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Distributions of Posterior Quantiles via Matching

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  • Anton Kolotilin
  • Alexander Wolitzky

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 $q$-quantiles for any $q \in (0,1)$. 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.

Suggested Citation

  • Anton Kolotilin & Alexander Wolitzky, 2024. "Distributions of Posterior Quantiles via Matching," Papers 2402.17142, arXiv.org.
    • Handle: RePEc:arx:papers:2402.17142
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    File URL: http://arxiv.org/pdf/2402.17142
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