The estimation of the parameters of a mixture of Gaussian densities is considered, within the framework of maximum likelihood. Due to unboundedness of the likelihood function, the maximum likelihood estimator fails to exist. We adopt a solution to likelihood function degeneracy which consists in penalizing the likelihood function. The resulting penalized likelihood function is then bounded over the parameter space and the existence of the penalized maximum likelihood estimator is granted. As original contribution we provide asymptotic properties, and in particular a consistency proof, for the penalized maximum likelihood estimator. Numerical examples are provided in the finite data case, showing the performances of the penalized estimator compared to the standard one.

Abstract—In this paper, we present a Hidden Markov Random Field (HMRF) data-fusion model. The proposed model is applied to the segmentation of natural images based on the fusion of colors and textons into Julesz ensembles. The corresponding Exploration/ Selection/Estimation (ESE) procedure for the estimation of the parameters is presented. This method achieves the estimation of the parameters of the Gaussian kernels, the mixture proportions, the region labels, the number of regions, and the Markov hyper-parameter. Meanwhile, we present a new proof of the asymptotic convergence of the ESE procedure, based on original finite time bounds for the rate of convergence. Index Terms—Bayesian estimation, color and texture segmentation, Exploration/Selection algorithm, Exploration/Selection/Estimation procedure, fusion of hidden Markov random field

A new Markov chain Monte Carlo method for the Bayesian analysis of finite mixture distributions with an unknown number of components is presented. The sampler is characterized by a state space consisting only of the number of components and the latent allocation variables. Its main advantage is that it can be used, with minimal changes, for mixtures of components from any parametric family, under the assumption that the component parameters can be integrated out of the model analytically. Artificial and real data sets are used to illustrate the method and mixtures of univariate and of multivariate normals are explicitly considered. The problem of label switching, when parameter inference is of interest, is addressed in a post-processing stage.

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This survey covers state-of-the-art Bayesian techniques for the estimation of mixtures. It complements the earlier Marin et al. (2005) by studying new types of distributions, the multinomial, latent class and t distributions. It also exhibits closed form solutions for Bayesian inference in some discrete setups. At last, it sheds a new light on the computation of Bayes factors via the approximation of Chib (1995).

This work considers probability models for partitions of a set of n elements using a predictive approach, i.e., models that are specified in terms of the conditional probability of either joining an already existing cluster or forming a new one. The inherent structure can be motivated by resorting to hierarchical models of either parametric or nonparametric nature. Parametric examples include the product partition models (PPMs) and the model-based approach of Dasgupta and Raftery (1998), while nonparametric alternatives include the Dirichlet Process, and more generally, the Species Sampling Models (SSMs). Under exchangeability, PPMs and SSMs induce the same type of partition structure. The methods are discussed in the context of outlier detection in normal linear regression models and of (univariate) density estimation.

In many different fields such as hydrology, telecommunications, physics of condensed matter and finance, the gaussian model results unsatisfactory and reveals difficulties in fitting data with skewness, heavy tails and multimodality. The use of stable distributions allows for modelling skewness and heavy tails but gives rise to inferential problems related to the estimation of the stable distribution's parameters. The aim of this work is to generalise the stable distribution framework by introducing a model that accounts also for multimodality. In particular we introduce a stable mixture model and a suitable reparameterisation of the mixture, which allow us to make inference on the mixture parameters. We use a full Bayesian approach and MCMC simulation techniques for the estimation of the posterior distribution.

“What the world needs is not dogma but an attitude of scientific inquiry combined with a belief that the torture of millions is not desirable, whether inflicted by Stalin or by a Deity imagined in the likeness of the believer.” – Bertrand Russell Declaration This dissertation is the result of work carried out by myself between October 2002 and December 2006. It includes nothing which is the outcome of work done in collaboration except where specifically indicated in the text. It contains approximately 56,900 words and does not exceed the limit of 65,000 words. It contains 10 figures, which does not exceed the limit

Abstract. In the past ten years there has been a dramatic increase of interest in the Bayesian analysis of finite mixture models. This is primarily because of the emergence of Markov chain Monte Carlo (MCMC) methods. Whilst MCMC provides a convenient way to draw inference from complicated statistical models, there are many, perhaps under appreciated, problems associated with the MCMC analysis of mixtures. The problems are mainly caused by the nonidentifiability of the components under symmetric priors, which leads to so called label switching in the MCMC output. This will mean that ergodic averages of component specific quantities will be identical and thus useless for inference. We review the solutions to the label switching problem, such as artificial identifiability constraints (e.g. Diebolt & Robert (1994)), relabelling algorithms (Stephens 1997a) and label invariant loss functions (Celeux, Hurn & Robert 2000). We also review various MCMC sampling schemes that have been suggested for mixture models and discuss posterior sensitivity to prior specification.