Pseudo-Marginal Approximation to the Free Energy in a Micro-Macro Markov Chain Monte Carlo Method

We introduce a generalised micro-macro Markov chain Monte Carlo (mM-MCMC) method with pseudo-marginal approximation to the free energy, that is able to accelerate sampling of the microscopic Gibbs distributions when there is a time-scale separation between the macroscopic dynamics of a reaction coordinate and the remaining microscopic degrees of freedom. The mM-MCMC method attains this efficiency by iterating four steps: i) Propose a new value of the reaction coordinate; ii) Accept or reject the macroscopic sample; iii) Run a biased simulation that creates a microscopic molecular instance that lies close to the newly sampled macroscopic reaction coordinate value; iv) Microscopic accept/reject step for the new microscopic sample. In the present paper, we eliminate the main computational bottleneck of earlier versions of this method: the necessity to have an accurate approximation of the free energy. We show that introduction of a pseudo-marginal approximation significantly reduces the computational cost of the microscopic accept/reject step, while still providing unbiased samples. We illustrate the method's behaviour on several molecular systems with low-dimensional reaction coordinates.