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
This results in N P (H |N, I) = P (H |En , N, I) P (En |N, I) n=0 N = n=0 N n (PB PR )n (1 − PB )N−n = (PB PR + 1 − PB )N = (1 − PB (1 − PR ))N . ✐ ✐ ✐ ✐ ✐ ✐ “9781107035904ar” — 2014/1/6 — 20:35 — page 36 — #50 ✐ 36 ✐ Bayesian inference The result makes perfect sense: the probability that the guest is ‘absorbed’ by a bar is PB (1 − PR ), therefore the probability that he overcomes a bar is given by 1 − PB (1 − PR ) and this has to happen N times. The relation to many physical problems is immediately obvious, be it damping of waves, propagation of particles in matter, diffusion, etc.
Suppose we want to estimate the mean value of a die. Throwing the die N times yields a sequence (a sample) of face values xi . If we compute the arithmetic mean (sample mean), we get an estimate of the mean of the probability distribution x : x= 1 N N xi . i=1 Later we will derive the validity of this approach and that the deviation of the sample mean from the true mean is – often but not always – given by Standard error of a sample of size N with individual standard deviation σ σ SE = √ . 2 Multivariate discrete random variables The following example will be used to guide the extension of the preceding definitions to more than one discrete random variable.
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. We assume that in nature there are in total 2N boxes, N of type 1 and N of type 2.
Bayesian Probability Theory: Applications in the Physical Sciences by von der Linden W., Dose V., von Toussaint U.