## Bell's derivation; socks and Jaynes

In regards to Jaynes’ view: Bell incorrectly factored a joint probability; it may be informative to analyze the data set presented by N. David Mermin in his article: “Is the moon there when nobody looks? Reality and the quantum theory.” The following represents the summary of the data.

A = Same Switch; A’ = Different Switch; B = Same Color; B’ = Different Color

P(A) = 14/45; P(B) = 24/45
P(B/A) =14/14
P(A’) = 31/45
P(B/A’) = 10/31

We can now calculate the probability of the lights flashing the same color. This should be done two ways for the purpose of resolving which argument is correct. Bell or Jaynes.

General Multiplication Rule (Dependent Events)

1. P( A and B) = P(A)*P(B/A) = (14/45)*(14/14) = .311
2. P(A’ and B) = P(A’)*P(B/A’) = (31/45)*(10/31) = .222

P(Same color) = .311 + .222 = .533

Specific Multiplication Rule (Independent Events)

3. P(A and B) = P(A)*P(B) = (14/45)*(24/45) = .166
4. P(A’ and B) = P(A’)*P(B) = (31/45)*(24/45) = .367

P(Same Color) = .166 + .367 = .533

Wow! Both methods give the same prediction of .533. This was unexpected and there may be an underlying reason for this. Mermin’s theoretical prediction for the lights flashing the same color is 1/3*1 + 2/3*1/4 = .500. The 45 runs closely match the theoretical. However, only the general multiplication rule aligns with the theoretical calculation term for term which tends to support Jaynes’ view. Assuming the above is correct with no mistakes, what do the above findings say about Bell’s derivation using the factored form of the joint probability and ultimately about Bell’s theorem?

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 Quote by rlduncan In regards to Jaynes’ view: Bell incorrectly factored a joint probability; it may be informative to analyze the data set presented by N. David Mermin in his article: “Is the moon there when nobody looks? Reality and the quantum theory.” The following represents the summary of the data. A = Same Switch; A’ = Different Switch; B = Same Color; B’ = Different Color P(A) = 14/45; P(B) = 24/45 P(B/A) =14/14 P(A’) = 31/45 P(B/A’) = 10/31 We can now calculate the probability of the lights flashing the same color. This should be done two ways for the purpose of resolving which argument is correct. Bell or Jaynes. General Multiplication Rule (Dependent Events) 1. P( A and B) = P(A)*P(B/A) = (14/45)*(14/14) = .311 2. P(A’ and B) = P(A’)*P(B/A’) = (31/45)*(10/31) = .222 P(Same color) = .311 + .222 = .533 Specific Multiplication Rule (Independent Events) 3. P(A and B) = P(A)*P(B) = (14/45)*(24/45) = .166 4. P(A’ and B) = P(A’)*P(B) = (31/45)*(24/45) = .367 P(Same Color) = .166 + .367 = .533 Wow! Both methods give the same prediction of .533. This was unexpected and there may be an underlying reason for this. Mermin’s theoretical prediction for the lights flashing the same color is 1/3*1 + 2/3*1/4 = .500. The 45 runs closely match the theoretical. However, only the general multiplication rule aligns with the theoretical calculation term for term which tends to support Jaynes’ view. Assuming the above is correct with no mistakes, what do the above findings say about Bell’s derivation using the factored form of the joint probability and ultimately about Bell’s theorem?
So let me see if I have this straight. If you apply the probability analysis (either dependent or independent in your example), you would predict .5333 (actually a minimum). The quantum prediction is .5 which agrees to actual experiments.

Well, I would say Bell's point works nicely. Focusing on his factorization is a mistake. Once you know of Bell, I think it is easier to simply require that counterfactual cases must have a probability >=0. Which is the requirement of realism, going back to EPR and the famous "elements of reality".

 Quote by IsometricPion If it isn't too much trouble, I would like to see the code that generated your results. Edit: If it is too long to post here, perhaps we could get in touch by e-mail.
I will post my code here if my shot in the dark completely missed - but I didn't yet automize the data treatment so I don't know yet (but I do see now that it's not clear-cut). For the moment it's simply a useful exercise for me, that helps me to better understand possible issues so that I find the right questions to ask.

