Conditional expectation of a Markov kernel given another: Some applications

A. García Nogales

Markov kernels play a decisive role in probability and mathematical statistics theories.
A Markov kernel is also an extension of the concepts of sigma-field and statistic, and concepts such as independence, sufficiency, completeness, ancillarity or conditional distribution have been extended previously to Markov kernels.

In this paper, we introduce the concept of conditional expectation of a Markov kernel given another, setting its first properties, the relationship with conditional distribution, as well as a representation theorem in terms of conditional expectations between random variables. As a consequence of a discrete example, an application to clinical diagnosis is provided, obtaining a optimality property of the predictive values of a diagnosis test.

In a statistical framework, these new tools are used to extend to Markov kernels the theorems of Rao-Blackwel and Lehmann-Scheffé. Two non trivial examples of sufficient Markov kernel are provided.

Palabras clave: Markov kernel, conditional expectation, clinical diagnosis, unbiased estimation.

L08.5 Procesos Estocásticos II
5 de septiembre de 2016  15:40
Aula 21.06

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