The supplementary materials and XPN Matlab code are publicly available at website: https://genome.unc.edu/xpn

No abstract

Discrete-state Markov random ®elds on regular arrays have played a signi®cant role in spatial statistics and image analysis. For example, they are used to represent objects against background in computer vision and pixel-based classi®cation of a region into different crop types in remote sensing. Convenience has generally favoured formulations that involve only pairwise interactions. Such models are in themselves unrealistic and, although they often perform surprisingly well in tasks such as the restoration of degraded images, they are unsatisfactory for many other purposes. In this paper, we consider particular forms of Markov random ®elds that involve higher-order interactions and therefore are better able to represent the large-scale properties of typical spatial scenes. Interpretations of the parameters are given and realizations from a variety of models are produced via Markov chain Monte Carlo. Potential applications are illustrated in two examples. The ®rst concerns Bayesian image analysis and con®rms that pairwise-interaction priors may perform very poorly for image functionals such as number of objects, even when restoration apparently works well. The second example describes a model for a geological dataset and obtains maximum-likelihood parameter estimates using Markov chain Monte Carlo. Despite the complexity of the formulation, realizations of the estimated model suggest that the representation is quite realistic.

Markov chain Monte Carlo algorithms generate samples from a target distribution by simulating a Markov chain. Large¯exibility exists in speci®cation of transition matrix of the chain. In practice, however, most algorithms used only allow small changes in the state vector in each iteration. This choice typically causes problems for multimodal distributions as moves between modes become rare and, in turn, results in slow convergence to the target distribution. In this paper we consider continuous distributions on R n and specify how optimization for local maxima of the target distribution can be incorporated in the speci®cation of the Markov chain. Thereby, we obtain a chain with frequent jumps between modes. We demonstrate the effectiveness of the approach in three examples. The ®rst considers a simple mixture of bivariate normal distributions, whereas the two last examples consider sampling from posterior distributions based on previously analysed data sets.