Beta-binomial conjugacy

A beta prior $\mathrm{Beta}(\alpha ,\beta )$ combined with $s$ Bernoulli successes and $f$ failures gives posterior $\mathrm{Beta}(\alpha +s,\beta +f)$.

Draw $P\sim \mathrm{Beta}(\alpha ,\beta )$, $X\mid P\sim \mathrm{Binomial}(n,P)$ and $Q\mid X\sim \mathrm{Beta}(\alpha +X,\beta +n-X)$, with $\alpha ,\beta >0$ and integer $n\ge 0$. The joint density of $(P,X,Q)$ is symmetric in $P,Q$, making them exchangeable random variables with identical Beta distribution marginal distributions. The posterior beta normalization constant depends on $X$ and must be retained. The correlation coefficient is $n/(n+\alpha +\beta )$: this constructs tunable dependence without altering either marginal.