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We introduce a generalised micro-macro Markov chain Monte Carlo (mM-MCMC)method with pseudo-marginal approximation to the free energy, that is able toaccelerate sampling of the microscopic Gibbs distributions when there is atime-scale separation between the macroscopic dynamics of a reaction coordinateand the remaining microscopic degrees of freedom. The mM-MCMC method attainsthis efficiency by iterating four steps: i) Propose a new value of the reactioncoordinate; ii) Accept or reject the macroscopic sample; iii) Run a biasedsimulation that creates a microscopic molecular instance that lies close to thenewly sampled macroscopic reaction coordinate value; iv) Microscopicaccept/reject step for the new microscopic sample. In the present paper, weeliminate the main computational bottleneck of earlier versions of this method:the necessity to have an accurate approximation of the free energy. We showthat introduction of a pseudo-marginal approximation significantly reduces thecomputational cost of the microscopic accept/reject step, while still providingunbiased samples. We illustrate the method's behaviour on several molecularsystems with low-dimensional reaction coordinates.

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