Beyond Binomial and Negative Binomial: Adaptation in Bernoulli Parameter Estimation

TL;DR

This paper introduces a novel resource allocation framework for Bernoulli parameter estimation, demonstrating significant improvements over traditional methods in active imaging scenarios through optimized trial allocation and stopping rules.

Contribution

It develops a trellis-based framework for adaptive trial stopping rules and extends the approach to estimating functions of Bernoulli parameters, outperforming classical binomial methods.

Findings

Up to 4.36 dB mean-squared error improvement in estimating p

Up to 1.80 dB improvement in estimating log p

Asymptotic optimality of the simplest stopping rule

Abstract

Estimating the parameter of a Bernoulli process arises in many applications, including photon-efficient active imaging where each illumination period is regarded as a single Bernoulli trial. Motivated by acquisition efficiency when multiple Bernoulli processes are of interest, we formulate the allocation of trials under a constraint on the mean as an optimal resource allocation problem. An oracle-aided trial allocation demonstrates that there can be a significant advantage from varying the allocation for different processes and inspires a simple trial allocation gain quantity. Motivated by realizing this gain without an oracle, we present a trellis-based framework for representing and optimizing stopping rules. Considering the convenient case of Beta priors, three implementable stopping rules with similar performances are explored, and the simplest of these is shown to asymptotically achieve the oracle-aided trial allocation. These approaches are further extended to estimating functions of a Bernoulli parameter. In simulations inspired by realistic active imaging scenarios, we demonstrate significant mean-squared error improvements: up to 4.36 dB for the estimation of p and up to 1.80 dB for the estimation of log p.