Some Examples on Conditional Expectation of Markov Kernels
A well-known property of conditional expectation states that if X,Y,Z are random variables such that Z is independent of (X,Y), then E(X∣Y,Z)=E(X∣Y). A Markov kernel can be considered as an extension of the concept of a random variable (and of a sub-sigma-algebra), and the authors have studied in previous works extensions to Markov kernels of certain probabilistic or statistical concepts (including independence and conditional expectation) and their properties. The main result of this communication considers the extension of the aforementioned result. Thinking about applications, a version in terms of densities is also presented. Finally, some examples from the field of clinical diagnosis are included with the intention of illustrating and delimiting this theorem.
Keywords: Conditional expectation Markov kernels.