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vanhees71 said:Perhaps it would help me to understand the Bayesian view, if you could explain how to test a probilistic theoretical statement empirically from this point of view.
Here's a simplified example. Suppose we have two competing theories about a coin: Theory A says that it is a fair coin, giving "heads" 1/2 of the time. Theory B says that it is a trick coin, weighted to give "heads" 2/3 of the time. To start off with, we don't have any reason for preferring one theory over the other, so we write:
[itex]P(A) = P(B) = \dfrac{1}{2}[/itex]
Now flip the coin 4 times, and suppose you get HHTT. Call this event E. We compute probabilities:
[itex]P(E|A) = 0.0625[/itex]
[itex]P(E|B) = 0.0494[/itex]
[itex]P(E) = P(E|A) P(A) + P(E|B) P(B) = 0.0560[/itex]
Now, the Bayesian rules say that we revise our likelihood of the two theories in light of this new information:
[itex]P'(A) = \dfrac{P(A) P(E|A)}{P(E)} = 0.558[/itex]
[itex]P'(B) = \dfrac{P(B) P(E|B)}{P(E)} = 0.441[/itex]
So based on this one experiment, the likelihood of theory A has risen, and the likelihood of B has fallen.