We address the problem of Bayesian structure learning for domains with hundreds of variables by employing non-parametric bootstrap, recursively. We propose a method that covers both model averaging and model selection in the same framework. The proposed method deals with the main weakness of constraint-based learning---sensitivity to errors in the independence tests---by a novel way of combining bootstrap with constraint-based learning. Essentially, we provide an algorithm for learning a tree, in which each node represents a scored CPDAG for a subset of variables and the level of the node corresponds to the maximal order of conditional independencies that are encoded in the graph. As higher order independencies are tested in deeper recursive calls, they benefit from more bootstrap samples, and therefore more resistant to the curse-of-dimensionality. Moreover, the re-use of stable low order independencies allows greater computational efficiency. We also provide an algorithm for sampling CPDAGs efficiently from their posterior given the learned tree. We empirically demonstrate that the proposed algorithm scales well to hundreds of variables, and learns better MAP models and more reliable causal relationships between variables, than other state-of-the-art-methods.
In this paper, we employ variational arguments to establish a connection between ensemble methods for Neural Networks and Bayesian inference. We consider an ensemble-based scheme where each model/particle corresponds to a perturbation of the data by means of parametric bootstrap and a perturbation of the prior. We derive conditions under which any optimization steps of the particles makes the associated distribution reduce its divergence to the posterior over model parameters. Such conditions do not require any particular form for the approximation and they are purely geometrical, giving insig…
In this paper we revisit the weighted likelihood bootstrap, a method that generates samples from an approximate Bayesian posterior of a parametric model. We show that the same method can be derived, without approximation, under a Bayesian nonparametric model with the parameter of interest defined as minimising an expected negative log-likelihood under an unknown sampling distribution. This interpretation enables us to extend the weighted likelihood bootstrap to posterior sampling for parameters minimizing an expected loss. We call this method the loss-likelihood bootstrap. We make a connection…
University of Trieste（的里雅斯特大学）；University of Milan-Bicocca（米兰比可卡大学）
Learning the structure of dependencies among multiple random variables is a problem of considerable theoretical and practical interest. Within the context of Bayesian Networks, a practical and surprisingly successful solution to this learning problem is achieved by adopting score-functions optimisation schema, augmented with multiple restarts to avoid local optima. Yet, the conditions under which such strategies work well are poorly understood, and there are also some intrinsic limitations to learning the directionality of the interaction among the variables. Following an early intuition of Fr…
University of Warwick（华威大学）；University College London（伦敦大学学院）
Simulator-based models are models for which the likelihood is intractable but simulation of synthetic data is possible. They are often used to describe complex real-world phenomena, and as such can often be misspecified in practice. Unfortunately, existing Bayesian approaches for simulators are known to perform poorly in those cases. In this paper, we propose a novel algorithm based on the posterior bootstrap and maximum mean discrepancy estimators. This leads to a highly-parallelisable Bayesian inference algorithm with strong robustness properties. This is demonstrated through an in-depth the…
In this article, we present data-subsetting algorithms that allow for the approximate and scalable implementation of the Bayesian bootstrap. They are analogous to two existing algorithms in the frequentist literature: the bag of little bootstraps (Kleiner et al., 2014) and the subsampled double bootstrap (SDB; Sengupta et al., 2016). Our algorithms have appealing theoretical and computational properties that are comparable to those of their frequentist counterparts. Additionally, we provide a strategy for performing lossless inference for a class of functionals of the Bayesian bootstrap, and b…
Channel Pruning, widely used for accelerating Convolutional Neural Networks, is an NP-hard problem due to the inter-layer dependency of channel redundancy. Existing methods generally ignored the above dependency for computation simplicity. To solve the problem, under the Bayesian framework, we here propose a layer-wise Recursive Bayesian Pruning method (RBP). A new dropout-based measurement of redundancy, which facilitate the computation of posterior assuming inter-layer dependency, is introduced. Specifically, we model the noise across layers as a Markov chain and target its posterior to refl…
For a Bayesian, the task to define the likelihood can be as perplexing as the task to define the prior. We focus on situations when the parameter of interest has been emancipated from the likelihood and is linked to data directly through a loss function. We survey existing work on both Bayesian parametric inference with Gibbs posteriors as well as Bayesian non-parametric inference. We then highlight recent bootstrap computational approaches to approximating loss-driven posteriors. In particular, we focus on implicit bootstrap distributions defined through an underlying push-forward mapping. We…
University of Oxford（牛津大学）；The Alan Turing Institute（艾伦·图灵研究所）
Increasingly complex datasets pose a number of challenges for Bayesian inference. Conventional posterior sampling based on Markov chain Monte Carlo can be too computationally intensive, is serial in nature and mixes poorly between posterior modes. Further, all models are misspecified, which brings into question the validity of the conventional Bayesian update. We present a scalable Bayesian nonparametric learning routine that enables posterior sampling through the optimization of suitably randomized objective functions. A Dirichlet process prior on the unknown data distribution accounts for mo…
We develop a weighted Bayesian Bootstrap (WBB) for machine learning and statistics. WBB provides uncertainty quantification by sampling from a high dimensional posterior distribution. WBB is computationally fast and scalable using only off-theshelf optimization software such as TensorFlow. We provide regularity conditions which apply to a wide range of machine learning and statistical models. We illustrate our methodology in regularized regression, trend filtering and deep learning. Finally, we conclude with directions for future research.
