Functional uniqueness and stability of Gaussian priors in optimal L1 estimation

2025-11-20

AI Summary

This paper investigates the functional uniqueness and stability of Gaussian priors in L1-optimal estimation—i.e., under absolute error loss, where the optimal estimator is the conditional median. Specifically, it addresses: *Under Gaussian noise, which priors induce approximately linear estimators?* Methodologically, the authors develop an analytical framework based on Hermite polynomial expansions, integrated with adjoint operator techniques and the Lévy metric. Their main contributions are: (i) a rigorous characterization showing that the Gaussian prior is the unique stable solution under L1 loss; (ii) quantitative stability bounds—under L2 loss, they derive an explicit convergence rate for priors approaching Gaussianity; and (iii) under L1 loss, they prove that if the posterior median is approximately linear, the prior must be strongly constrained to a Gaussian distribution, ensuring both uniqueness and stability. These results advance the theoretical foundations of Bayesian robustness and loss-function–prior compatibility.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Bayesian Learning
• Intelligent Robots: State Estimation

Application Category

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

Abstract

This paper studies the functional uniqueness and stability of Gaussian priors in optimal $L^1$ estimation. While it is well known that the Gaussian prior uniquely induces linear conditional means under Gaussian noise, the analogous question for the conditional median (i.e., the optimal estimator under absolute-error loss) has only recently been settled. Building on the prior work establishing this uniqueness, we develop a quantitative stability theory that characterizes how approximate linearity of the optimal estimator constrains the prior distribution. For $L^2$ loss, we derive explicit rates showing that near-linearity of the conditional mean implies proximity of the prior to the Gaussian in the Lévy metric. For $L^1$ loss, we introduce a Hermite expansion framework and analyze the adjoint of the linearity-defining operator to show that the Gaussian remains the unique stable solution. Together, these results provide a more complete functional-analytic understanding of linearity and stability in Bayesian estimation under Gaussian noise.

Problem

• Analyzes uniqueness of Gaussian priors in L1 estimation
• Develops stability theory for near-linear optimal estimators
• Compares L1 and L2 loss in Bayesian estimation stability

Innovation

• Quantitative stability theory for approximate linearity constraints
• Lévy metric proximity analysis for L2 loss estimation
• Hermite expansion framework for L1 loss stability analysis