This paper builds on recent developments in Bayesian network (BN) structure learning under the controversial assumption that the input variables are dependent. This assumption can be viewed as a learning constraint geared towards cases where the input variables are known or assumed to be dependent. It addresses the problem of learning multiple disjoint subgraphs that do not enable full propagation of evidence. This problem is highly prevalent in cases where the sample size of the input data is low with respect to the dimensionality of the model, which is often the case when working with real data. The paper presents a novel hybrid structure learning algorithm, called SaiyanH, that addresses this issue. The results show that this constraint helps the algorithm to estimate the number of true edges with higher accuracy compared to the state-of-the-art. Out of the 13 algorithms investigated, the results rank SaiyanH 4th in reconstructing the true DAG, with accuracy scores lower by 8.1% (F1), 10.2% (BSF), and 19.5% (SHD) compared to the top ranked algorithm, and higher by 75.5% (F1), 118% (BSF), and 4.3% (SHD) compared to the bottom ranked algorithm. Overall, the results suggest that the proposed algorithm discovers satisfactorily accurate connected DAGs in cases where other algorithms produce multiple disjoint subgraphs that often underfit the true graph.
University of Michigan（密歇根大学）；DEVCOM Army Research Laboratory（DEVCOM陆军研究实验室）；BI Norwegian Business School（BI挪威商学院）；University of Brescia（布雷西亚大学）
In second-order uncertain Bayesian networks, the conditional probabilities are only known within distributions, i.e., probabilities over probabilities. The delta-method has been applied to extend exact first-order inference methods to propagate both means and variances through sum-product networks derived from Bayesian networks, thereby characterizing epistemic uncertainty, or the uncertainty in the model itself. Alternatively, second-order belief propagation has been demonstrated for polytrees but not for general directed acyclic graph structures. In this work, we extend Loopy Belief Propagat…
This paper presents a new deterministic approximation technique in Bayesian networks. This method, "Expectation Propagation", unifies two previous techniques: assumed-density filtering, an extension of the Kalman filter, and loopy belief propagation, an extension of belief propagation in Bayesian networks. All three algorithms try to recover an approximate distribution which is close in KL divergence to the true distribution. Loopy belief propagation, because it propagates exact belief states, is useful for a limited class of belief networks, such as those which are purely discrete. Expectatio…
Approximate Bayesian inference methods provide a powerful suite of tools for finding approximations to intractable posterior distributions. However, machine learning applications typically involve selecting actions, which -- in a Bayesian setting -- depend on the posterior distribution only via its contribution to expected utility. A growing body of work on loss-calibrated approximate inference methods has therefore sought to develop posterior approximations sensitive to the influence of the utility function. Here we introduce loss-calibrated expectation propagation (Loss-EP), a loss-calibrate…
We propose a novel algorithm to solve the expectation propagation relaxation of Bayesian inference for continuous-variable graphical models. In contrast to most previous algorithms, our method is provably convergent. By marrying convergent EP ideas from (Opper&Winther 05) with covariance decoupling techniques (Wipf&Nagarajan 08, Nickisch&Seeger 09), it runs at least an order of magnitude faster than the most commonly used EP solver.
KTH Royal Institute of Technology（瑞典皇家理工学院）；Stockholm University（斯德哥尔摩大学）
Unsupervised learning of hierarchical representations has been one of the most vibrant research directions in deep learning during recent years. In this work we study biologically inspired unsupervised strategies in neural networks based on local Hebbian learning. We propose new mechanisms to extend the Bayesian Confidence Propagating Neural Network (BCPNN) architecture, and demonstrate their capability for unsupervised learning of salient hidden representations when tested on the MNIST dataset.
