Simulating Markov random fields with a conclique-based Gibbs sampler

TL;DR

This paper introduces a conclique-based Gibbs sampler for simulating Markov random fields more efficiently, significantly reducing computation time compared to traditional single-site updates, especially in models with bounded neighborhoods.

Contribution

The paper proposes a novel conclique-based Gibbs sampling method for MRFs, enabling faster simulation by updating groups of non-neighboring observations simultaneously.

Findings

Order-of-magnitude speed improvements in simulations.

Effective for spatial and network MRF models with bounded neighborhoods.

Validated through numerical studies demonstrating performance gains.

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. Fast simulation from such MRF models is often an important consideration, particularly when repeated generation of large numbers of data sets is required. However, a standard Gibbs strategy for simulating from MRF models involves single-site updates, performed with the conditional univariate distribution of each observation in a sequential manner, whereby a complete Gibbs iteration may become computationally involved even for moderate 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 independently updating all observations per conclique at once. The speed improvement depends on the number of concliques relative to the sample size for simulation, and order-of-magnitude speed increases are possible with many MRF models (e.g., having appropriately bounded neighborhoods). 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.