Sequentially interacting Markov chain Monte Carlo methods

TL;DR

The paper introduces SIMCMC, a new sequential sampling method that combines MCMC and interacting sequences to improve estimates iteratively, with proven convergence and applications in Bayesian time series.

Contribution

It proposes SIMCMC, an alternative to SMC, enabling iterative improvement of estimates using interacting non-Markovian sequences that mimic independent MH chains.

Findings

SIMCMC converges under realistic assumptions.

Demonstrated effectiveness in Bayesian time series analysis.

Provides an MCMC-like iterative estimation process.

Abstract

Sequential Monte Carlo (SMC) is a methodology for sampling approximately from a sequence of probability distributions of increasing dimension and estimating their normalizing constants. We propose here an alternative methodology named Sequentially Interacting Markov Chain Monte Carlo (SIMCMC). SIMCMC methods work by generating interacting non-Markovian sequences which behave asymptotically like independent Metropolis-Hastings (MH) Markov chains with the desired limiting distributions. Contrary to SMC, SIMCMC allows us to iteratively improve our estimates in an MCMC-like fashion. We establish convergence results under realistic verifiable assumptions and demonstrate its performance on several examples arising in Bayesian time series analysis.