Conditional Expectation of a Markov Kernel Given Another with Some Applications in Statistical Inference and Disease Diagnosis

TL;DR

This paper introduces the concept of conditional expectation for Markov kernels, extending classical statistical theorems and providing applications in clinical diagnosis and statistical inference.

Contribution

It develops the theory of conditional expectation for Markov kernels and applies it to extend key statistical theorems like Rao-Blackwell and Lehmann-Scheffé.

Findings

Optimality of predictive values in clinical diagnosis

Extension of Rao-Blackwell and Lehmann-Scheffé theorems to Markov kernels

New results on completeness of sufficient statistics

Abstract

Markov kernels play a decisive role in probability and mathematical statistics theories, and are an extension of the concepts of sigma-field and statistic. Concepts such as independence, sufficiency, completeness, ancillarity or conditional distribution have been extended previously to Markov kernels. In this paper, the concept of conditional expectation of a Markov kernel given another is introduced, setting its first properties. An application to clinical diagnosis is provided, obtaining {an} optimality property of the predictive values of a diagnosis test. In a statistical framework, this new probabilistic tool is used to extend to Markov kernels the theorems of {Rao-Blackwell} and Lehmann-Scheff\'e. A result about the completeness of a sufficient statistic is obtained in passing by properly enlarging the family of probabilities. As a final statistical scholium, a generalization of a result about the completeness of the family of nonrandomized estimators is given.