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Bayesian models are a powerful tool for studying complex data, allowing the analyst to encode rich hierarchical dependencies and leverage prior information. Most importantly, they facilitate a complete characterization of uncertainty through the posterior distribution. Practical posterior computation is commonly performed via MCMC, which can be computationally infeasible for high dimensional models with many observations. In this article we discuss the potential to improve posterior computation using ideas from machine learning. Concrete future directions are explored in vignettes on normalizing flows, Bayesian coresets, distributed Bayesian inference, and variational inference.

We show that the posterior distribution of parameters in a hidden Markov model with parametric emission distributions and discrete and known state space is asymptotically normal. The main novelty of our proof is that it is based on a testing condition and the sequence of test functions is obtained using an optimal transportation inequality., 22 pages

Divide-and-conquer Bayesian methods consist of three steps: dividing the data into smaller computationally manageable subsets, running a sampling algorithm in parallel on all the subsets, and combining parameter draws from all the subsets. The combined parameter draws are used for efficient posterior inference in massive data settings. A major restriction of existing divide-and-conquer methods is that their first two steps assume that the observations are independent. We address this problem by developing a divide-and-conquer method for Bayesian inference in parametric hidden Markov models, where the state space is known and finite. Our main contributions are two-fold. First, after partitioning the data into smaller blocks of consecutive observations, we modify the likelihood for performing posterior computations on the subsets such that the posterior variances of the subset and true posterior distributions have the same asymptotic order. Second, if the number of subsets is chosen appropriately depending on the mixing properties of the hidden Markov chain, then we show that the subset posterior distributions defined using the modified likelihood are asymptotically normal as the subset sample size tends to infinity. The latter result also implies that we can use any existing combination algorithm in the third step. We show that the combined posterior distribution obtained using one such algorithm is close to the true posterior distribution in 1-Wasserstein distance under widely used regularity assumptions. Our numerical results show that the proposed method provides an accurate approximation of the true posterior distribution than its competitors in diverse simulation studies and a real data analysis., 51 pages and 8 tables

International Classification of Disease (ICD) codes are widely used for encoding diagnoses in electronic health records (EHR). Automated methods have been developed over the years for predicting biomedical responses using EHR that borrow information among diagnostically similar patients. Relatively less attention has been paid to developing patient similarity measures that model the structure of ICD codes and the presence of multiple chronic conditions, where a chronic condition is defined as a set of ICD codes. Motivated by this problem, we first develop a type of string kernel function for measuring similarity between a pair of subsets of ICD codes, which uses the definition of chronic conditions. Second, we extend this similarity measure to define a family of covariance functions on subsets of ICD codes. Using this family, we develop Gaussian process (GP) priors for Bayesian nonparametric regression and classification using diagnoses and other demographic information as covariates. Markov chain Monte Carlo (MCMC) algorithms are used for posterior inference and predictions. The proposed methods are free of any tuning parameters and are well-suited for automated prediction of continuous and categorical biomedical responses that depend on chronic conditions. We evaluate the practical performance of our method on EHR data collected from 1660 patients at the University of Iowa Hospitals and Clinics (UIHC) with six different primary cancer sites. Our method has better sensitivity and specificity than its competitors in classifying different primary cancer sites and estimates the marginal associations between chronic conditions and primary cancer sites., Comment: 8 tables, 1 figure

We devise survey-weighted pseudo posterior distribution estimators under two-stage informative sampling of both primary clusters and secondary nested units for a one-way analysis of variance (ANOVA) population generating model as a simple canonical case where population model random effects are defined to be coincident with the primary clusters, for example student performance based on a survey of schools and students such as the 2000 OECD Programme for International Student Assessment (PISA). We consider estimation on an observed informative sample under both an augmented pseudo likelihood that co-samples the random effects, as well as an integrated likelihood that marginalizes out the random effects from the survey-weighted augmented pseudo likelihood. This paper includes a theoretical exposition that enumerates easily verified conditions for which estimation under the augmented pseudo posterior is guaranteed to be consistent at the true generating parameters. We reveal in simulation that both approaches produce asymptotically unbiased estimation of the generating hyperparameters for the random effects when a key condition on the sum of within cluster weighted residuals is met. We present a comparison with two frequentist alternatives, an expectation-maximization approach and a composite likelihood method that requires pairwise sampling weights., Comment: 45 pages, 12 figures

Varying coefficient models (VCMs) are widely used for estimating nonlinear regression functions for functional data. Their Bayesian variants using Gaussian process priors on the functional coefficients, however, have received limited attention in massive data applications, mainly due to the prohibitively slow posterior computations using Markov chain Monte Carlo (MCMC) algorithms. We address this problem using a divide-and-conquer Bayesian approach. We first create a large number of data subsamples with much smaller sizes. Then, we formulate the VCM as a linear mixed-effects model and develop a data augmentation algorithm for obtaining MCMC draws on all the subsets in parallel. Finally, we aggregate the MCMC-based estimates of subset posteriors into a single Aggregated Monte Carlo (AMC) posterior, which is used as a computationally efficient alternative to the true posterior distribution. Theoretically, we derive minimax optimal posterior convergence rates for the AMC posteriors of both the varying coefficients and the mean regression function. We provide quantification on the orders of subset sample sizes and the number of subsets. The empirical results show that the combination schemes that satisfy our theoretical assumptions, including the AMC posterior, have better estimation performance than their main competitors across diverse simulations and in a real data analysis.

