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160b_bayes

# 160b_bayes - Spring 2010 Pstat160b Handout Bayesian...

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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 ( α, β ) , α, β > 0 and given P = p , X follows a Binomial B ( n, p ) distribution. f P ( p ) = 1 B ( α,β ) p α - 1 (1 - p ) β - 1 0 p 1 0 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, α = β = 1), to different beliefs about p . To see, this recall the following facts: The mode (most likely value, or location
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