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.

翻译：我们研究了在$L^1$损失下用于泊松参数估计的先验分布。具体而言，我们构建了一族新的先验分布，其最优贝叶斯估计量（条件中位数）可以是满足特定正则性条件的任意预设递增函数。在线性估计量的情形下，该族先验不同于通常的共轭先验（即伽马分布）。我们的先验分布是通过匹配特定矩条件的极限过程构建的。这些结果首次明确描述了满足线性条件中位数性质的、超越共轭先验的分布族；更广泛地说，对于泊松噪声模型，它们能够给出任意预设的条件中位数。