= Posterior predictive repeated-success identity
{title2=$\mathbb P(Y_1=\cdots=Y_k=1\mid D)=\mathbb E[\pi(\Theta)^k\mid D]$}
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 $k\geq1$, <Jensen's inequality> and $0\leq\pi\leq1$ give $(\mathbb E[\pi\mid D])^k\leq\mathbb E[\pi^k\mid D]\leq\mathbb E[\pi\mid D]$. For two distinct future trials the posterior <covariance> is $\mathbb E[\pi^2\mid D]-(\mathbb E[\pi\mid D])^2=\operatorname{Var}(\pi\mid D)$. Thus shared parameter uncertainty can induce positive dependence even when trials are conditionally independent.
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