Dynamic MCMC Sampling
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The Markov chain Monte Carlo (MCMC) methods are the primary tools for sampling from graphical models, e.g. Markov random fields (MRF). Traditional MCMC sampling algorithms are focused on a classic static setting, where the input is fixed. In this paper we study the problem of sampling from an MRF when the graphical model itself is changing dynamically with time. The problem is well motivated by the growing volume and velocity of data in today's applications of the MCMC methods. For the two major MCMC approaches, respectively for the approximate and perfect sampling, namely, the Gibbs sampling and the coupling from the past (CFTP), we give dynamic versions for the respective MCMC sampling algorithms. On MRF with n variables and bounded maximum degrees, these dynamic sampling algorithms can maintain approximate or perfect samples, while the MRF is dynamically changing. Furthermore, our algorithms are efficient with Õ(n) space cost, and Õ(^2n) incremental time cost upon each local update to the input MRF, as long as certain decay conditions are satisfied in each step by natural couplings of the corresponding single-site chains. These decay conditions were well known in the literature of couplings for rapid mixing of Markov chains, and now for the first time, are used to imply efficient dynamic sampling algorithms. Consequently, we have efficient dynamic sampling algorithms for the following models: (1) general MRF satisfying the Dobrushin-Shlosman condition (for approximate sampling); (2) Ising model with temperature β where e^-2|β|> 1-2/Δ+1 (for both approximate and perfect samplings); (3) hardcore model with fugacity λ<2/Δ-2 (for both approximate and perfect samplings); (4) proper q-coloring with: q>2Δ (for approximate sampling); or q>2Δ^2+3Δ (for perfect sampling).
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