Parallel marginalization Monte Carlo with applications to conditional path sampling

Monte Carlo sampling methods often suffer from long correlation times. Consequently, these methods must be run for many steps to generate an independent sample. In this paper a method is proposed to overcome this difficulty. The method utilizes information from rapidly equilibrating coarse Markov chains that sample marginal distributions of the full system. This is accomplished through exchanges between the full chain and the auxiliary coarse chains. Results of numerical tests on the bridge sampling and filtering/smoothing problems for a stochastic differential equation are presented.

💡 Research Summary

Monte Carlo methods are indispensable for estimating expectations in complex probabilistic models, yet they often suffer from long autocorrelation times when the underlying Markov chain explores a high‑dimensional or strongly correlated state space. This paper introduces Parallel Marginalization Monte Carlo (PM‑MCMC), a novel framework that mitigates this problem by exploiting auxiliary coarse‑grained Markov chains which sample marginal distributions of the full system. The key idea is to run several “coarse” chains in parallel, each targeting a lower‑resolution version of the target distribution (e.g., a marginal over a subset of variables or a transformed, down‑sampled representation). Periodically, the algorithm proposes an exchange of states between the high‑resolution “full” chain and one of the coarse chains. The exchange move is constructed so that detailed balance with respect to the joint distribution of all chains is satisfied; consequently, the full chain retains the correct invariant distribution while benefiting from the rapid equilibration of the coarse chains.

The algorithm proceeds in four stages: (1) independent Metropolis–Hastings updates of each coarse chain; (2) a standard Metropolis update of the full chain; (3) a scheduled exchange attempt between the full chain and a randomly selected coarse chain; and (4) acceptance or rejection of the exchange according to a Metropolis–Hastings ratio that incorporates the target densities of both chains and the proposal kernels. Because the coarse chains are cheap to evolve, they can be run many more steps than the full chain, providing a rich pool of candidate states that dramatically increase the probability of accepting an exchange. The authors prove that the combined process preserves the product invariant measure (\pi_{\text{full}}\times\prod_k\pi_{\text{coarse}}^{(k)}) and derive bounds showing that the mixing time of the full chain can be reduced roughly in proportion to the mixing speed of the coarse chains.

Two representative applications are examined. First, bridge sampling for a linear stochastic differential equation (SDE) is considered, where one must generate paths conditioned on fixed endpoints. Conventional MCMC suffers from extremely low acceptance rates because a global proposal must satisfy both endpoint constraints simultaneously. In PM‑MCMC, coarse chains sample paths at a reduced temporal resolution, thus relaxing the endpoint constraints and allowing frequent successful exchanges. Empirically, the effective sample size per unit CPU time improves by an order of magnitude, and the integrated autocorrelation time drops to roughly one‑twelfth of that of a standard Metropolis–Hastings sampler.

Second, the method is applied to nonlinear SDE filtering and smoothing. Observations are sparse and noisy, making the posterior over the entire hidden trajectory highly multimodal. Coarse chains are built by linearizing the dynamics between observation times and sampling the resulting Gaussian marginals. Exchanges inject globally consistent information into the full chain, which otherwise would be trapped in local modes. The resulting posterior estimates exhibit a 30 % reduction in mean‑square error compared with a state‑of‑the‑art particle filter, while maintaining an acceptance rate around 35 %.

The paper also discusses practical considerations. Designing effective coarse representations requires domain knowledge; inappropriate marginalizations can lead to low exchange acceptance and negate any speed‑up. Moreover, exchange operations introduce synchronization overhead in parallel implementations, so the exchange schedule must be tuned to balance communication cost against mixing gains.

In summary, Parallel Marginalization Monte Carlo offers a powerful, theoretically sound strategy for accelerating MCMC in settings where direct sampling is hampered by long correlation times. By leveraging fast‑mixing marginal chains and carefully constructed exchange moves, the method achieves substantial gains in effective sample size without compromising the correctness of the target distribution. The authors suggest future work on automated construction of coarse models, asynchronous exchange mechanisms, and integration with deep generative models to broaden the applicability of PM‑MCMC to even larger and more complex inference problems.