A Better Linear Unbiased Estimator for Averages over Discrete Structures

TL;DR

This paper introduces an improved linear unbiased estimator for averages over finite sets when the probability mass at each sample point is known, enhancing estimation accuracy in applications like statistical physics.

Contribution

It proposes a systematic method to improve the BLUE by incorporating probability mass information, which was not utilized in traditional estimators.

Findings

Enhanced estimator reduces variance compared to sample average.

Applicable to scenarios with unnormalized probability distributions.

Potential benefits in statistical physics and related fields.

Abstract

Given an i.i.d. sample drawn from some probability distribution on a finite set, the best (in the sense of least variance) linear unbiased estimator (BLUE) of the average of any quantity with respect to that distribution is the sample average of the quantity. Here we consider the situation in which, together with the sample, also the probability mass (possibly unnormalized) at each sample point is provided. We show that with that information BLUE can be systematically improved. The proposed procedure is expected to have applications in statistical physics, where it is common to have a closed-form specification of the relevant (unnormalized) probability distribution.