Abstract not found

With reference to a specific data set, we consider how to perform a flexible non-parametric Bayesian analysis of an inhomogeneous point pattern modelled by a Markov point process, with a location dependent first order term and pairwise interaction only. A priori we assume that the first order term is a shot noise process, and the interaction function for a pair of points depends only on the distance between the two points and is a piecewise linear function modelled by a marked Poisson process. Simulation of the resulting posterior using a Metropolis-Hastings algorithm in the “conventional ” way involves evaluating ratios of unknown normalising constants. We avoid this problem by applying a new auxiliary variable technique introduced by Møller, Pettitt, Reeves & Berthelsen (2006). In the present setting the auxiliary variable used is an example of a partially ordered Markov point process model.

We summarize and discuss the current state of spatial point process theory and directions for future research, making an analogy with generalized linear models and random effect models, and illustrating the theory with various examples of applications. In particular, we consider Poisson, Gibbs, and Cox process models, diagnostic tools and model checking, Markov chain Monte Carlo algorithms, computational methods for likelihood-based inference, and quick non-likelihood approaches to inference.

Abstract. For frequentist inference, the efficacy of the closed-form (CF) likelihood approximation of Aït-Sahalia (2002, 2007) in financial modeling has been widely demonstrated. Bayesian inference, however, requires the use of MCMC, and the CF likelihood can become inaccurate when the parameters θ are far from the MLE. Due to numerical stability problems, the samplers can therefore become stuck when (typically) in the tails of the posterior distribution. It may be possible to address this problem by using numerical integration to estimate the intractable normalizers in the CF likelihood, but determining the limiting distribution after using such approximations remains an open research question. Auxiliary variables have been used in conjunction with MCMC to address intractable normalizers (see Møller et al. (2006)), but choosing such variables is not trivial. We propose a MCMC algorithm (called EMCMC) that addresses the intractable normalizers in the CF likelihood which 1) is easy to implement, 2) yields a sampler with the correct limiting distribution, and 3) greatly increases the stability of the sampler compared to using the unnormalized CF likelihood in a standard Metropolis-Hastings algorithm. The efficacy of our approach is demonstrated in a simulation study of the Cox-Ingersoll-Ross (CIR) model. Keywords. Bayesian inference, diffusion process, closed-form likelihood, EMCMC. 1

The k-nearest-neighbour procedure is a well-known deterministic method used in supervised classification. While it has been superseded by more recent methods developed in machine learning, it remains an essential tool for classifiers. This paper proposes a reassessment of this approach as a statistical technique derived from a proper probabilistic model; in particular, we modify the assessment made in a previous analysis of this method undertaken by Holmes and Adams (2002, 2003) where the underlying probabilistic model is not completely well-defined. Once clear probabilistic bases of the k-nearest-neighbour procedure are established, we proceed to the derivation of practical computational tools to conduct Bayesian inference on the parameters of the corresponding model. In particular, we assess the difficulties inherent to pseudo-likelihood and to path sampling approximations of a missing normalising constant, and propose a perfect sampling strategy to implement a correct MCMC sampler associated with our model. Illustrations of the performance of the corresponding Bayesian classifier are provided for two benchmark datasets, demonstrating in particular the limitations of the pseudo-likelihood approximation in this set-up.

apport de recherche

The k-nearest-neighbour procedure is a well-known deterministic method used in supervised classification. This paper proposes a reassessment of this approach as a statistical technique derived from a proper probabilistic model; in particular, we modify the assessment made in a previous analysis of this method undertaken by Holmes and Adams (2002, 2003), and evaluated by Manocha and Girolami (2007), where the underlying probabilistic model is not completely well-defined. Once a clear probabilistic basis for the k-nearest-neighbour procedure is established, we derive computational tools for conducting Bayesian inference on the parameters of the corresponding model. In particular, we assess the difficulties inherent to pseudo-likelihood and to path sampling approximations of an intractable normalising constant, and propose a perfect sampling strategy to implement a correct MCMC sampler associated with our model. If perfect sampling is not available, we suggest using a Gibbs sampling approximation. Illustrations of the performance of the corresponding Bayesian classifier are provided for several benchmark datasets, demonstrating in particular the limitations of the pseudo-likelihood approximation in this set-up.