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

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

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P (n|N, I) The likelihood is the binomial distribution. Next we need to specify the prior probability P (Eq |N, I). Here the knowledge that N draws have been performed with replacement is not of any help. Based on our prior knowledge each value of q is equally likely, which is appropriately described by a uniform prior p(Eq |I) = 1(0 ≤ q ≤ 1). 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.

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. In a company, the height and weight of employees have been measured. 1. As a matter of fact, height and weight are actually continuous quantities, but we introduce a discretization, which is frequently very useful.

This is a crucial step in serious hypothesis testing, as we will discuss in Part IV [p. 255]. In the coin example it corresponds to assigning values to the prior PDF P (q|N, A, I). The present background information is encoded as P (q|A, I) = δ(q − 1/2) for A = H 1(0 ≤ q ≤ 1) for A = H . That means for us a coin is only fair if the probability q is precisely 1/2. Moreover, as an alternative, H , everything is conceivable. This might be too extreme in real-world applications, as it implies that if we collect all coins available on earth, and sort out the fair ones, we are left with a collection of coins with q uniformly spread over the interval [0, 1].

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