Simulating Markov random fields with a conclique-based Gibbs sampler

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.