Abstract
In the current work we present two generalizations of the Parallel Tempering algorithm in the context of discrete-time Markov chain Monte Carlo methods for Bayesian inverse problems. These generalizations use state-dependent swapping rates, inspired by the so-called continuous time Infinite Swapping algorithm presented in Plattner et al. (J Chem Phys 135(13):134111, 2011). We analyze the reversibility and ergodicity properties of our generalized PT algorithms. Numerical results on sampling from different target distributions, show that the proposed methods significantly improve sampling efficiency over more traditional sampling algorithms such as Random Walk Metropolis, preconditioned Crank–Nicolson, and (standard) Parallel Tempering.
Highlights
Modern computational facilities and recent advances in computational techniques have made the use of Markov Chain Monte Carlo (MCMC) methods feasible for some large-scale Bayesian inverse problems (BIP), where the goal is to characterize the posterior distribution of a set of parameters θ of a computational model which describes some physical phenomena, conditioned on some measured data y
We propose to either (i) swap states in the chains at every iteration of the algorithm in such a way that the swap is accepted with probability one, which we will refer to as the Unweighted Generalized Parallel Tempering (UGPT), or (ii), swap dynamics at every step of the algorithm
We can see how both GPT algorithms provide a gain over random walk Metropolis (RWM), PT and Pairwise State Dependent Parallel Tempering (PSDPT) algorithms, with the Weighted Generalized Parallel Tempering (WGPT) algorithm providing the largest gain
Summary
Modern computational facilities and recent advances in computational techniques have made the use of Markov Chain Monte Carlo (MCMC) methods feasible for some large-scale Bayesian inverse problems (BIP), where the goal is to characterize the posterior distribution of a set of parameters θ of a computational model which describes some physical phenomena, conditioned on some (usually indirectly) measured data y. Some computational difficulties are prone to arise when dealing with difficult to explore posteriors, i.e., posterior distributions that are multi-modal, or that concentrate around a non-linear, lower-dimensional manifold, since some of the more commonly-used Markov transition kernels in MCMC algorithms, such as random walk Metropolis (RWM) or preconditioned Crank-Nicholson (pCN), are not well-suited in such situations. This in turn can make the computational time needed to properly explore these compli-. By proposing to exchange states between chains that target posteriors at different temperatures, it is possible for the chain of interest (i.e., the one targeting μ) to mix faster, and to avoid the undesirable behavior of some MCMC sam-
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