Monte Carlo algorithms, such as Markov chain Monte Carlo (MCMC) and Hamiltonian Monte Carlo (HMC), are routinely used for Bayesian inference in generalized linear models; however, these algorithms are prohibitively slow in massive data settings because they require multiple passes through the full data in every iteration. Addressing this problem, we develop a scalable extension of these algorithms using the divide-and-conquer (D&C) technique that divides the data into a sufficiently large number of subsets, draws parameters in parallel on the subsets using a \textit{powered} likelihood, and produces Monte Carlo draws of the parameter by combining parameter draws obtained from each subset. These combined parameter draws play the role of draws from the original sampling algorithm. Our main contributions are two-fold. First, we demonstrate through diverse simulated and real data analyses that our distributed algorithm is comparable to the current state-of-the-art D&C algorithm in terms of statistical accuracy and computational efficiency. Second, providing theoretical support for our empirical observations, we identify regularity assumptions under which the proposed algorithm leads to asymptotically optimal inference. We illustrate our methodology through normal linear and logistic regressions, where parts of our D&C algorithm are analytically tractable., 24 pages, 3 Tables

We propose a three-step divide-and-conquer strategy within the Bayesian paradigm that delivers massive scalability for any spatial process model. We partition the data into a large number of subsets, apply a readily available Bayesian spatial process model on every subset, in parallel, and optimally combine the posterior distributions estimated across all the subsets into a pseudo-posterior distribution that conditions on the entire data. The combined pseudo posterior distribution replaces the full data posterior distribution for predicting the responses at arbitrary locations and for inference on the model parameters and spatial surface. Based on distributed Bayesian inference, our approach is called "Distributed Kriging" (DISK) and offers significant advantages in massive data applications where the full data are stored across multiple machines. We show theoretically that the Bayes $L_2$-risk of the DISK posterior distribution achieves the near optimal convergence rate in estimating the true spatial surface with various types of covariance functions, and provide upper bounds for the number of subsets as a function of the full sample size. The model-free feature of DISK is demonstrated by scaling posterior computations in spatial process models with a stationary full-rank and a nonstationary low-rank Gaussian process (GP) prior. A variety of simulations and a geostatistical analysis of the Pacific Ocean sea surface temperature data validate our theoretical results., Comment: 29 pages, including 4 figures and 5 tables

Divide-and-conquer based methods for Bayesian inference provide a general approach for tractable posterior inference when the sample size is large. These methods divide the data into smaller subsets, sample from the posterior distribution of parameters in parallel on all the subsets, and combine posterior samples from all the subsets to approximate the full data posterior distribution. The smaller size of any subset compared to the full data implies that posterior sampling on any subset is computationally more efficient than sampling from the true posterior distribution. Since the combination step takes negligible time relative to sampling, posterior computations can be scaled to massive data by dividing the full data into a sufficiently large number of data subsets. One such approach relies on the geometry of posterior distributions estimated across different subsets and combines them through their barycenter in a Wasserstein space of probability measures. We provide theoretical guarantees on the accuracy of approximation that are valid in many applications. We show that the geometric method approximates the full data posterior distribution better than its competitors across diverse simulations and reproduces known results when applied to a movie ratings database., Comment: 43 pages, 7 figures, and 11 tables. The updated revision will appear in JMLR

The plasmid hik31 operon (P3, slr6039-slr6041) is located on the pSYSX plasmid in Synechocystis sp. PCC 6803. A P3 mutant (ΔP3) had a growth defect in the dark and a pigment defect that was worsened by the addition of glucose. The glucose defect was from incomplete metabolism of the substrate, was pH dependent, and completely overcome by the addition of bicarbonate. Addition of organic carbon and nitrogen sources partly alleviated the defects of the mutant in the dark. Electron micrographs of the mutant revealed larger cells with division defects, glycogen limitation, lack of carboxysomes, deteriorated thylakoids and accumulation of polyhydroxybutyrate and cyanophycin. A microarray experiment over two days of growth in light-dark plus glucose revealed downregulation of several photosynthesis, amino acid biosynthesis, energy metabolism genes; and an upregulation of cell envelope and transport and binding genes in the mutant. ΔP3 had an imbalance in carbon and nitrogen levels and many sugar catabolic and cell division genes were negatively affected after the first dark period. The mutant suffered from oxidative and osmotic stress, macronutrient limitation, and an energy deficit. Therefore, the P3 operon is an important regulator of central metabolism and cell division in the dark.

The promise of Bayesian methods for big data sets has not fully been realized due to the lack of scalable computational algorithms. For massive data, it is necessary to store and process subsets on different machines in a distributed manner. We propose a simple, general, and highly efficient approach, which first runs a posterior sampling algorithm in parallel on different machines for subsets of a large data set. To combine these subset posteriors, we calculate the Wasserstein barycenter via a highly efficient linear program. The resulting estimate for the Wasserstein posterior (WASP) has an atomic form, facilitating straightforward estimation of posterior summaries of functionals of interest. The WASP approach allows posterior sampling algorithms for smaller data sets to be trivially scaled to huge data. We provide theoretical justification in terms of posterior consistency and algorithm efficiency. Examples are provided in complex settings including Gaussian process regression and nonparametric Bayes mixture models.