Matti Vihola (University of Jyväskylä)

On the stability and convergence of adaptive MCMC 

Adaptive MCMC algorithms tune the proposal distribution of the Metropolis-Hastings Markov kernel continuously using the simulated history of the chain. There are many applications where AMCMC algorithms are empirically shown to improve over the traditional MCMC methods. Due to the non-Markovian nature of the adaptation, however, the analysis of these algorithms requires more care than the traditional methods.

Most results on adaptive MCMC in the literature are based on assumptions that require the adaptation process to be `stable' (in a certain sense).

Such stability can rarely be established unless the process is modified by introducing specific stabilisation structures. The most straightforward strategy is to constrain the adaptation within certain pre-defined limits.

Such limits may sometimes be difficult to choose in practice, and the algorithms are generally sensitive to these parameters. In the worst case, poor choices can render the algorithms useless.

This talk focuses on the recent stability and ergodicity results obtained for adaptive MCMC algorithms without such constraints. The key idea behind the results is that the ergodic averages can converge even if the Markov kernels gradually `lose' their ergodic properties. The new approach enables to show some sufficient conditions for stability and ergodicity for random walk Metropolis algorithms: the seminal Adaptive Metropolis algorithm and an algorithm adjusting the scale of the proposal distribution based on the observed acceptance probability.  The results assume only verifiable conditions on the target distribution and the functional.