Bayes estimator for multinomial parameters and Bhattacharyya distances

TL;DR

This paper derives Bayesian estimators for multinomial parameters using Bhattacharyya-based loss functions, introduces a quantum generalization, and applies the results to find minimax estimators for binomial parameters.

Contribution

It provides the first derivation of Bayes estimators under Bhattacharyya-based loss functions and explores a quantum generalization as an open problem.

Findings

Derived Bayes estimators for multinomial parameters under Bhattacharyya loss.

Formulated a quantum generalization of the estimator problem.

Applied the estimators to obtain minimax estimators for binomial parameters.

Abstract

We derive the Bayes estimator for the parameters of a multinomial distribution under two loss functions ($1-B$ and $1-B^2$) that are based on the Bhattacharyya coefficient $B(\vec{p},\vec{q}) = \sum{\sqrt{p_kq_k}}$. We formulate a non-commutative generalization relevant to quantum probability theory as an open problem. As an example application, we use our solution to find minimax estimators for a binomial parameter under Bhattacharyya loss ($1-B^2$).