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

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

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G. 90% of the probability mass, 90% of the probability mass is in the interval nmax N L−1 . 1 must hold. Therefore 1 I90% = [nmax , nmax 10 L−1 ]. 29]nmax . The result is quite satisfactory, since in such problems we are usually interested only in the order of magnitude. 3 Ockham’s razor Let us now evaluate two theoretical models, M1 and M2 , in the light of experimental data D. The corresponding odds ratio is given by o= P (M (1) |D, I) P (D|M (1) , I) P (M (1) |I) = . (2) P (M |D, I) P (D|M (2) , I) P (M (2) |I) Bayes factor prior odds If both models have no adjustable parameters this is the end of the story and we have the competing factors: prior odds (oP ) versus Bayes factor (oBF ).

By construction (I) the balls are drawn one after the other with replacement. g. c = {g, g, r, g}. 20) with ng = 3 in the sequence c. We have used the sum rule to specify the probability for red in a single trial P (r|N, q, I) = 1 − q. 20) is valid for arbitrary colour sequences. As a matter of fact, we are not interested in the probability for a specific colour sequence, but rather in the probability that there are ng green balls. The two probabilities can be linked through the marginalization rule P (ng |N, q1 , I) = P (ng |c, N, q1 , I)P (c|N, q, I).

X(N ) , where the ith random variable is enumerated by ni ∈ Mi . Given a function Y = f X (1) , . . , X (N ) , the mean value of that function is given by ✐ ✐ ✐ ✐ ✐ ✐ “9781107035904ar” — 2014/1/6 — 20:35 — page 20 — #34 ✐ 20 ✐ Basic definitions for frequentist statistics and Bayesian inference Mean value of a function of several random variables f X(1) , . . , X(N ) := ··· n1 ∈M1 nN ∈MN f Xn(1) , . . ,nN . 11) mass in kg . The aver(height in m)2 age body mass index of the employees is given by the mean of the function f (m, h) = hm2 .

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