Distributions of posterior quantiles via matching

We offer a simple analysis of the problem of choosing a statistical experiment to optimize the induced distribution of posterior medians or, more generally, q-quantiles for any q ∈ (0,1). We show that a single experiment-the q-quantile matching experiment-implements all implementable distributions of posterior q-quantiles, with different distributions spanned by different selections from the sets of posterior q-quantiles. A dense subset of implementable distributions of posterior q-quantiles can be uniquely implemented by perturbing the q-quantile matching experiment. A linear functional is optimized over distributions of posterior q-quantiles by taking the optimal selection from each set of posterior q-quantiles. The q-quantile matching experiment is the only experiment that simultaneously implements all implementable distributions of posterior q-quantiles.