Probability

Download Bayesian Probability Theory: Applications in the Physical by von der Linden W., Dose V., von Toussaint U. PDF

By von der Linden W., Dose V., von Toussaint U.

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2 [p. 100]): P (Eq |n, N, I) = 1(0 ≤ q ≤ 1) q n (1 − q)N−n . 12c) [p. 100], q = n+1 N +2 which is Laplace’s law of succession. Laplace then applied the result to the probability that the sun will rise the next day, if it has risen each day in the past N days. He assumes N = 182 623 days (5000 years) as the most ancient epoch of history. Then, since n = N he concludes that the probability the sun will not rise tomorrow is P = 1/1 826 214. Laplace concludes ‘... it is a bet of 1 826 214 to one that it will rise again tomorrow.

Both boxes have the same total number M of tickets, which (α) corresponds to the normalization. M has to be chosen such that n1 := M/L(α) are integers, as they specify how often label 1 occurs in boxes of type α. Now we consider the following task. A single ticket is selected at random from an unknown box and it carries the integer value 1. Based on this information, we have to infer which type of box it came from. To this end, we identify the types of box with models M (α) and compute the odds ratio o= P (n|M (1) , I) P (M (1) |I) .

Q = qα : The fraction q of green balls is qα . N: The sample has size N. ng : The sample contains ng green balls. I: Background information. 9) [p. 12]: pα := P (Uα |ng , N, I) = P (ng |Uα , N, I) P (Uα |N, I) . 23) The first term in the numerator, P (ng |Uα , N, I), is the forward probability that a sample of size N from urn Uα contains ng green balls. 21)). The second term in the numerator, P (Uα |N, I), ✐ ✐ ✐ ✐ ✐ ✐ “9781107035904ar” — 2014/1/6 — 20:35 — page 32 — #46 ✐ 32 ✐ Basic definitions for frequentist statistics and Bayesian inference is the prior probability that urn Uα had been selected.

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