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Register a new userComputational Solutions for Bayesian Inference in Mixture Models
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Gertraud Malsiner-Walli
Publication date
2018
fields
Mathematical Statistics
and research's language is
English
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This chapter surveys the most standard Monte Carlo methods available for simulating from a posterior distribution associated with a mixture and conducts some experiments about the robustness of the Gibbs sampler in high dimensional Gaussian settings. This is a chapter prepared for the forthcoming Handbook of Mixture Analysis.
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We introduce computational causal inference as an interdisciplinary field across causal inference, algorithms design and numerical computing. The field aims to develop software specializing in causal inference that can analyze massive datasets with a variety of causal effects, in a performant, general, and robust way. The focus on software improves research agility, and enables causal inference to be easily integrated into large engineering systems. In particular, we use computational causal inference to deepen the relationship between causal inference, online experimentation, and algorithmic decision making. This paper describes the new field, the demand, opportunities for scalability, open challenges, and begins the discussion for how the community can unite to solve challenges for scaling causal inference and decision making.
In this paper, we introduce efficient ensemble Markov Chain Monte Carlo (MCMC) sampling methods for Bayesian computations in the univariate stochastic volatility model. We compare the performance of our ensemble MCMC methods with an improved version of a recent sampler of Kastner and Fruwirth-Schnatter (2014). We show that ensemble samplers are more efficient than this state of the art sampler by a factor of about 3.1, on a data set simulated from the stochastic volatility model. This performance gain is achieved without the ensemble MCMC sampler relying on the assumption that the latent process is linear and Gaussian, unlike the sampler of Kastner and Fruwirth-Schnatter.
We study the class of state-space models and perform maximum likelihood estimation for the model parameters. We consider a stochastic approximation expectation-maximization (SAEM) algorithm to maximize the likelihood function with the novelty of using approximate Bayesian computation (ABC) within SAEM. The task is to provide each iteration of SAEM with a filtered state of the system, and this is achieved using an ABC sampler for the hidden state, based on sequential Monte Carlo (SMC) methodology. It is shown that the resulting SAEM-ABC algorithm can be calibrated to return accurate inference, and in some situations it can outperform a version of SAEM incorporating the bootstrap filter. Two simulation studies are presented, first a nonlinear Gaussian state-space model then a state-space model having dynamics expressed by a stochastic differential equation. Comparisons with iterated filtering for maximum likelihood inference, and Gibbs sampling and particle marginal methods for Bayesian inference are presented.
We address the problem of providing inference from a Bayesian perspective for parameters selected after viewing the data. We present a Bayesian framework for providing inference for selected parameters, based on the observation that providing Bayesian inference for selected parameters is a truncated data problem. We show that if the prior for the parameter is non-informative, or if the parameter is a fixed unknown constant, then it is necessary to adjust the Bayesian inference for selection. Our second contribution is the introduction of Bayesian False Discovery Rate controlling methodology,which generalizes existing Bayesian FDR methods that are only defined in the two-group mixture model.We illustrate our results by applying them to simulated data and data froma microarray experiment.
In this article, we discuss two specific classes of models - Gaussian Mixture Copula models and Mixture of Factor Analyzers - and the advantages of doing inference with gradient descent using automatic differentiation. Gaussian mixture models are a popular class of clustering methods, that offers a principled statistical approach to clustering. However, the underlying assumption, that every mixing component is normally distributed, can often be too rigid for several real life datasets. In order to to relax the assumption about the normality of mixing components, a new class of parametric mixture models that are based on Copula functions - Gaussian Mixuture Copula Models were introduced. Estimating the parameters of the proposed Gaussian Mixture Copula Model (GMCM) through maximum likelihood has been intractable due to the positive semi-positive-definite constraints on the variance-covariance matrices. Previous attempts were limited to maximizing a proxy-likelihood which can be maximized using EM algorithm. These existing methods, even though easier to implement, does not guarantee any convergence nor monotonic increase of the GMCM Likelihood. In this paper, we use automatic differentiation tools to maximize the exact likelihood of GMCM, at the same time avoiding any constraint equations or Lagrange multipliers. We show how our method leads a monotonic increase in likelihood and converges to a (local) optimum value of likelihood. In this paper, we also show how Automatic Differentiation can be used for inference with Mixture of Factor Analyzers and advantages of doing so. We also discuss how this method also has all the properties such as monotonic increase in likelihood and convergence to a local optimum. Note that our work is also applicable to special cases of these two models - for e.g. Simple Copula models, Factor Analyzer model, etc.
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