Linearity-Inducing Priors for Poisson Parameter Estimation Under $L^{1}$ Loss

TL;DR

This paper introduces a new family of prior distributions for Poisson parameter estimation under $L^1$ loss, enabling flexible Bayesian estimators with prescribed median properties beyond traditional conjugate priors.

Contribution

The authors construct a novel family of priors that allow the Bayesian estimator's median to match any increasing function satisfying specific conditions, extending beyond gamma conjugate priors.

Findings

New prior family matches prescribed median functions

Explicit construction via moment-matching limits

First description of non-conjugate priors with affine median property

Abstract

We study prior distributions for Poisson parameter estimation under $L^1$ loss. Specifically, we construct a new family of prior distributions whose optimal Bayesian estimators (the conditional medians) can be any prescribed increasing function that satisfies certain regularity conditions. In the case of affine estimators, this family is distinct from the usual conjugate priors, which are gamma distributions. Our prior distributions are constructed through a limiting process that matches certain moment conditions. These results provide the first explicit description of a family of distributions, beyond the conjugate priors, that satisfy the affine conditional median property; and more broadly for the Poisson noise model they can give any arbitrarily prescribed conditional median.