By von der Linden W., Dose V., von Toussaint U.
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Extra info for Bayesian Probability Theory: Applications in the Physical Sciences
L(1) . Similarly, in a type-2 box, there are tickets labelled by the integers 1, 2, . . , L(2) . 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.
First we need to specify all relevant propositions. • • • • • Uα : Urn α has been selected. 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.
P (N|nL , L, I) ≈ 1(nmax ≤ N < Nmax ) (L − 1) nL−1 max N The probability for N given the sample information is largest for N = nmax and decays for larger values as N −L . Next we compute the mean N , based on the same approximations: 1 N = Z Nmax NN −L N=nmax ≈ nmax 1 ≈ Z Nmax N −L+1 dN N=nmax 1 1+ . L−2 This result is only sensible for sample sizes L > 2. For samples of size 1 and 2 the posterior probability depends on the chosen cutoff value Nmax and neither the norm nor the first moment exists.
Bayesian Probability Theory: Applications in the Physical Sciences by von der Linden W., Dose V., von Toussaint U.