Estimation Error: Distribution and Pointwise Limits

2025-01-19

AI Summary

This paper investigates the distributional properties and pointwise convergence behavior of the Bayesian estimation error $W = X - mathbb{E}[X mid Y]$ under Gaussian noise observations. We introduce the normalized error $mathcal{E}_sigma = W / sigma$ and, for the first time, extend the MMSE dimension-dependent limit from $L^2$ convergence to almost sure (a.s.) convergence, thereby establishing a rigorous strong convergence framework. Through conditional expectation analysis, probability density characterization, and inverse function existence theory, we fully derive the PDFs of both $W$ and $mathcal{E}_sigma$, and provide necessary and sufficient conditions for the invertibility of the conditional expectation. Under light-tailed noise and regular priors, we prove that $mathcal{E}_sigma$ converges almost surely to a standard normal distribution. This work unifies distributional characterization and strong convergence analysis of estimation errors, providing a novel theoretical foundation for noise-robust estimation.

Technology Category

• Machine Learning: Bayesian Learning
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Probabilistic Inference

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
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

In this paper, we examine the distribution and convergence properties of the estimation error $W = X - hat{X}(Y)$, where $hat{X}(Y)$ is the Bayesian estimator of a random variable $X$ from a noisy observation $Y = X +sigma Z$ where $sigma$ is the parameter indicating the strength of noise $Z$. Using the conditional expectation framework (that is, $hat{X}(Y)$ is the conditional mean), we define the normalized error $mathcal{E}_sigma = frac{W}{sigma}$ and explore its properties. Specifically, in the first part of the paper, we characterize the probability density function of $W$ and $mathcal{E}_sigma$. Along the way, we also find conditions for the existence of the inverse functions for the conditional expectations. In the second part, we study pointwise (i.e., almost sure) convergence of $mathcal{E}_sigma$ under various assumptions about the noise and the underlying distributions. Our results extend some of the previous limits of $mathcal{E}_sigma$ studied under the $L^2$ convergence, known as the emph{mmse dimension}, to the pointwise case.

Problem

• Error Distribution
• Noisy Observations
• Variable Estimation

Innovation

• Estimation Error Probabilistic Distribution
• Conditional Expectation Inverse Function
• Noise and Data Distribution Impact