Bell's factorization possible", and I don't think Jaynes was either. As I've said in previous posts, I think Jaynes was saying that requiring there to be some such set of hidden variables λ might not be the correct definition of "local realism".

 

[lugita15]

This may just be because I don't grasp Jaynes' argument, but it seems to me that there is no need to go deep in the weeds concerning the mathematics of conditional probabilities. As far as I know, proofs of Bell's theorem (except Bell's original) generally do not even depend on the notion of conditional probability. What is Jaynes' fundamental explanation for the experimental fact that there seem to be nonlocal correlations between measurements of entangled particles, of a kind that is different than the correlations that could arise just from the local sharing of hidden variables between the two particles? Phrased in this way, all the thorny issues of Bayesian probability inference and the like go out the window.

 

[PeterDonis]

This may just be because I don't grasp Jaynes' argument, but it seems to me that there is no need to go deep in the weeds concerning the mathematics of conditional probabilities. As far as I know, proofs of Bell's theorem (except Bell's original) generally do not even depend on the notion of conditional probability.

I believe there are Bell-type results that do not involve probabilities. For example, suppose there were a scenario in which "local realism" would require, not just that probabilities obey certain inequalities, but that certain measurement results simply could *not* happen at all, whereas QM would predict that they could. I seem to remember reading about one such scenario constructed by Roger Penrose using spin-3/2 particles in The Emperor's New Mind, but I don't have my copy handy to check. As far as I can see, Jaynes' arguments wouldn't apply at all to such a scenario.

 

[Delta Kilo]

Well of course, if you have a completely deterministic theory (which Bell's "local realistic" theory is), and you have complete knowledge of initial conditions, then you have complete knowledge of outcomes. I was talking about the case (which is the only case of real interest if we're trying to compare a "local realistic" theory in Bell's sense with QM) where we don't know λ, since that's the case Bell and Jaynes are discussing.

Well, the theory doesn't have to be deterministic. λ can include any number of random variables (in fact expected to include some). Since factorization works for any given value of λ, we don't need to know it, we just need to make sure λ exists. Specifically, we assume there exists probability distribution ρ(λ) independent from a and b: ρ(λ|ab) = ρ(λ) (of course A(a,λ) and B(b,λ) must exist as well, they only make sense together).

It is hard to tell where the breakdown occurs but we can guess. According to Bell, λ can be thought of as all relevant laws of physics and all relevant initial conditions with the exception of values a and b. If only we allow A to depend on b: A=A(a,b,λ), then everything clicks into place and we have a working model (QM). That suggests it's not a problem with general setup but specifically with factorizing a and b out of λ, that is local realism assumption.

 

[harrylin]

[..] 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.[..]

Yes, and that's why Bell didn't use electrons for his argument. But he dropped Bertlmann's socks and instead he gave an illustration with Booles's Lille and Lyon. However, some of us shortly discussed a Lille-Lyon counter example in a thread that I started a long time ago, but none of us appreciated it much; perhaps Lille-Lyon doesn't catch the detector setting aspect well. It would be more interesting to try an adapted variant of Bertlmann's socks.
So, here's the intro of an example that I had in mind. It's a shot in the dark as I don't know the outcome concerning Bell vs. Jaynes (likely it will support Bell which would "weaken" Jaynes, but I can imagine that it could by chance "invalidate" Bell):

A group of QM students get classes from Prof. Bertlmann. It's an intensive course with Morning class, Afternoon class and Evening class. The students wonder if Bell's story could actually be true and Bertlmann really wears different socks. However Bertlmann happens to wear long trousers and when he goes to sit behind his desk, his socks are out of sight.

Never mind, one student knows a little electronics and makes two devices with LED's to illuminate the socks and light detectors to determine if the sock is light or dark. He hides them on both sides under the desk, aiming at where Bertlmann's socks should appear. With a wireless control he can secretly do a measurement with the press of a button and the result is then indicated by two LED's that are visible for the students, but out of sight for Bertlmann. The next morning he fiddles a bit with the settings and then they wait for Bertlmann [to be continued]

Would such a scenario correspond to post #32 of IsometricPion? I intend to let Morning, Afternoon and Evening be selected by the students, as a and b.

 

[PeterDonis]

Yes, and that's why Bell didn't use electrons for his argument.

