Expectation propagation as a way of life: A framework for Bayesian inference on partitioned data

A common approach for Bayesian computation with big data is to partition the data into smaller pieces, perform local inference for each piece separately, and finally combine the results to obtain an approximation to the global posterior. Looking at this from the bottom up, one can perform separate analyses on individual sources of data and then combine these in a larger Bayesian model. In either case, the idea of distributed modeling and inference has both conceptual and computational appeal, but from the Bayesian perspective there is no general way of handling the prior distribution: if the prior is included in each separate inference, it will be multiply-counted when the inferences are combined; but if the prior is itself divided into pieces, it may not provide enough regularization for each separate computation, thus eliminating one of the key advantages of Bayesian methods. To resolve this dilemma, we propose expectation propagation (EP) as a general prototype for distributed Bayesian inference. The central idea is to factor the likelihood according to the data partitions, and to iteratively combine each factor with an approximate model of the prior and all other parts of the data, thus producing an overall approximation to the global posterior at convergence. In this paper, we give an introduction to EP and an overview of some recent developments of the method, with particular emphasis on its use in combining inferences from partitioned data. In addition to distributed modeling of large datasets, our unified treatment also includes hierarchical modeling of data with a naturally partitioned structure. The paper describes a general algorithmic framework, rather than a specific algorithm, and presents an example implementation for it.

Bayesian leave-one-out cross-validation approximations for Gaussian latent variable models

The future predictive performance of a Bayesian model can be estimated using Bayesian cross-validation. In this article, we consider Gaussian latent variable models where the integration over the latent values is approximated using the Laplace method or expectation propagation (EP). We study the properties of several Bayesian leave-one-out (LOO) cross-validation approximations that in most cases can be computed with a small additional cost after forming the posterior approximation given the full data. Our main objective is to assess the accuracy of the approximative LOO cross-validation estimators. That is, for each method (Laplace and EP) we compare the approximate fast computation with the exact brute force LOO computation. Secondarily, we evaluate the accuracy of the Laplace and EP approximations themselves against a ground truth established through extensive Markov chain Monte Carlo simulation. Our empirical results show that the approach based upon a Gaussian approximation to the LOO marginal distribution (the so-called cavity distribution) gives the most accurate and reliable results among the fast methods.