Bayesian updating for a fill rate, the Beta-Binomial

Question

You model whether a resting order fills as a coin with unknown probability $p$ , so successes are $\text{Binomial}$ . You place a Beta prior $p \sim \text{Beta}(\alpha, \beta)$ with $\alpha = 2$ , $\beta = 2$ , then observe $8$ fills out of $10$ orders.

Derive the posterior, give its mean, and interpret the prior's weight.

Accepted Answer

Likelihood. For s successes in n trials, p(	ext{data}\mid p) \ \propto\ p^{s}(1-p)^{\,n-s} = p^{8}(1-p)^{2}. Prior. The Beta(\alpha,\beta) density is p(p) \propto p^{\alpha-1}(1-p)^{\beta-1}. Posterior. Multiply and collect powers: p(p\mid	ext{data}) \ \propto\ p^{\alpha-1}(1-p)^{\beta-1}\cdot p^{s}(1-p)^{n-s} = p^{\alpha+s-1}(1-p)^{\beta+n-s-1}. This is a Beta kernel, so p \mid 	ext{data} \ \sim\ 	ext{Beta}(\alpha + s,\ \beta + n - s) = 	ext{Beta}(2+8,\ 2+2) = 	ext{Beta}(10, 4). Beta is conjugate to the Binomial: the posterior stays Beta. You simply add successes to \alpha and failures…

Bayesian updating for a fill rate, the Beta-Binomial

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