Andee Kaplan
Duke University

A Fast Sampler for Data Simulation from Spatial, and Other, Markov Random Fields

For spatial and network data, a model may be formulated on the basis of a Markov random field (MRF) structure and the specification of a conditional distribution for each observation. This piece-wise conditional approach often provides an attractive alternative to directly specifying a full joint data distribution, which may be difficult for large correlated data. At issue, fast simulation of data 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 reference distributions for statistics). However, the standard Gibbs strategy for simulating data from a spatial MRF models involves individual-site updates from conditional distributions, which is often challenging and computationally slow even for one complete iteration of relatively small sample size. As a remedy, we describe a fast way to simulate from MRF models, based on the concept of "concliques", (i.e., groups of non-neighboring observations). The proposed simulation scheme is computationally fast due to its ability to lower the number of steps necessary to complete one iteration of a Gibbs sampler. We motivate the simulation method, formally establish its validity, and assess its computational performance through numerical studies, where speed advantages are shown. In addition to numerical evidence, we also present formal results that show the proposed Gibbs sampler for simulating MRF data is geometrically ergodic (i.e., exhibits fast convergence rates) for simulating data from many commonly used spatial MRF models. Such general convergence results are typically unusual for spatial data generation but made possible here through the proposed sampling scheme.

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