Deriving useful bounds on the mixing of high dimensional Markov chain Monte Carlo (MCMC) algorithms still seems impossibly difficult. In the analysis of their performance we therefore restrict ourselves to answering simpler proxy questions, such as how to optimally
choose design parameters in an MCMC method. We will review some classical results on how to optimally scale the proposal variance in the class of Metropolis-Hastings algorithms. Then we will approach the same problem with a novel axiomatic framework enabling a more
detailed analysis. We will look at counterexamples exhibiting anomalous optimal scaling rates and identifying exact smoothness assumptions required for classical scaling. We will discuss mathematical background, intuitive understanding and further applications of the
The talk is based on joint work with Wilfrid Kendall.