Distributions of Posterior Quantiles via Matching
Abstract: We offer a simple analysis of the problem of choosing a statistical experiment to optimize the induced distribution of posterior medians, or more generally -quantiles for any . We show that all implementable distributions of the posterior -quantile are implemented by a single experiment, the -quantile matching experiment, which pools pairs of states across the -quantile of the prior in a positively assortative manner, with weight on the lower state in each pair. A dense subset of implementable distributions of posterior -quantiles can be uniquely implemented by perturbing the -quantile matching experiment. A linear functional is optimized over distributions of posterior -quantiles by taking the optimal selection from each set of -quantiles induced by the -quantile matching experiment. The -quantile matching experiment is the only experiment that simultaneously implements all implementable distributions of the posterior -quantile.
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