160b_bayes - Spring 2010 Pstat160b Handout Bayesian...

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Unformatted text preview: Spring 2010 Pstat160b Handout Bayesian Statistics example: Binomial/Beta conjugate prior Set-up: Binomial trials (say coin tosses) where the probability of success p is unknown. This is a typical example in statistics where one would want to estimate p , but here we will think of p as random (so we denote it by P ) and quantify the uncertainty (prior knowledge, or lack of) by assuming that P follows a Beta distribution, P B ( , ) , , > 0 and given P = p , X follows a Binomial B ( n,p ) distribution. f P ( p ) = 1 B ( , ) p - 1 (1- p ) - 1 p 1 otherwise f X | P = p ( k ) = n k p k (1- p ) n- k k = 0 , ,n The reason for choosing the Beta distribution is its following advantages: first, its range is [0 , 1], so it can be interpreted as a probability (for p , the probability of success in Bernoulli trials); by varying the parameters (see graphs in handout 1) we can model from no prior knowledge (uniform distribution, = = 1), to different beliefs about...
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This note was uploaded on 01/10/2011 for the course STAT PStat 160b taught by Professor Bonnet during the Spring '10 term at UCSB.

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