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

## Best Answer

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.