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Functional uniqueness and stability of Gaussian priors in optimal L1 estimation

Leighton Barnes, Alex Dytso

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.

Functional uniqueness and stability of Gaussian priors in optimal L1 estimation

Abstract

This paper studies the functional uniqueness and stability of Gaussian priors in optimal 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 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 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.

Paper Structure

This paper contains 13 sections, 11 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Past Work
  3. Contributions
  4. Technical Preliminaries
  5. Monotonicity of Bayesian Estimators
  6. Optimality and Orthogonality Conditions
  7. Hermite Functions
  8. The Adjoint Operator $T_a^*$
  9. Main Results
  10. Stability in the $L^2$ Case
  11. Stability in the $L^1$ Case
  12. Conclusion
  13. Auxiliary Results

Key Result

Lemma 1

Let $X$ be a continuous random variable. Then the conditional median $y \mapsto \mathsf{med}(X|Y=y)$ is a non-decreasing function.

Theorems & Definitions (21)

  • Lemma 1
  • proof
  • Proposition 1
  • Proposition 2
  • proof
  • Remark 1
  • Proposition 3
  • Proposition 4
  • proof
  • Proposition 5
  • ...and 11 more