Bayesian models provide recursive inference naturally because they can formally reconcile new data and existing scientific information. However, popular use of Bayesian methods often avoids priors that are based on exact posterior distributions resulting from former studies. Two existing Recursive Bayesian methods are: Prior- and Proposal-Recursive Bayes. Prior-Recursive Bayes uses Bayesian updating, fitting models to partitions of data sequentially, and provides a way to accommodate new data as they become available using the posterior from the previous stage as the prior in the new stage bas…
University of Bristol（布里斯托大学）；Spotify（声田）；University of Bath（巴斯大学）；University of Cambridge（剑桥大学）
We present a novel approach to Bayesian inference and general Bayesian computation that is defined through a sequential decision loop. Our method defines a recursive partitioning of the sample space. It neither relies on gradients nor requires any problem-specific tuning, and is asymptotically exact for any density function with a bounded domain. The output is an approximation to the whole density function including the normalisation constant, via partitions organised in efficient data structures. Such approximations may be used for evidence estimation or fast posterior sampling, but also as b…
National University of Singapore（新加坡国立大学）；Massachusetts Institute of Technology（麻省理工学院）
This paper presents a recursive reasoning formalism of Bayesian optimization (BO) to model the reasoning process in the interactions between boundedly rational, self-interested agents with unknown, complex, and costly-to-evaluate payoff functions in repeated games, which we call Recursive Reasoning-Based BO (R2-B2). Our R2-B2 algorithm is general in that it does not constrain the relationship among the payoff functions of different agents and can thus be applied to various types of games such as constant-sum, general-sum, and common-payoff games. We prove that by reasoning at level 2 or more a…
Systems that are based on recursive Bayesian updates for classification limit the cost of evidence collection through certain stopping/termination criteria and accordingly enforce decision making. Conventionally, two termination criteria based on pre-defined thresholds over (i) the maximum of the state posterior distribution; and (ii) the state posterior uncertainty are commonly used. In this paper, we propose a geometric interpretation over the state posterior progression and accordingly we provide a point-by-point analysis over the disadvantages of using such conventional termination criteri…
Clustering procedures typically estimate which data points are clustered together, a quantity of primary importance in many analyses. Often used as a preliminary step for dimensionality reduction or to facilitate interpretation, finding robust and stable clusters is often crucial for appropriate for downstream analysis. In the present work, we consider Bayesian nonparametric (BNP) models, a particularly popular set of Bayesian models for clustering due to their flexibility. Because of its complexity, the Bayesian posterior often cannot be computed exactly, and approximations must be employed. …
A major focus of causal inference is the estimation of heterogeneous average treatment effects (HTE) - average treatment effects within strata of another variable of interest such as levels of a biomarker, education, or age strata. Inference involves estimating a stratum-specific regression and integrating it over the distribution of confounders in that stratum - which itself must be estimated. Standard practice involves estimating these stratum-specific confounder distributions independently (e.g. via the empirical distribution or Rubin's Bayesian bootstrap), which becomes problematic for spa…
Accurate diagnosis of disease is of great importance in clinical practice and medical research. The receiver operating characteristic (ROC) surface is a popular tool for evaluating the discriminatory ability of continuous diagnostic test outcomes when there exist three ordered disease classes (e.g., no disease, mild disease, advanced disease). We propose the Bayesian bootstrap, a fully nonparametric method, for conducting inference about the ROC surface and its functionals, such as the volume under the surface. The proposed method is based on a simple, yet interesting, representation of the RO…
We propose a novel Bayesian method to solve the maximization of a time-dependent expensive-to-evaluate oracle. We are interested in the decision that maximizes the oracle at a finite time horizon, when relatively few noisy evaluations can be performed before the horizon. Our recursive, two-step lookahead expected payoff ($\texttt{r2LEY}$) acquisition function makes nonmyopic decisions at every stage by maximizing the estimated expected value of the oracle at the horizon. $\texttt{r2LEY}$ circumvents the evaluation of the expensive multistep (more than two steps) lookahead acquisition function …