Quote by DrChinese
 Originally Posted by harrylin It's somewhat combining Bell's sock illustration with his Lille-Lyon illustration, but in a way that in principle could be really tested in the living room.[..]
The Lille-Lyon demonstration is kind of a joke to me, because it exploits the fair sampling assumption. As I am fond to say, you could use the same logic to assert that the true speed of light is 1 meter per second rather than c. The missing ingredient is always an explanation of WHY the true value is one thing and the observed value is something else. As a scientist, I don't see how you are supposed to ignore your recorded results in favor of something which is pulled out of the air.
Sorry you lost me here; Bell presented that example to defend his separation of terms. What is your issue with it?

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 Quote by harrylin Sorry you lost me here; Bell presented that example to defend his separation of terms. What is your issue with it?
I thought you were using it to demonstrate that classical data can violate a Bell Inequality. If you weren't intending that, then my apologies. But if you were, then I will say it is not a suitable analogy. A suitable analogy would be one like particle spin or polarization.

 Quote by DrChinese I thought you were using it to demonstrate that classical data can violate a Bell Inequality. If you weren't intending that, then my apologies. But if you were, then I will say it is not a suitable analogy. A suitable analogy would be one like particle spin or polarization.
Bell was using it to make it plausible that classical data must obey his method of probability analysis. I mentioned why I find both Lille/Lyon and particle spin useless for illustrating such things as particle spin in post #55. For me Lille-lyon is too difficult to analyse and it doesn't include the detection aspects well. What do you find unsuited about Lille-Lyon?

 Quote by rlduncan In regards to Jaynes’ view: Bell incorrectly factored a joint probability; it may be informative to analyze the data set presented by N. David Mermin in his article: “Is the moon there when nobody looks? Reality and the quantum theory.” [..]
Now that you bring it up, I was going to bring up Mermin as a separate topic but perhaps the answer on my question is very simple: can anyone tell me how his equality of 0.5 follows from (or, as he presents it, is) Bell's inequality?

 Quote by DrChinese So let me see if I have this straight. If you apply the probability analysis (either dependent or independent in your example), you would predict .5333 (actually a minimum). The quantum prediction is .5 which agrees to actual experiments. Well, I would say Bell's point works nicely. Focusing on his factorization is a mistake. Once you know of Bell, I think it is easier to simply require that counterfactual cases must have a probability >=0. Which is the requirement of realism, going back to EPR and the famous "elements of reality".

The data shows that the events A and B are dependent not independent, an assumption made by Bell. The P(A)*P(B/A) ≠ P(A)*P(B). Can you exlain how Bell got it right using an invalid assumption?

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 Quote by harrylin Now that you bring it up, I was going to bring up Mermin as a separate topic but perhaps the answer on my question is very simple: can anyone tell me how his equality of 0.5 follows from (or, as he presents it, is) Bell's inequality?
Rather than examining Mermin, you might want to look at Nick Herbert's exposition here. It's written in a style like Mermin's, but the example used is even simpler. This example was the one Bell used in talks to popular audiences, as he said that it was simplest known Bell inequality.

 Quote by rlduncan [..] The data shows that the events A and B are dependent not independent, an assumption made by Bell. The P(A)*P(B/A) ≠ P(A)*P(B). Can you exlain how Bell got it right using an invalid assumption?
It appears that you don't have lambda in your analysis. That is however necessary to test his assumption (see the discussion on the first page of this thread).

 Quote by lugita15 Rather than examining Mermin, you might want to look at Nick Herbert's exposition here. It's written in a style like Mermin's, but the example used is even simpler. This example was the one Bell used in talks to popular audiences, as he said that it was simplest known Bell inequality.
Thanks, that may very well provide the answer on my Mermin question and it looks very interesting.

PS: I think that Herbert's proof deserves to be a separate topic - it looks really good and no need for a lambda!

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 Quote by harrylin Thanks, that may very well provide the answer on my Mermin question and it looks very interesting. PS: I think that Herbert's proof deserves to be a separate topic - it looks really good and no need for a lambda!
Yes, it would be nice to have a thread on Herbert's proof.

 Quote by harrylin [..] I think that Herbert's proof deserves to be a separate topic - it looks really good and no need for a lambda!
 Quote by lugita15 Yes, it would be nice to have a thread on Herbert's proof.
So, I started that topic here: