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84 CH A P T E R 5: Binomial Proportion via Mathematical Analysis 5.2.2 The Posterior Beta Now that we have determined a convenient prior for the Bernoulli likelihood function, let's figure out exactly what the posterior distribution is when we apply Bayes' rule (Equation 4.4, p. 57). Suppose we have a set of data compris- ing N flips with z heads. Substituting the Bernoulli likelihood (Equation 5.3) and the beta prior distribution (Equation 5.4) into Bayes' rule yields p(|z, N) = p(z, N| )p( )/p(z, N) = z (1 - ) (N-z) (a-1) (1 - ) (b-1) / [B(a, b)p(z, N)] = ((z+a)-1) (1 - ) ((N-z+b)-1) [B(a, b) p(z, N)] B(z + a, N - z + b) In that sequence of equations, you probably followed the collection of powers of and of (1 - ), but you may have balked at the transition, underbraced in the denominator, from B(a, b)p(z, N) to B(z + a, N - z + b). This transition was not made via some elaborate analysis of integrals. Instead, the transition was (5.7)