# Markov Process – Can Markov Matrices Be Ranked in Stochastic Order for Better Analysis?

markov-processstochastic-ordering

I am familiar with the concept of stochastic ordering for two random variables and how we can say if a markov matrix is stochastically monotone. What im interested in is if there is a concept for ranking two separate markov matrices.

To illustrate suppose we have two stochastically monotone Markov matricies $$A$$ and $$B$$ which preserve the ordering of $$x\succsim y$$. Under what circumstances can we say (if any) that matrix $$A$$ is preferred to matrix $$B$$ in stochastic order?

Note: The definitions im using are from this slide deck: http://polaris.imag.fr/jean-marc.vincent/index.html/Slides/asmta09.pdf

Suppose $$A$$ and $$B$$ are two Markov kernels. One example where a "matrix $$A$$ is preferred to matrix $$B$$" is when, under certain conditions, $$A(x,C) \ge B(x,C)$$ for any $$x,C$$ such that $$x \not\in C$$. This is a Peskun ordering, written as $$A \succeq_P B$$. I think of this intuitively as "$$A$$ just moves around more than $$B$$."

If these transition kernels correspond to MCMC samplers, and if you're able to show that $$A \succeq_P B$$, then $$A$$ is a superior algorithm in the sense that it gives you estimators with smaller asymptotic variance, among other things.