Abstract not found

This paper develops mixture models for spatially indexed data. We confine attention to the case of finite, typically irregular, patterns of points or regions with prescribed spatial relationships, and to problems where it is only the weights in the mixture that vary from one location to another. Our specific focus is on Poisson distributed data, and applications in disease mapping. We work in a Bayesian framework, with the Poisson parameters drawn from gamma priors, and an unknown number of components. We propose two alternative models for spatially-dependent weights, based on transformations of autoregressive gaussian processes: in one (the Logistic normal model), the mixture component labels are exchangeable, in the other (the Grouped continuous model), they are ordered. Reversible jump Markov chain Monte Carlo algorithms for posterior inference are developed. Finally, the performance of both of these formulations is examined on synthetic data and real data on mortality from rare disease.

The Wang-Landau algorithm is an adaptive Markov chain Monte Carlo algorithm to calculate the spectral density for a physical system. A remarkable feature of the algorithm is that it is not trapped by local energy minima, which is very important for systems with rugged energy landscapes. This feature has led to many successful applications of the algo-rithm in statistical physics and biophysics. However, there does not exist rigorous theory to support its convergence, and the estimates produced by the algorithm can only reach a limited statistical accuracy. In this paper, we propose the stochastic approximation Monte Carlo (SAMC) algorithm, which overcomes the shortcomings of the Wang-Landau algorithm. We establish a theorem concerning its convergence. The estimates produced by SAMC can be improved continuously as the simulation goes on. SAMC also extends applications of the Wang-Landau algorithm to continuum systems. The potential uses of SAMC in statistics are discussed through two classes of applications, importance sampling and model selection. The results show that SAMC can work as a general importance sampling algorithm and a

We develop computational procedures for a class of Bayesian nonparametric and semiparametric multiplicative intensity models incorporating kernel mixtures of spatial weighted gamma measures. A key feature of our approach is that explicit expressions for posterior distributions of these models share many common structural features with the posterior distributions of Bayesian hierarchical models using the Dirichlet process. Using this fact, along with an approximation for the weighted gamma process, we show that with some care, one can adapt efficient algorithms used for the Dirichlet process to this setting. We discuss blocked Gibbs sampling procedures and Pólya urn Gibbs samplers. We illustrate our methods with applications to proportional hazard models, Poisson spatial regression models, recurrent events, and panel count data.

While learning the maximum likelihood value of parameters of an undirected graphical model is hard, modelling the posterior distribution over parameters given data is harder. Yet, undirected models are ubiquitous in computer vision and text modelling (e.g. conditional random fields). But where Bayesian approaches for directed models have been very successful, a proper Bayesian treatment of undirected models in still in its infant stages. We propose a new method for approximating the posterior of the parameters given data based on the Laplace approximation. This approximation requires the computation of the covariance matrix over features which we compute using the linear response approximation based in turn on loopy belief propagation. We develop the theory for conditional and “unconditional ” random fields with or without hidden variables. In the conditional setting we introduce a new variant of bagging suitable for structured domains. Here we run the loopy max-product algorithm on a “super-graph ” composed of graphs for individual models sampled from the posterior and connected by constraints. Experiments on real world data validate the proposed methods. 1

Variational methods for model comparison have become popular in the neural computing/machine learning literature. In this paper we explore their application to the Bayesian analysis of mixtures of Gaussians. We also consider how the Deviance Information Criterion, or DIC, devised by Spiegelhalter et al. (2002), can be extended to these types of model by exploiting the use of variational approximations. We illustrate the results of using variational methods for model selection and the calculation of a DIC using real and simulated data. Using the variational approximation, one can simultaneously estimate component parameters and the model complexity. It turns out that, if one starts o# with a large number of components, superfluous components are eliminated as the method converges to a solution, thereby leading to an automatic choice of model complexity, the appropriateness of which is reflected in the DIC values.

The mixture model likelihood function is invariant with respect to permutation of the components of the mixture. If functions of interest are permutation sensitive, as in classification applications, then interpretation of the likelihood function requires valid inequality constraints and a very large sample may be required to resolve ambiguities. If functions of interest are permutation invariant, as in prediction applications, then there are no such problems of interpretation. Contrary to assessments in some recent publications, simple and widely used Markov chain Monte Carlo (MCMC) algorithms with data augmentation reliably recover the entire posterior distribution. 1 1

The problem of marginal density estimation for a multivariate density function f(x) can be generally stated as a problem of density function estimation for a random vector λ(x) of dimension lower than that of x. In this paper, we propose a technique, the so-called continuous Contour Monte Carlo (CCMC) algorithm, for solving this problem. CCMC can be viewed as a continuous version of the contour Monte Carlo (CMC) algorithm recently proposed in the literature. CCMC abandons the use of sample space partitioning and incorporates the techniques of kernel density estimation into its simulations. CCMC is more general than other marginal density estimation algorithms. First, it works for any density functions, even for those having a rugged or unbalanced energy landscape. Second, it works for any transformation λ(x) regardless of the availability of the analytical form of the inverse transformation. In this paper, CCMC is applied to estimate the unknown normalizing constant function for a spatial autologistic model, and the estimate is then used in a Bayesian analysis for the spatial autologistic model in place of the true normalizing constant function. Numerical results on the US cancer mortality data indicate that the Bayesian method can produce much more accurate estimates than the MPLE and MCMLE methods for the parameters of the spatial autologistic model.

This paper presents a new sampling algorithm for approximating functions of variables representable as undirected graphical models of arbitrary connectivity with pairwise potentials, as well as for estimating the notoriously difficult partition function of the graph. The algorithm fits into the framework of sequential Monte Carlo methods rather than the more widely used MCMC, and relies on constructing a sequence of intermediate distributions which get closer to the desired one. While the idea of using "tempered" proposals is known, we construct a novel sequence of target distributions where, rather than dropping a global temperature parameter, we sequentially couple individual pairs of variables that are, initially, sampled exactly from a spanning tree of the variables. We present

Abstract not found