Bayesian predictive densities as an interpretation of a class of Skew--Student $t$ distributions with application to medical data

TL;DR

This paper introduces a Bayesian interpretation of skew--Student t distributions through hierarchical models, demonstrating improved predictive density estimators for medical data applications.

Contribution

It provides a novel Bayesian framework linking skew--Student t distributions to hierarchical normal models with unknown covariance, enhancing predictive density estimation.

Findings

Bayesian interpretation aligns with known skew--Student t distributions.

Proposed estimators outperform regular Bayesian estimators in risk.

Application to medical data shows practical effectiveness.

Abstract

This paper describes a new Bayesian interpretation of a class of skew--Student $t$ distributions. We consider a hierarchical normal model with unknown covariance matrix and show that by imposing different restrictions on the parameter space, corresponding Bayes predictive density estimators under Kullback-Leibler loss function embrace some well-known skew--Student $t$ distributions. We show that obtained estimators perform better in terms of frequentist risk function over regular Bayes predictive density estimators. We apply our proposed methods to estimate future densities of medical data: the leg-length discrepancy and effect of exercise on the age at which a child starts to walk.