Exact Bayesian inference via data augmentation

Data augmentation is a common tool in Bayesian statistics, especially in the application of MCMC. Data augmentation is used where direct computation of the posterior density, π(θ|x), of the parameters θ, given the observed data x, is not possible. We show that for a range of problems, it is possible to augment the data by y, such that, π(θ|x,y) is known, and π(y|x) can easily be computed. In particular, π(y|x) is obtained by collapsing π(y,θ|x) through integrating out θ. This allows the exact computation of π(θ|x) as a mixture distribution without recourse to approximating methods such as MCMC. Useful byproducts of the exact posterior distribution are the marginal likelihood of the model and the exact predictive distribution

Efficient data augmentation for fitting stochastic epidemic models to prevalence data

Stochastic epidemic models describe the dynamics of an epidemic as a disease spreads through a population. Typically, only a fraction of cases are observed at a set of discrete times. The absence of complete information about the time evolution of an epidemic gives rise to a complicated latent variable problem in which the state space size of the epidemic grows large as the population size increases. This makes analytically integrating over the missing data infeasible for populations of even moderate size. We present a data augmentation Markov chain Monte Carlo (MCMC) framework for Bayesian estimation of stochastic epidemic model parameters, in which measurements are augmented with subject-level disease histories. In our MCMC algorithm, we propose each new subject-level path, conditional on the data, using a time-inhomogeneous continuous-time Markov process with rates determined by the infection histories of other individuals. The method is general, and may be applied, with minimal modifications, to a broad class of stochastic epidemic models. We present our algorithm in the context of multiple stochastic epidemic models in which the data are binomially sampled prevalence counts, and apply our method to data from an outbreak of influenza in a British boarding school

Conjugate Bayes for probit regression via unified skew-normal distributions

Regression models for dichotomous data are ubiquitous in statistics. Besides being useful for inference on binary responses, these methods serve also as building blocks in more complex formulations, such as density regression, nonparametric classification and graphical models. Within the Bayesian framework, inference proceeds by updating the priors for the coefficients, typically set to be Gaussians, with the likelihood induced by probit or logit regressions for the responses. In this updating, the apparent absence of a tractable posterior has motivated a variety of computational methods, including Markov Chain Monte Carlo routines and algorithms which approximate the posterior. Despite being routinely implemented, Markov Chain Monte Carlo strategies face mixing or time-inefficiency issues in large p and small n studies, whereas approximate routines fail to capture the skewness typically observed in the posterior. This article proves that the posterior distribution for the probit coefficients has a unified skew-normal kernel, under Gaussian priors. Such a novel result allows efficient Bayesian inference for a wide class of applications, especially in large p and small-to-moderate n studies where state-of-the-art computational methods face notable issues. These advances are outlined in a genetic study, and further motivate the development of a wider class of conjugate priors for probit models along with methods to obtain independent and identically distributed samples from the unified skew-normal posterior

Discriminative Nonparametric Latent Feature Relational Models with Data Augmentation

We present a discriminative nonparametric latent feature relational model (LFRM) for link prediction to automatically infer the dimensionality of latent features. Under the generic RegBayes (regularized Bayesian inference) framework, we handily incorporate the prediction loss with probabilistic inference of a Bayesian model; set distinct regularization parameters for different types of links to handle the imbalance issue in real networks; and unify the analysis of both the smooth logistic log-loss and the piecewise linear hinge loss. For the nonconjugate posterior inference, we present a simple Gibbs sampler via data augmentation, without making restricting assumptions as done in variational methods. We further develop an approximate sampler using stochastic gradient Langevin dynamics to handle large networks with hundreds of thousands of entities and millions of links, orders of magnitude larger than what existing LFRM models can process. Extensive studies on various real networks show promising performance.Comment: Accepted by AAAI 201