I'm not sure what you mean by this. He certainly used electrons to derive the *quantum* probabilities, which are what I was talking about in the passage you quoted. Bertlmann's socks, and the heart attacks in Lille and Lyon, are stipulated to be classical objects; there is nothing in their behavior corresponding to the behavior of electrons that undergo successive spin measurements in different directions. That's the point.

 

[harrylin]

I'm not sure what you mean by this. He certainly used electrons to derive the *quantum* probabilities, which are what I was talking about in the passage you quoted. Bertlmann's socks, and the heart attacks in Lille and Lyon, are stipulated to be classical objects; there is nothing in their behavior corresponding to the behavior of electrons that undergo successive spin measurements in different directions. That's the point.

That's my (and I think also your) point: it didn't make much sense for Bell to use electrons as example to defend the validity of his separation of terms; he had to use an example that we can understand - and he chose to use Lille-Lyon for that.
Now, his socks example is too simple, and none of us appreciated his Lille-Lyon example much when De Raedt presented a variant of it as counter example. And I think that we all agree that Jayne's example is also insufficient. Thus, it may be more instructive to improve Bertlmann's socks example into something like Lille-Lyon. My example keeps the physical separation and adds complexity as well as a certain "weirdness" of observed correlations at varying detector parameters. Only thing I was extremely busy until today so I have not yet worked out the probabilities. It's just a shot in the dark.

 

[harrylin]

[..] 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).

I am now starting to study the outcomes of my little thought experiment in spreadsheet and it immediately gets interesting as I can now put much more meaning to the symbols and how they are used. Do you agree that the bold term should also apply on my example?

But then I encounter trouble! For what Bell next does (in his socks paper; it's instant in his first paper), is to multiply that term with dλ ρ(λ) [eq.11+12]. It looks to me that for every increment dλ there is a single λ, which appears to be a fixed set of variables because of Bell's "probability distribution" ρ(λ). That sounds pretty much fixed to me for the total experiment of many runs. If not, can someone please explain what the "probability distribution" ρ(λ) exactly means?

 

[IsometricPion]

Do you agree that the bold term should also apply on my example?

Yes, assuming a local realistic theory for predicting the color of Bertlmann's socks (which I would say is the only intuitive kind in such ordinary situations).

It looks to me that for every increment dλ there is a single λ, which appears to be a fixed set of variables because of Bell's "probability distribution" ρ(λ). That sounds pretty much fixed to me for the total experiment of many runs. If not, can someone please explain what the "probability distribution" ρ(λ) exactly means?

I think you're correct. In Bell's eq. 11 it is assumed that one knows the values of the variables that make up λ. His eq. 12 incorporates the fact that in actual experiments λ is not known so he multiplies the joint outcome probability by ρ(λ), the probability density for λ, and integrates with respect to λ to removing it from the equations. There is nothing that intrinsically prevents ρ(λ) from varying between runs in "real life". However, if one is numerically simulating the ensemble distribution of results of an experiment (which is what I assume you are doing), it should not be allowed to vary between runs.

 

[harrylin]

Yes, assuming a local realistic theory for predicting the color of Bertlmann's socks (which I would say is the only intuitive kind in such ordinary situations).

Yes, the measurements do not affect each other ("no action at a distance"). BTW, he is in reality wearing ordinary socks.

I think you're correct. In Bell's eq. 11 it is assumed that one knows the values of the variables that make up λ. His eq. 12 incorporates the fact that in actual experiments λ is not known so he multiplies the joint outcome probability by ρ(λ), the probability density for λ, and integrates with respect to λ to removing it from the equations. There is nothing that intrinsically prevents ρ(λ) from varying between runs in "real life". However, if one is numerically simulating the ensemble distribution of results of an experiment (which is what I assume you are doing), it should not be allowed to vary between runs.

I'm about to start doing that (I need to add more columns and add a function etc.). So, I'm puzzled by your last remark; why should "real life" not be allowed in a Bell type calculation of reality?

PS. I guess that he wants to calculate the outcome for any (a, b) combination for all possible "real life" λ (thus all possible x), taking in account their frequency of occurrence. It seems plausible that λ (thus (x1,x2)) is different from one set of pair measurements to the next, and now it looks to me that Bell does account for that possibility (but can one treat anything as just a number?). And I suppose that according to Bell the total function of λ (thus X) cannot vary from one total experiment to the next, as the results are reproducible. Is that what you mean?