 Posterior predictive repeated-success identity

If future Bernoulli random variables are conditionally independent given a shared parameter $\Theta$, with success probability $\pi (\Theta )$, their all-success conditional probability is $\pi (\Theta {)}^k$. Taking a posterior expectation proves the identity. For $k\ge 1$, Jensen's inequality and $0\le \pi \le 1$ give $(\mathbb{E}[\pi \mid D]{)}^k\le \mathbb{E}[{\pi }^k\mid D]\le \mathbb{E}[\pi \mid D]$. For two distinct future trials the posterior covariance is $\mathbb{E}[{\pi }^2\mid D]-(\mathbb{E}[\pi \mid D]{)}^2=\mathrm{Var}(\pi \mid D)$. Thus shared parameter uncertainty can induce positive dependence even when trials are conditionally independent.