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Register a new userMCMC for Normalized Random Measure Mixture Models
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Added by
Stefano Favaro
Publication date
2013
fields
Mathematical Statistics
and research's language is
English
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This paper concerns the use of Markov chain Monte Carlo methods for posterior sampling in Bayesian nonparametric mixture models with normalized random measure priors. Making use of some recent posterior characterizations for the class of normalized random measures, we propose novel Markov chain Monte Carlo methods of both marginal type and conditional type. The proposed marginal samplers are generalizations of Neals well-regarded Algorithm 8 for Dirichlet process mixture models, whereas the conditional sampler is a variation of those recently introduced in the literature. For both the marginal and conditional methods, we consider as a running example a mixture model with an underlying normalized generalized Gamma process prior, and describe comparative simulation results demonstrating the efficacies of the proposed methods.
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Continuous mixtures of distributions are widely employed in the statistical literature as models for phenomena with highly divergent outcomes; in particular, many familiar heavy-tailed distributions arise naturally as mixtures of light-tailed distributions (e.g., Gaussians), and play an important role in applications as diverse as modeling of extreme values and robust inference. In the case of social networks, continuous mixtures of graph distributions can likewise be employed to model social processes with heterogeneous outcomes, or as robust priors for network inference. Here, we introduce some simple families of network models based on continuous mixtures of baseline distributions. While analytically and computationally tractable, these models allow more flexible modeling of cross-graph heterogeneity than is possible with conventional baseline (e.g., Bernoulli or $U|man$ distributions). We illustrate the utility of these baseline mixture models with application to problems of multiple-network ERGMs, network evolution, and efficient network inference. Our results underscore the potential ubiquity of network processes with nontrivial mixture behavior in natural settings, and raise some potentially disturbing questions regarding the adequacy of current network data collection practices.
We study the problem of sampling from the power posterior distribution in Bayesian Gaussian mixture models, a robust version of the classical posterior. This power posterior is known to be non-log-concave and multi-modal, which leads to exponential mixing times for some standard MCMC algorithms. We introduce and study the Reflected Metropolis-Hastings Random Walk (RMRW) algorithm for sampling. For symmetric two-component Gaussian mixtures, we prove that its mixing time is bounded as $d^{1.5}(d + Vert theta_{0} Vert^2)^{4.5}$ as long as the sample size $n$ is of the order $d (d + Vert theta_{0} Vert^2)$. Notably, this result requires no conditions on the separation of the two means. En route to proving this bound, we establish some new results of possible independent interest that allow for combining Poincar{e} inequalities for conditional and marginal densities.
Determining the number G of components in a finite mixture distribution is an important and difficult inference issue. This is a most important question, because statistical inference about the resulting model is highly sensitive to the value of G. Selecting an erroneous value of G may produce a poor density estimate. This is also a most difficult question from a theoretical perspective as it relates to unidentifiability issues of the mixture model. This is further a most relevant question from a practical viewpoint since the meaning of the number of components G is strongly related to the modelling purpose of a mixture distribution. We distinguish in this chapter between selecting G as a density estimation problem in Section 2 and selecting G in a model-based clustering framework in Section 3. Both sections discuss frequentist as well as Bayesian approaches. We present here some of the Bayesian solutions to the different interpretations of picking the right number of components in a mixture, before concluding on the ill-posed nature of the question.
Traditional Bayesian random partition models assume that the size of each cluster grows linearly with the number of data points. While this is appealing for some applications, this assumption is not appropriate for other tasks such as entity resolution, modeling of sparse networks, and DNA sequencing tasks. Such applications require models that yield clusters whose sizes grow sublinearly with the total number of data points -- the microclustering property. Motivated by these issues, we propose a general class of random partition models that satisfy the microclustering property with well-characterized theoretical properties. Our proposed models overcome major limitations in the existing literature on microclustering models, namely a lack of interpretability, identifiability, and full characterization of model asymptotic properties. Crucially, we drop the classical assumption of having an exchangeable sequence of data points, and instead assume an exchangeable sequence of clusters. In addition, our framework provides flexibility in terms of the prior distribution of cluster sizes, computational tractability, and applicability to a large number of microclustering tasks. We establish theoretical properties of the resulting class of priors, where we characterize the asymptotic behavior of the number of clusters and of the proportion of clusters of a given size. Our framework allows a simple and efficient Markov chain Monte Carlo algorithm to perform statistical inference. We illustrate our proposed methodology on the microclustering task of entity resolution, where we provide a simulation study and real experiments on survey panel data.
In Chib (1995), a method for approximating marginal densities in a Bayesian setting is proposed, with one proeminent application being the estimation of the number of components in a normal mixture. As pointed out in Neal (1999) and Fruhwirth-Schnatter (2004), the approximation often fails short of providing a proper approximation to the true marginal densities because of the well-known label switching problem (Celeux et al., 2000). While there exist other alternatives to the derivation of approximate marginal densities, we reconsider the original proposal here and show as in Berkhof et al. (2003) and Lee et al. (2008) that it truly approximates the marginal densities once the label switching issue has been solved.
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