Weak convergence of Markov chain sampling methods and annealing algorithms to diffusions

On Sampling Methods and Annealing Algorithms

Convergence Rates of Markov Chain Approximation Methods for Controlled Regime-Switching Diffusions with Stopping

This work summarizes our recent work on rates of convergence of Markov chain approximation methods for controlled switching diffusions, in which both continuous dynamics and discrete events coexist. The discrete events are formulated by continuous-time Markov chains to delineate random environment and other random factors that cannot be represented by diffusion processes. The cost function is o...

Convergence rates of Markov chain approximation methods for controlled diffusions with stopping

This work is concerned with rates of convergence of numerical methods using Markov chain approximation for controlled diffusions with stopping (the first exit time from a bounded region). In lieu of considering the associated finite difference schemes for Hamilton-Jacobi-Bellman (HJB) equations, a purely probabilistic approach is used. There is an added difficulty due to the boundary condition,...

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. Aft...

Assessing Convergence of Markov Chain Monte Carlo Algorithms

We motivate the use of convergence diagnostic techniques for Markov Chain Monte Carlo algorithms and review various methods proposed in the MCMC literature. A common notation is established and each method is discussed with particular emphasis on implementational issues and possible extensions. The methods are compared in terms of their interpretability and applicability and recommendations are...