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.

Bayesian finite mixtures with an unknown number of components: The allocation sampler

A natural Bayesian approach for mixture models with an unknown number of components is to take the usual finite mixture model with symmetric Dirichlet weights, and put a prior on the number of components—that is, to use a mixture of finite mixtures (MFM). The most commonly used method of inference for MFMs is reversible jump Markov chain Monte Carlo, but it can be nontrivial to design good reversible jump moves, especially in high-dimensional spaces. Meanwhile, there are samplers for Dirichlet process mixture (DPM) models that are relatively simple and are easily adapted to new applications. It turns out that, in fact, many of the essential properties of DPMs are also exhibited by MFMs—an exchangeable partition distribution, restaurant process, random measure representation, and stick-breaking representation—and crucially, the MFM analogues are simple enough that they can be used much like the corresponding DPM properties. Consequently, many of the powerful methods developed for inference in DPMs ...

/pdf/mixture-models-with-a-prior-on-the-number-of-components-5awm6z5ek1.pdf

Mixture Models With a Prior on the Number of Components

A natural Bayesian approach for mixture models with an unknown number of components is to take the usual finite mixture model with Dirichlet weights, and put a prior on the number of components---that is, to use a mixture of finite mixtures (MFM). While inference in MFMs can be done with methods such as reversible jump Markov chain Monte Carlo, it is much more common to use Dirichlet process mixture (DPM) models because of the relative ease and generality with which DPM samplers can be applied. In this paper, we show that, in fact, many of the attractive mathematical properties of DPMs are also exhibited by MFMs---a simple exchangeable partition distribution, restaurant process, random measure representation, and in certain cases, a stick-breaking representation. Consequently, the powerful methods developed for inference in DPMs can be directly applied to MFMs as well. We illustrate with simulated and real data, including high-dimensional gene expression data.

Mixture models with a prior on the number of components

We consider approximate Bayesian model choice for model selection problems that involve models whose Fisher-information matrices may fail to be invertible along other competing submodels. Such singular models do not obey the regularity conditions underlying the derivation of Schwarz's Bayesian information criterion (BIC) and the penalty structure in BIC generally does not reflect the frequentist large-sample behavior of their marginal likelihood. While large-sample theory for the marginal likelihood of singular models has been developed recently, the resulting approximations depend on the true parameter value and lead to a paradox of circular reasoning. Guided by examples such as determining the number of components of mixture models, the number of factors in latent factor models or the rank in reduced-rank regression, we propose a resolution to this paradox and give a practical extension of BIC for singular model selection problems.

A Bayesian information criterion for singular models

Method: The objective of the present article is to propose and evaluate a probabilistic approach based on Bayesian networks for modelling non-homogeneous and non-linear gene regulatory processes. The method is based on a mixture model, using latent variables to assign individual measurements to different classes. The practical inference follows the Bayesian paradigm and samples the network structure, the number of classes and the assignment of latent variables from the posterior distribution with Markov Chain Monte Carlo (MCMC), using the recently proposed allocation sampler as an alternative to RJMCMC.

Results: We have evaluated the method using three criteria: network reconstruction, statistical significance and biological plausibility. In terms of network reconstruction, we found improved results both for a synthetic network of known structure and for a small real regulatory network derived from the literature. We have assessed the statistical significance of the improvement on gene expression time series for two different systems (viral challenge of macrophages, and circadian rhythms in plants), where the proposed new scheme tends to outperform the classical BGe score. Regarding biological plausibility, we found that the inference results obtained with the proposed method were in excellent agreement with biological findings, predicting dichotomies that one would expect to find in the studied systems.

Availability: Two supplementary papers on theoretical (T) and experi-mental (E) aspects and the datasets used in our study are available from http://www.bioss.ac.uk/associates/marco/supplement/

Contact:[email protected], [email protected]

/pdf/modelling-non-stationary-gene-regulatory-processes-with-a-o5njk8v0ae.pdf

Modelling non-stationary gene regulatory processes with a non-homogeneous Bayesian network and the allocation sampler

A new method for the computation of the posterior distribution of the number k of components in a finite mixture is presented. Two aspects of prior specification are also studied: an argument is made for the use of a Poisson(1) distribution as the prior for k; and methods are given for the selection of hyperparameter values in the mixture of normals model, with natural conjugate priors on the components parameters.

Agostino Nobile

Papers

Bayesian finite mixtures: a note on prior specification and posterior computation