Pseudo-marginal approximation to the free energy in a micro-macro Markov chain Monte Carlo method

We introduce a generalized 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, and (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 free energy. We show that the 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 behavior on several molecular systems with low-dimensional reaction coordinates.