Darren Wilkinson

Statistical applications of the Potts model

Stage 4 project, 2024/25

Darren Wilkinson

The Potts model is a discrete Markov random field studied in statistical mechanics that has numerous potential applications in statistics. The model represents a probability distribution over colourings of nodes of a graph, with adjacent nodes more likely to be assigned the same colour than two nodes picked at random. The model can be used as a Bayesian prior distribution for clustering on graphs, including lattices, and hence has potential applications in image analysis and the detection of community structure in network data.

Potential areas for more in-depth study include:

• MCMC simulation via a Gibbs sampler
• Parallel simulation strategies
• Perfect simulation
• Estimating the partition function
• Parallel tempering
• Inference using data
• The “exchange algorithm” for fully Bayesian inference
• Applications to Bayesian image segmentation, colour quantisation, and “painting-by-numbers”
• Applications to graph community detection, with and without covariates

Pre-requisites

You must have taken MATH3421 (Bayesian computation and modelling), in addition to several other courses in statistics and probability at stages II and III. You must be very comfortable programming in a language such as R or Python.

Some relevant resources

Books

• Winkler, G. (2003) Image Analysis, Random Fields and Markov Chain Monte Carlo Methods, Springer.

Papers

• Wu, F. Y. (1982) The Potts model, Rev. Mod. Phys., 54, 235.
• Beaudin, L. (2007) A Review of the Potts Model, Rose-Hulman Undergraduate Mathematics Journal, 8(1):13.
• Murray, I., Ghahramani, Z., MacKay, D. (2006) MCMC for doubly-intractable distributions, Proceedings of UAI 2006.
• Wilkinson, D. J. (2005) Parallel Bayesian Computation, Chapter 16 in E. J. Kontoghiorghes (ed.) Handbook of Parallel Computing and Statistics, Marcel Dekker/CRC Press, 481-512.
• Li, H. et al (2012) Community structure detection based on Potts model and network’s spectral characterization, EPL, 97, 48005.
• Propp, J. G., Wilson, D. B. (1996) Exact sampling with coupled Markov chains and applications to statistical mechanics, Random Structures and Algorithms, 9, 223-252.
• Kendall, W. S. (2004) Notes on perfect simulation, Warwick statistics technical report.
• Childs, A. M. et al (2001) Exact sampling from nonattractive distributions using summary states, Phys. Rev. E, 63:036113.
• Caimo, A., Friel, N. (2011) Bayesian inference for exponential random graph models, Social Networks, 33: 41–55.
• Lee, C., Wilkinson, D. J. (2019) A Review of Stochastic Block Models and Extensions for Graph Clustering, Applied Network Science, 4:122.

Web pages, blog posts, etc.

• Potts model
• Markov random field
• The Ising model and Gibbs sampling (and also this PDF)
• Random cluster model
• Exponential family random graph models
• Gibbs sampling
• Coupling from the past
• Parallel tempering
• Swendsen-Wang algorithm
• Partition function
• Community structure
• PottsUtils R package
• Parallel tempering and MCMCMC
• Marginal likelihood from tempered Bayesian posteriors

A realisation from a Potts model on a square lattice with 4 states, close to criticality.