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

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.