Optimal scaling of MCMC beyond Metropolis

The problem of optimally scaling the proposal distribution in a Markov chain Monte Carlo algorithm is critical to the quality of the generated samples. Much work has gone into obtaining such results for various Metropolis–Hastings (MH) algorithms. Recently, acceptance probabilities other than MH are being employed in problems with intractable target distributions. There are few resources available on tuning the Gaussian proposal distributions for this situation. We obtain optimal scaling results for a general class of acceptance functions, which includes Barker’s and lazy MH. In particular, optimal values for Barker’s algorithm are derived and found to be significantly different from that obtained for the MH algorithm. Our theoretical conclusions are supported by numerical simulations indicating that when the optimal proposal variance is unknown, tuning to the optimal acceptance probability remains an effective strategy.

Efficient Bernoulli factory Markov chain Monte Carlo for intractable posteriors

Accept-reject-based Markov chain Monte Carlo algorithms have traditionally utilized acceptance probabilities that can be explicitly written as a function of the ratio of the target density at the two contested points. This feature is rendered almost useless in Bayesian posteriors with unknown functional forms. We introduce a new family of Markov chain Monte Carlo acceptance probabilities that has the distinguishing feature of not being a function of the ratio of the target density at the two points. We present two stable Bernoulli factories that generate events within this class of acceptance probabilities. The efficiency of our methods relies on obtaining reasonable local upper or lower bounds on the target density, and we present two classes of problems where such bounds are viable: Bayesian inference for diffusions, and Markov chain Monte Carlo on constrained spaces. The resulting portkey Barker’s algorithms are exact and computationally more efficient that the current state of the art.