Statistical applications of the Potts model

Stage 4 project, 2024/25

Supervisor: Darren Wilkinson

Project outline

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

  • 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 III), 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

Papers

Web pages, blog posts, etc.

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