Motivated by the study of Anglo-Saxon settlements locations, where administrative clusters involving a variety of complementary names tend to appear, we develop a Bayesian Cluster Model that, given a multi-type (k-type) point process, looks for clusters formed by points of different types. We obtain a multimodal, intractable posterior distribution on the space of matchings contained in a k-partite hypergraph. Such distribution is closely related to the posterior distributions arising in multi-target tracking and data association problems.

We develop an efficient Metropolis-Hastings algorithm to sample from the posterior distribution. We consider the problem of choosing an optimal proposal distribution to sample from product-form measures on the space of matchings contained in a bipartite graph. Simulated Tempering techniques are used to overcome multimodality. A multiple proposal scheme is developed to allow for parallel programming. Finally, Convergence Diagnostic techniques are used to assess the improvements in mixing given by such techniques. This allows us to study the Anglo-Saxon placenames locations dataset. First results seem to support the hypothesis of the settlements being organized into administrative clusters.

*PhD work supervised by Wilfrid Kendall*

**Keywords:** Bayesian cluster models; Complementary clustering; MCMC; Optimal proposal distribution; k-dimensional assignment problem; Probabilistic data association; Anglo-Saxon placenames locations