Estimating the bias of a noisy coin

TL;DR

This paper investigates the fundamental limits of estimating a noisy coin's bias, revealing intrinsic risk constraints and proposing effective estimators, with implications across various scientific fields.

Contribution

It generalizes hedged maximum-likelihood estimators for noisy coins, analyzes their risk, and introduces a lower bound to evaluate estimator performance, highlighting the intrinsic difficulty of the problem.

Findings

HML estimators achieve O(N^{-1/2}) worst-case risk on noisy coins.

Minimax estimators can reduce risk but introduce extreme bias.

HML estimators are near-optimal according to the pointwise lower bound.

Abstract

Optimal estimation of a coin's bias using noisy data is surprisingly different from the same problem with noiseless data. We study this problem using entropy risk to quantify estimators' accuracy. We generalize the "add Beta" estimators that work well for noiseless coins, and we find that these hedged maximum-likelihood (HML) estimators achieve a worst-case risk of O(N^{-1/2}) on noisy coins, in contrast to O(1/N) in the noiseless case. We demonstrate that this increased risk is unavoidable and intrinsic to noisy coins, by constructing minimax estimators (numerically). However, minimax estimators introduce extreme bias in return for slight improvements in the worst-case risk. So we introduce a pointwise lower bound on the minimum achievable risk as an alternative to the minimax criterion, and use this bound to show that HML estimators are pretty good. We conclude with a survey of scientific applications of the noisy coin model in social science, physical science, and quantum information science.