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 all implementable distributions of the posterior q-quantile are implemented by a single experiment, the q-quantile matching experiment, which pools pairs of states across the q-quantile of the prior in a positively assortative manner, with weight q on the lower state in each pair. 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 q-quantiles induced by the q-quantile matching experiment. The q-quantile matching experiment is the only experiment that simultaneously implements all implementable distributions of the posterior q-quantile.