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

2025-05-27

AI Summary

This paper addresses the Bayesian median estimation problem under the Poisson model with an $L^1$ loss function, aiming to construct a novel family of prior distributions such that the posterior conditional median exactly realizes any pre-specified strictly increasing function—including affine functions. Departing from the conventional conjugate Gamma prior, we explicitly construct a non-conjugate prior family via a limiting moment-matching approach grounded in Bayesian decision theory. This ensures full design flexibility and precise control over the conditional median. The proposed method overcomes the restrictive functional forms imposed by conjugacy, yielding a computationally tractable and theoretically rigorous framework for robust parameter estimation under Poisson noise. Its essential innovations lie in achieving exact $L^1$ risk minimization and endowing the median estimator with complete functional plasticity—enabling arbitrary monotonic shaping of the posterior median while preserving statistical coherence and computational feasibility.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Security and Privacy: Large-scale security measurements

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.

Problem

• Construct priors for Poisson parameter estimation under L1 loss
• Develop non-conjugate priors enabling prescribed affine estimators
• Provide explicit distributions for arbitrary conditional medians in Poisson models

Innovation

• Constructs new prior distributions for Poisson estimation
• Enables arbitrary prescribed increasing conditional medians
• Uses limiting process matching moment conditions