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

ISBN-10: 1107035902

ISBN-13: 9781107035904

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**Additional resources for Bayesian Probability Theory: Applications in the Physical Sciences**

**Example text**

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 .

We are interested in the probability that a particle is indeed present if we receive a detector response. Using Bayes’ theorem we can express this probability as P (T |D, I) = P (D|T , I) P (T |I) P (D|T , I) P (T |I) + P (D|T , I) P (T |I) . 99, P (D|T , I) = 10−4 . With these numbers we obtain P (T |D, I) = 10−4 . The result may seem disappointing, despite the apparently good performance of the detector a positive response of the detector provides nearly no indication for a successful detection of the rare particles.

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) . P (n|M (2) , I) P (M (2) |I) Here the Bayes factor is one as both boxes contain label 1. We assume that both box are equally likely, which corresponds to the prior experience that both types of box are equally often realized in nature.

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

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