Northeastern University（东北大学）；University of Pittsburgh（匹兹堡大学）
Recursive Bayesian inference (RBI) provides optimal Bayesian latent variable estimates in real-time settings with streaming noisy observations. Active RBI attempts to effectively select queries that lead to more informative observations to rapidly reduce uncertainty until a confident decision is made. However, typically the optimality objectives of inference and query mechanisms are not jointly selected. Furthermore, conventional active querying methods stagger due to misleading prior information. Motivated by information theoretic approaches, we propose an active RBI framework with unified in…
Online Bayesian system identification in multivariate autoregressive models via message passing
Eindhoven Artificial Intelligence Systems Institute（埃因霍温人工智能系统研究所）；Department of Electrical Engineering, Eindhoven University of Technology（电子工程系，埃因霍温理工大学）
We propose Posterior Bootstrap, a set of algorithms extending Weighted Likelihood Bootstrap, to properly incorporate prior information and address the problem of model misspecification in Bayesian inference. We consider two approaches to incorporating prior knowledge: the first is based on penalization of the Weighted Likelihood Bootstrap objective function, and the second uses pseudo-samples from the prior predictive distribution. We also propose methodology for hierarchical models, which was not previously known for methods based on Weighted Likelihood Bootstrap. Edgeworth expansions guide t…
University of Maryland, College Park（马里兰大学帕克分校）；Amazon Web Services（亚马逊云科技）；Amazon.com, Inc.（亚马逊公司）
Bayesian learning of model parameters in neural networks is important in scenarios where estimates with well-calibrated uncertainty are important. In this paper, we propose Bayesian quantized networks (BQNs), quantized neural networks (QNNs) for which we learn a posterior distribution over their discrete parameters. We provide a set of efficient algorithms for learning and prediction in BQNs without the need to sample from their parameters or activations, which not only allows for differentiable learning in QNNs, but also reduces the variance in gradients. We evaluate BQNs on MNIST, Fashion-MN…
Many recent invertible neural architectures are based on coupling block designs where variables are divided in two subsets which serve as inputs of an easily invertible (usually affine) triangular transformation. While such a transformation is invertible, its Jacobian is very sparse and thus may lack expressiveness. This work presents a simple remedy by noting that subdivision and (affine) coupling can be repeated recursively within the resulting subsets, leading to an efficiently invertible block with dense, triangular Jacobian. By formulating our recursive coupling scheme via a hierarchical …
Recent findings suggest that abnormal operating conditions of equipment in the oil and gas supply chain represent a large fraction of anthropogenic methane emissions. Thus, effective mitigation of emissions necessitates rapid identification and repair of sources caused by faulty equipment. In addition to advances in sensing technology that allow for more frequent surveillance, prompt and cost-effective identification of sources requires computational frameworks that provide automatic fault detection. Here, we present a changepoint detection algorithm based on a recursive Bayesian scheme that a…
We introduce a new sequential methodology to calibrate the fixed parameters and track the stochastic dynamical variables of a state-space system. The proposed method is based on the nested hybrid filtering (NHF) framework of [1], that combines two layers of filters, one inside the other, to compute the joint posterior probability distribution of the static parameters and the state variables. In particular, we explore the use of deterministic sampling techniques for Gaussian approximation in the first layer of the algorithm, instead of the Monte Carlo methods employed in the original procedure.…
Identifying changes in the generative process of sequential data, known as changepoint detection, has become an increasingly important topic for a wide variety of fields. A recently developed approach, which we call EXact Online Bayesian Changepoint Detection (EXO), has shown reasonable results with efficient computation for real time updates. The method is based on a \textit{forward} recursive message-passing algorithm. However, the detected changepoints from these methods are unstable. We propose a new algorithm called Lagged EXact Online Bayesian Changepoint Detection (LEXO) that improves t…
