There is a lack of simple and scalable algorithms for uncertainty quantification. Bayesian methods quantify uncertainty through posterior and predictive distributions, but it is difficult to rapidly estimate summaries of these distributions, such as quantiles and intervals. Variational Bayes approximations are widely used, but may badly underestimate posterior covariance. Typically, the focus of Bayesian inference is on point and interval estimates for one-dimensional functionals of interest. In small scale problems, Markov chain Monte Carlo algorithms remain the gold standard, but such algorithms face major problems in scaling up to big data. Various modifications have been proposed based on parallelization and approximations based on subsamples, but such approaches are either highly complex or lack theoretical support and/or good performance outside of narrow settings. We propose a very simple and general posterior interval estimation algorithm, which is based on running Markov chain Monte Carlo in parallel for subsets of the data and averaging quantiles estimated from each subset. We provide strong theoretical guarantees and illustrate performance in several applications.
Divide-and-conquer strategies for Monte Carlo algorithms are an increasingly popular approach to making Bayesian inference scalable to large data sets. In its simplest form, the data are partitioned across multiple computing cores and a separate Markov chain Monte Carlo algorithm on each core targets the associated partial posterior distribution, which we refer to as a sub-posterior, that is the posterior given only the data from the segment of the partition associated with that core. Divide-and-conquer techniques reduce computational, memory and disk bottle-necks, but make it difficult to rec…
Few Bayesian methods for analyzing high-dimensional sparse survival data provide scalable variable selection, effect estimation and uncertainty quantification. Such methods often either sacrifice uncertainty quantification by computing maximum a posteriori estimates, or quantify the uncertainty at high (unscalable) computational expense. We bridge this gap and develop an interpretable and scalable Bayesian proportional hazards model for prediction and variable selection, referred to as SVB. Our method, based on a mean-field variational approximation, overcomes the high computational cost of MC…
The Cut posterior and related Semi-Modular Inference are Generalised Bayes methods for Modular Bayesian evidence combination. Analysis is broken up over modular sub-models of the joint posterior distribution. Model-misspecification in multi-modular models can be hard to fix by model elaboration alone and the Cut posterior and SMI offer a way round this. Information entering the analysis from misspecified modules is controlled by an influence parameter $η$ related to the learning rate. This paper contains two substantial new methods. First, we give variational methods for approximating the Cut …
Several structure-learning algorithms for staged trees, asymmetric extensions of Bayesian networks, have been proposed. However, these either do not scale efficiently as the number of variables considered increases, a priori restrict the set of models, or they do not find comparable models to existing methods. Here, we define an alternative algorithm based on a totally ordered hyperstage. We demonstrate how it can be used to obtain a quadratically-scaling structural learning algorithm for staged trees that restricts the model space a-posteriori. Through comparative analysis, we show that throu…
The paper addresses joint sparsity selection in the regression coefficient matrix and the error precision (inverse covariance) matrix for high-dimensional multivariate regression models in the Bayesian paradigm. The selected sparsity patterns are crucial to help understand the network of relationships between the predictor and response variables, as well as the conditional relationships among the latter. While Bayesian methods have the advantage of providing natural uncertainty quantification through posterior inclusion probabilities and credible intervals, current Bayesian approaches either r…
CISPA Helmholtz Center for Information Security（CISPA亥姆霍兹信息安全中心）；University of Oxford（牛津大学）
We propose a new method to approximate the posterior distribution of probabilistic programs by means of computing guaranteed bounds. The starting point of our work is an interval-based trace semantics for a recursive, higher-order probabilistic programming language with continuous distributions. Taking the form of (super-/subadditive) measures, these lower/upper bounds are non-stochastic and provably correct: using the semantics, we prove that the actual posterior of a given program is sandwiched between the lower and upper bounds (soundness); moreover the bounds converge to the posterior (com…
We develop scalable methods for producing conformal Bayesian predictive intervals with finite sample calibration guarantees. Bayesian posterior predictive distributions, $p(y \mid x)$, characterize subjective beliefs on outcomes of interest, $y$, conditional on predictors, $x$. Bayesian prediction is well-calibrated when the model is true, but the predictive intervals may exhibit poor empirical coverage when the model is misspecified, under the so called ${\cal{M}}$-open perspective. In contrast, conformal inference provides finite sample frequentist guarantees on predictive confidence interva…
Optimization-based samplers such as randomize-then-optimize (RTO) [2] provide an efficient and parallellizable approach to solving large-scale Bayesian inverse problems. These methods solve randomly perturbed optimization problems to draw samples from an approximate posterior distribution. "Correcting" these samples, either by Metropolization or importance sampling, enables characterization of the original posterior distribution. This paper focuses on the scalability of RTO to problems with high- or infinite-dimensional parameters. We introduce a new subspace acceleration strategy that makes t…
The automation of posterior inference in Bayesian data analysis has enabled experts and nonexperts alike to use more sophisticated models, engage in faster exploratory modeling and analysis, and ensure experimental reproducibility. However, standard automated posterior inference algorithms are not tractable at the scale of massive modern datasets, and modifications to make them so are typically model-specific, require expert tuning, and can break theoretical guarantees on inferential quality. Building on the Bayesian coresets framework, this work instead takes advantage of data redundancy to s…
Xidian University（西安电子科技大学）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
To build a flexible and interpretable model for document analysis, we develop deep autoencoding topic model (DATM) that uses a hierarchy of gamma distributions to construct its multi-stochastic-layer generative network. In order to provide scalable posterior inference for the parameters of the generative network, we develop topic-layer-adaptive stochastic gradient Riemannian MCMC that jointly learns simplex-constrained global parameters across all layers and topics, with topic and layer specific learning rates. Given a posterior sample of the global parameters, in order to efficiently infer th…
Generalized linear models (GLMs) -- such as logistic regression, Poisson regression, and robust regression -- provide interpretable models for diverse data types. Probabilistic approaches, particularly Bayesian ones, allow coherent estimates of uncertainty, incorporation of prior information, and sharing of power across experiments via hierarchical models. In practice, however, the approximate Bayesian methods necessary for inference have either failed to scale to large data sets or failed to provide theoretical guarantees on the quality of inference. We propose a new approach based on constru…
We present an extensible software framework, hIPPYlib, for solution of large-scale deterministic and Bayesian inverse problems governed by partial differential equations (PDEs) with infinite-dimensional parameter fields (which are high-dimensional after discretization). hIPPYlib overcomes the prohibitive nature of Bayesian inversion for this class of problems by implementing state-of-the-art scalable algorithms for PDE-based inverse problems that exploit the structure of the underlying operators, notably the Hessian of the log-posterior. The key property of the algorithms implemented in hIPPYl…
