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So, heuristically, observation independence is the condition that separated measurements are statistically independent.

I have lots of questions about this! (Actually, I have lots of questions about the theoretical foundation of using statistics at all in physics, but anyways...)

The first question is for a precise statement of this condition.

Presumably, you could say something like, for separated measurements A and B, we have P(A = a) = P(A = a | B = b), but that leads into my next question:

Classically, what is the theoretical justification for observation independence? Is there a proof that classical physics is observation independent?

The naive statement I gave above would seem to fail in certain simple cases. For example, if we packaged a red ball and a green ball, randomly mixed them up, and shipped them off to different cities, we would have that P(A = green) = 0.5, but P(A = green | B = red) = 1.

The usual answer is that the given scenario is not the complete description of the physical situation, in which the universe "knows" which ball went which way... but why would we ever be speaking about probabilities in the first place?

Deterministically, I suppose we have a trivial proof of observation independence. Does it make sense to ask about nondeterministic classical physics?

The impression I get is that, classically, we use statistics to describe our "lack of knowledge", but I cannot see any reason to think that lack of knowledge should obey the observation independence postulate.

Moving onto the topic I really wanted to discuss! (Though I would still like to have good answers to the previous questions)

When discussing hidden variable theories, the observation independence postulate is often invoked (indirectly, through the Bell-locality criterion). Isn't the point of a HVT that the probabilities describe lack of knowledge? If so, then why would we ever think that the observation independence postulate should be applicable?

Consider this thought experiment:

Suppose we have any system governed by some probability distribution.

We make a measurement of the system.

We write down two copies of the outcome of the measurement, and ship them off to different cities.

Now, consider the two observers in the different cities who open the envelopes and read the results...

If we apply the observation independence postulate, it is easy to see that the observations must have a definite outcome. (because they must be the same, and this is the only way they can be statistically independent)

So, it seems that observation independence forces the notion of a statistical collapse somewhere along the way, I suppose either when the original measurement occured, or when the duplication of information occurred.

My gut tells me that if we carry this thought experiment out to its logical conclusion, we must eventually conclude that all interactions are deterministic, and even if the original system somehow happened to be governed by an actual probability law, that it is quickly forced to choose one way or the other by the environment.

So, I shall be so bold as to claim that observation independence, as a fundamental postulate, is effectively equivalent to determinism!

Furthermore, it seems fruitless to attempt to develop any sort of statistical theory in which the observation independence postulate is satisfied!

(Ack, I didn't think I was headed in this direction! I wonder what others think of it)

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# Observation Independence - what and why?

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