Simulating Markov Random Fields With a Conclique-Based Gibbs Sampler

Abstract

For spatial and network data, we consider models formed from a Markov random field (MRF) structure and the specification of a conditional distribution for each observation. At issue, fast simulation from such MRF models is often an important consideration, particularly when repeated generation of large numbers of data sets is required (e.g., for approximating sampling distributions). However, a standard Gibbs strategy for simulating from MRF models involves single-updates, performed with the conditional distribution of each observation in a sequential manner, whereby a Gibbs iteration may become computationally involved even for relatively small samples. As an alternative, we describe a general way to simulate from MRF models using Gibbs sampling with "concliques" (i.e., groups of non-neighboring observations). Compared to standard Gibbs sampling, this simulation scheme can be much faster by reducing Gibbs steps and by independently updating all observations per conclique at once. We detail the simulation method, establish its validity, and assess its computational performance through numerical studies, where speed advantages are shown for several spatial and network examples. arXiv:1808.04739v1 [stat.CO]

Random Fields in Physics, Biology and Data Science

“…Gao and Gormley implemented a Gibbs sampling scheme based on CRFs weighted via neural scoring factors (implemented as parameters in factor graphs) with applications to Natural Language Processing (NLP) [136]. MCMC has also been used, in the context of Gibbs random fields in data pre-processing, to reduce the computational burden of data intensive signal processing [137,138]. Gibbs sampling can also be applied in parallel within the context of Gaussian MRFs on large grids or lattice models [139].…”

Prediction interval methods for reliability data

“…Model formulation in a conditional, component-wise fashion provides an alternative to direct specification of a full joint data distribution, and often such conditional distributions depend functionally on small subsets of "neighboring" observations. Using the latter property, we describe here how plug-in and calibration-bootstrap prediction methods may be extended to such MRF models; as an attractive feature, fast simulation from such models is also possible when implementing the bootstrap (Kaplan et al (2020))…”

“…Besag et al (1991)) with the fitted full conditionals f i (•|x(N i ); θ). Additionally, fast algorithms to Gibbs sample from MRF models are available based on concliques (Kaplan et al (2020)).…”

Rank-based inference for conditional independence and its applications

“…Note that our data generating process is solely based on the node-conditional distributions, instead of covariance matrices that are broadly used in other studies on conditional dependence. For a more efficient generating process in large spaces, such as 1000 ˆ1000, one may consider updates in step 2 and 3 with a conclique-based Gibbs sampler (Kaplan et al, 2020).…”