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Bell's derivation; socks and Jaynesby harrylin
Tags: bell socks jaynes 
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#55
Mar612, 07:23 AM

P: 3,181

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. 


#56
Mar612, 08:52 AM

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#57
Mar612, 11:04 AM

P: 3,181

Now, his socks example is too simple, and none of us appreciated his LilleLyon 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 LilleLyon. 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. 


#58
Mar612, 04:41 PM

P: 3,181

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? 


#59
Mar612, 08:16 PM

P: 177




#60
Mar712, 01:27 AM

P: 3,181

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? 


#61
Mar712, 06:37 PM

P: 177

It is essentially the difference between a timeaverage and an ensemble average. There is nothing preventing one from asserting from the start that ρ(λ) is the same for all experiments and experimental runs, it is merely a (reasonable) restriction on the set of hidden variable theories under consideration (which is almost certain to be necessary in order to make the analysis tractible). 


#62
Mar812, 05:39 PM

P: 3,181




#63
Mar912, 12:25 AM

P: 177




#64
Mar912, 01:39 AM

P: 272

Well, ρ(λ) should not change from one run to another, otherwise you won't get repeatable results (I mean repeatable statistics for long runs of course, not repeatable single outcomes). If ρ(λ) does vary, it just means some random factor ζ has not been accounted for, it needs to be lumped into λ'={λ,ζ}, then ρ becomes joint distribution ρ(λ') = ρ(λ,ζ).



#65
Mar912, 04:20 AM

P: 3,181

@ Delta Kilo : Yes, that sounds reasonable, but I have a problem already with one full statistical experiment.
And certainly Bell didn't sufficiently defend properly that his integration is compatible with all possible types of λ. He simply writes in his socks paper: "We have to consider then some probability distribution ρ(λ)", but he doesn't prove the validity of that claim. So, it may be best that I now give my example together with a small selection of results (later today I hope), and then try to work it out, perhaps with the help of some of you. *PS: I'm now rereading Jaynes and it does look as if his eq.15 exactly points at the problem that I now encounter. 


#66
Mar912, 06:00 AM

P: 3,181

Anyway, I'm here still at the start of Bell's derivation and which corresponds to Jaynes point 1. 


#67
Mar912, 06:38 AM

P: 272

BTW what are x_{A} and x_{B} exactly? 


#68
Mar912, 07:25 AM

P: 3,181




#69
Mar912, 08:22 AM

P: 272




#70
Mar912, 01:39 PM

P: 3,181




#71
Mar912, 05:46 PM

P: 272

Bell does not say that explicitly but it follows from



#72
Mar1012, 05:36 AM

P: 3,181

PS. Oops, there was still something wrong with it... maybe later! 


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