Bayesian updating with a Beta prior

Question

A coin has unknown head probability p. Your prior is p \sim 	ext{Beta}(a, b). You flip n times and observe k heads. Derive the posterior distribution of p. With a uniform prior and 7 heads in 10 flips, what is the posterior mean, and how does it compare to the MLE?

Accepted Answer

The update. Ignore constants and track only the p-dependence: \pi(p \mid 	ext{data}) \propto \underbrace{p^{a-1}(1-p)^{b-1}}{	ext{prior}} 	imes \underbrace{p^{k}(1-p)^{n-k}}{	ext{likelihood}} = p^{a+k-1}(1-p)^{b+n-k-1}. This is the kernel of a Beta density, so p \mid 	ext{data} \sim 	ext{Beta}(a + k,\; b + n - k). The Beta family is conjugate to the binomial likelihood: the posterior stays in the family, and updating is just "add successes to a, failures to b." The prior acts exactly like a + b pseudo-observations seen before the experiment. The numbers. Uniform prior = 	ext{Beta}(1,1)…

Bayesian updating with a Beta prior

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