Information-geometric Markov Chain Monte Carlo methods using Diffusions

Recent work incorporating geometric ideas in Markov chain Monte Carlo is reviewed in order to highlight these advances and their possible application in a range of domains beyond Statistics. A full exposition of Markov chains and their use in Monte Carlo simulation for Statistical inference and molecular dynamics is provided, with particular emphasis on methods based on Langevin diffusions. After this geometric concepts in Markov chain Monte Carlo are introduced. A full derivation of the Langevin diffusion on a Riemannian manifold is given, together with a discussion of appropriate Riemannian metric choice for different problems. A survey of applications is provided, and some open questions are discussed.Comment: 22 pages, 2 figure

MCMC inference for Markov Jump Processes via the Linear Noise Approximation

Bayesian analysis for Markov jump processes is a non-trivial and challenging problem. Although exact inference is theoretically possible, it is computationally demanding thus its applicability is limited to a small class of problems. In this paper we describe the application of Riemann manifold MCMC methods using an approximation to the likelihood of the Markov jump process which is valid when the system modelled is near its thermodynamic limit. The proposed approach is both statistically and computationally efficient while the convergence rate and mixing of the chains allows for fast MCMC inference. The methodology is evaluated using numerical simulations on two problems from chemical kinetics and one from systems biology

Quantification of coarse-graining error in Langevin and overdamped Langevin dynamics

In molecular dynamics and sampling of high dimensional Gibbs measures coarse-graining is an important technique to reduce the dimensionality of the problem. We will study and quantify the coarse-graining error between the coarse-grained dynamics and an effective dynamics. The effective dynamics is a Markov process on the coarse-grained state space obtained by a closure procedure from the coarse-grained coefficients. We obtain error estimates both in relative entropy and Wasserstein distance, for both Langevin and overdamped Langevin dynamics. The approach allows for vectorial coarse-graining maps. Hereby, the quality of the chosen coarse-graining is measured by certain functional inequalities encoding the scale separation of the Gibbs measure. The method is based on error estimates between solutions of (kinetic) Fokker-Planck equations in terms of large-deviation rate functionals

Coarse-Grained Kinetic Computations for Rare Events: Application to Micelle Formation

We discuss a coarse-grained approach to the computation of rare events in the context of grand canonical Monte Carlo (GCMC) simulations of self-assembly of surfactant molecules into micelles. The basic assumption is that the {\it computational} system dynamics can be decomposed into two parts -- fast (noise) and slow (reaction coordinates) dynamics, so that the system can be described by an effective, coarse grained Fokker-Planck (FP) equation. While such an assumption may be valid in many circumstances, an explicit form of FP equation is not always available. In our computations we bypass the analytic derivation of such an effective FP equation. The effective free energy gradient and the state-dependent magnitude of the random noise, which are necessary to formulate the effective Fokker-Planck equation, are obtained from ensembles of short bursts of microscopic simulations {\it with judiciously chosen initial conditions}. The reaction coordinate in our micelle formation problem is taken to be the size of a cluster of surfactant molecules. We test the validity of the effective FP description in this system and reconstruct a coarse-grained free energy surface in good agreement with full-scale GCMC simulations. We also show that, for very small clusters, the cluster size seizes to be a good reaction coordinate for a one-dimensional effective description. We discuss possible ways to improve the current model and to take higher-dimensional coarse-grained dynamics into account

Langevin and Hamiltonian based Sequential MCMC for Efficient Bayesian Filtering in High-dimensional Spaces

Nonlinear non-Gaussian state-space models arise in numerous applications in statistics and signal processing. In this context, one of the most successful and popular approximation techniques is the Sequential Monte Carlo (SMC) algorithm, also known as particle filtering. Nevertheless, this method tends to be inefficient when applied to high dimensional problems. In this paper, we focus on another class of sequential inference methods, namely the Sequential Markov Chain Monte Carlo (SMCMC) techniques, which represent a promising alternative to SMC methods. After providing a unifying framework for the class of SMCMC approaches, we propose novel efficient strategies based on the principle of Langevin diffusion and Hamiltonian dynamics in order to cope with the increasing number of high-dimensional applications. Simulation results show that the proposed algorithms achieve significantly better performance compared to existing algorithms