 Posterior predictive probability

A posterior predictive probability averages a future-event probability over the Bayesian posterior. For a Bernoulli distribution with unknown parameter $p$, it is $\mathbb{E}[p\mid \mathcal{D}]$. Under Beta-binomial conjugacy with posterior $\mathrm{Beta}(a,b)$, the predictive success probability is $a/(a+b)$.