= Beta-binomial exchangeable coupling
{c}
{title2=$\operatorname{Corr}(P,Q)=n/(n+\alpha+\beta)$}
Draw $P\sim\operatorname{Beta}(\alpha,\beta)$, $X\mid P\sim\operatorname{Binomial}(n,P)$ and $Q\mid X\sim\operatorname{Beta}(\alpha+X,\beta+n-X)$, with $\alpha,\beta>0$ and integer $n\ge0$. 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.
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