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

Leighton P. Barnes, Alex Dytso, H. Vincent Poor

TL;DR

This work addresses Poisson parameter estimation under $L^1$ loss by showing that any prescribed increasing function $f$ can be realized as the conditional median $ ext{med}(X|Y=y)$ via a carefully constructed prior, using a limiting moment-matching approach. In particular, affine functions $f(y)=ay+b$ with $0<a<1/e$ are achievable, yielding a non-conjugate prior that induces a linear median under Poisson noise. The core method reduces the problem to an equivalent moment condition and builds distributions through truncated constructions and total-variation convergence, offering an explicit alternative to the gamma conjugate prior in this setting. The results broaden the Bayesian toolkit for Poisson estimation and suggest extensions to other noise models and loss functions.

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.

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

TL;DR

This work addresses Poisson parameter estimation under loss by showing that any prescribed increasing function can be realized as the conditional median via a carefully constructed prior, using a limiting moment-matching approach. In particular, affine functions with are achievable, yielding a non-conjugate prior that induces a linear median under Poisson noise. The core method reduces the problem to an equivalent moment condition and builds distributions through truncated constructions and total-variation convergence, offering an explicit alternative to the gamma conjugate prior in this setting. The results broaden the Bayesian toolkit for Poisson estimation and suggest extensions to other noise models and loss functions.

Abstract

We study prior distributions for Poisson parameter estimation under 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.

Paper Structure

This paper contains 10 sections, 7 theorems, 47 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Paper Outline and Contribution
  3. Main Results
  4. Proof
  5. Equivalent Moment Condition
  6. Proof of the Truncated Version of Theorem \ref{['thm:moment_theorem']}
  7. Extension to the General Case
  8. Proof of Theorem \ref{['thm:main_theorem']}
  9. Examples
  10. Conclusion

Key Result

Theorem 1

Suppose that $f:\mathbb{N}_0 \to \mathbb{R}_{+}$ satisfies the following properties: Then, there exists a distribution $P_X$ supported on such that for all $y \in \mathbb{N}_0$

Figures (3)

  • Figure 1: Plot of the conditional median in expression \ref{['eq:Conditional_median_poisson']} minus the conditional mean in \ref{['eq:linearconditionalmean']} for $a=b=\frac{1}{2}$. The difference decays like $O(1/y)$ as shown in chen1986bounds.
  • Figure 2: Comparison of the gamma cdf (red), which induces a linear conditional mean, to the distribution in \ref{['eq:construcion_alg']}.
  • Figure 3: Conditional medians induced by the distribution in \ref{['eq:construcion_alg']} with $a=b=0.3$.

Theorems & Definitions (7)

  • Theorem 1
  • Corollary 1
  • Lemma 1
  • Theorem 2
  • Lemma 2
  • Theorem 3
  • Lemma 3