Sufficiency and Completeness for Markov Kernels: Some Non-Trivial Examples

P. Perez Fernandez, A. García Nogales

Markov kernels play a decisive role in probability theory and mathematical statistics,
and they extend the classical notions of sigma-fields and statistics. Concepts such as
independence, sufficiency, completeness, ancillarity, and conditional distribution have
previously been generalized to the framework of Markov kernels.
In this communication, we revisit from an earlier work the notion of the conditional
expectation of one Markov kernel given another, together with its fundamental
properties. Within a statistical context, we also recall from Heyer’s book the concept of
sufficiency for Markov kernels, as well as the extension—developed in a previous
paper—of the notion of completeness to Markov kernels.
In this communication we present several nontrivial examples of sufficient or complete
Markov kernels, in the sense that they are not associated with any statistic.

Palabras clave: sufficiency complete Markov kernel.

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