Spring 2010Pstat160bHandoutBayesian Statistics example: Binomial/Beta conjugate priorSet-up: Binomial trials (say coin tosses) where the probability of successpis unknown. This is a typicalexample in statistics where one would want to estimatep, but here we will think ofpas random (so wedenote it byP) and quantify the uncertainty (prior knowledge, or lack of) by assuming thatPfollows aBeta distribution,P∼ B(α, β), α, β >0 and givenP=p,Xfollows a BinomialB(n, p) distribution.fP(p)=1B(α,β)pα-1(1-p)β-10≤p≤10otherwisefX|P=p(k)=nkpk(1-p)n-kk= 0,· · ·, nThe reason for choosing the Beta distribution is its following advantages: first, its range is [0,1], so it can beinterpreted as a probability (forp, 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), todifferent beliefs aboutp. To see, this recall the following facts: The mode (most likely value, or location
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