MCMC computations for Bayesian mixture models using repulsive point processes

Abstract

Repulsive mixture models have recently gained popularity for Bayesian cluster detection. Compared to more traditional mixture models, there is empirical evidence suggesting that repulsive mixture models produce a smaller number of well separated clusters. The most commonly used methods for posterior inference either require to fix a priori the number of components or are based on reversible jump MCMC computation. We present a general framework for mixture models, when the prior of the 'cluster centres' is a finite point process depending on a hyperparameter -not only a Poisson or determinantal point process (DPP) as previously considered in the literature but also a repulsive point process specified by a density which depends on an intractable normalizing constant. By investigating the posterior characterization of this class of mixture models, we derive a MCMC algorithm which avoids the well-known difficulties associated to reversible jump MCMC computation. In particular, when the point process density involves an intractable normalizing constant, we use an ancillary variable method which eliminate the problem of having a ratio of normalizing constants in the Hastings ratio when making posterior simulations for full conditional of the hyperparameter. The ancillary variable method relies on a perfect simulation algorithm, and we demonstrate this is fast because the number of components is typically small. In several simulation studies and an application on sociological data, we illustrate the advantage of our new methodology over existing methods, and we compare the use of a DPP or a repulsive Gibbs point process prior model.

Determinantal shot noise Cox processes

“…Likelihood-based inference is complicated because of the unobserved process of cluster centres. Møller and Waagepetersen (2004) showed how a missing data MCMC approach can be used for maximum likelihood estimation in the special case of the Thomas process, and it may be simpler but still rather complicated to use a MCMC Bayesian setting along similar lines as in Beraha et al (2022). We propose instead to exploit the parametric expressions of the intensity and of the pcf or K-function given in Section 3.2 when estimating γ and θ.…”

“…In fact, all results and statistical methods used in this paper will apply for the DGSNCP when k α is replaced by Ek Ay in all expressions to follow. The DGSNCP may most naturally be treated in a MCMC Bayesian setting using a similar approach as in Beraha et al (2022) and the references therein.…”

Aspects of Statistical Analysis of Spatial Point Patterns

“…Likelihood based inference is complicated because of the unobserved process of cluster centres. showed how a missing data MCMC approach can be used for maximum likelihood estimation in the special case of the Thomas process, and it may be simpler but still rather complicated to use a MCMC Bayesian setting along similar lines as in Beraha et al (2022). We propose instead to exploit the parametric expressions of the intensity and of the pcf or K-function given in Section 3.2 when estimating γ and θ.…”

“…In fact, all results and statistical methods used in this paper will apply for the DGSNCP when k α is replaced by Ek A y in all expressions to follow. The DGSNCP may most naturally be treated in a MCMC Bayesian setting using a similar approach as in Beraha et al (2022) and the references therein.…”