On the Bayesness, minimaxity, and admissibility of point estimators of allelic frequencies

TL;DR

This paper applies decision theory to derive and analyze Bayes and minimax estimators for allelic frequencies, considering different loci types and loss functions, and explores their admissibility and statistical properties.

Contribution

It introduces a comprehensive framework for estimating allelic frequencies using decision theory, including derivation of Bayes and minimax estimators for biallelic and multiallelic loci with multiple loss functions.

Findings

Bayes estimators under SEL and KLL are identical.

Certain estimators are admissible and minimax, with smaller variance than MLE.

Estimators incorporate population variability, reflecting real evolutionary dynamics.

Abstract

In this paper, decision theory was used to derive Bayes and minimax decision rules to estimate allelic frequencies and to explore their admissibility. Decision rules with uniformly smallest risk usually do not exist and one approach to solve this problem is to use the Bayes principle and the minimax principle to find decision rules satisfying some general optimality criterion based on their risk functions. Two cases were considered, the simpler case of biallelic loci and the more complex case of multiallelic loci. For each locus, the sampling model was a multinomial distribution and the prior was a Beta (biallelic case) or a Dirichlet (multiallelic case) distribution. Three loss functions were considered: squared error loss (SEL), Kulback-Leibler loss (KLL) and quadratic error loss (QEL). Bayes estimators were derived under these three loss functions and were subsequently used to find minimax estimators using results from decision theory. The Bayes estimators obtained from SEL and KLL turned out to be the same. Under certain conditions, the Bayes estimator derived from QEL led to an admissible minimax estimator (which was also equal to the maximum likelihood estimator). The SEL also allowed finding admissible minimax estimators. Some estimators had uniformly smaller variance than the MLE and under suitable conditions the remaining estimators also satisfied this property. In addition to their statistical properties, the estimators derived here allow variation in allelic frequencies, which is closer to the reality of finite populations exposed to evolutionary forces.