This paper makes two contributions to Bayesian machine learning algorithms. Firstly, we propose stochastic natural gradient expectation propagation (SNEP), a novel alternative to expectation propagation (EP), a popular variational inference algorithm. SNEP is a black box variational algorithm, in that it does not require any simplifying assumptions on the distribution of interest, beyond the existence of some Monte Carlo sampler for estimating the moments of the EP tilted distributions. Further, as opposed to EP which has no guarantee of convergence, SNEP can be shown to be convergent, even wh…
University of Regina（里贾纳大学）；Aalborg University（奥尔堡大学）
As Bayesian networks are applied to larger and more complex problem domains, search for flexible modeling and more efficient inference methods is an ongoing effort. Multiply sectioned Bayesian networks (MSBNs) extend the HUGIN inference for Bayesian networks into a coherent framework for flexible modeling and distributed inference.Lazy propagation extends the Shafer-Shenoy and HUGIN inference methods with reduced space complexity. We apply the Shafer-Shenoy and lazy propagation to inference in MSBNs. The combination of the MSBN framework and lazy propagation provides a better framework for mod…
Consider the situation where some evidence e has been entered to a Bayesian network. When performing conflict analysis, sensitivity analysis, or when answering questions like "What if the finding on X had been y instead of x?" you need probabilities P (e'| h), where e' is a subset of e, and h is a configuration of a (possibly empty) set of variables. Cautious propagation is a modification of HUGIN propagation into a Shafer-Shenoy-like architecture. It is less efficient than HUGIN propagation; however, it provides easy access to P (e'| h) for a great deal of relevant subsets e'.
Max Planck Institute for Molecular Genetics（马克斯·普朗克分子遗传学研究所）
We present a Bayesian method for feature selection in the presence of grouping information with sparsity on the between- and within group level. Instead of using a stochastic algorithm for parameter inference, we employ expectation propagation, which is a deterministic and fast algorithm. Available methods for feature selection in the presence of grouping information have a number of short-comings: on one hand, lasso methods, while being fast, underestimate the regression coefficients and do not make good use of the grouping information, and on the other hand, Bayesian approaches, while accura…
This paper presents a Bayesian image segmentation model based on Potts prior and loopy belief propagation. The proposed Bayesian model involves several terms, including the pairwise interactions of Potts models, and the average vectors and covariant matrices of Gauss distributions in color image modeling. These terms are often referred to as hyperparameters in statistical machine learning theory. In order to determine these hyperparameters, we propose a new scheme for hyperparameter estimation based on conditional maximization of entropy in the Potts prior. The algorithm is given based on loop…
In recent years Bayesian networks (BNs) with a mixture of continuous and discrete variables have received an increasing level of attention. We present an architecture for exact belief update in Conditional Linear Gaussian BNs (CLG BNs). The architecture is an extension of lazy propagation using operations of Lauritzen & Jensen [6] and Cowell [2]. By decomposing clique and separator potentials into sets of factors, the proposed architecture takes advantage of independence and irrelevance properties induced by the structure of the graph and the evidence. The resulting benefits are illustrated by…
We describe expectation propagation for approximate inference in dynamic Bayesian networks as a natural extension of Pearl s exact belief propagation.Expectation propagation IS a greedy algorithm, converges IN many practical cases, but NOT always.We derive a DOUBLE - loop algorithm, guaranteed TO converge TO a local minimum OF a Bethe free energy.Furthermore, we show that stable fixed points OF (damped) expectation propagation correspond TO local minima OF this free energy, but that the converse need NOT be the CASE .We illustrate the algorithms BY applying them TO switching linear dynamical s…
Bayesian neural networks (BNNs) have developed into useful tools for probabilistic modelling due to recent advances in variational inference enabling large scale BNNs. However, BNNs remain brittle and hard to train, especially: (1) when using deep architectures consisting of many hidden layers and (2) in situations with large weight variances. We use signal propagation theory to quantify these challenges and propose self-stabilising priors. This is achieved by a reformulation of the ELBO to allow the prior to influence network signal propagation. Then, we develop a stabilising prior, where the…
Recently, researchers have demonstrated that loopy belief propagation - the use of Pearls polytree algorithm IN a Bayesian network WITH loops OF error- correcting codes.The most dramatic instance OF this IS the near Shannon - limit performance OF Turbo Codes codes whose decoding algorithm IS equivalent TO loopy belief propagation IN a chain - structured Bayesian network. IN this paper we ask : IS there something special about the error - correcting code context, OR does loopy propagation WORK AS an approximate inference schemeIN a more general setting? We compare the marginals computed using l…
Aalborg University（奥尔堡大学）；University of Copenhagen（哥本哈根大学）
We propose Unity Smoothing (US) for handling inconsistencies between a Bayesian network model and new unseen observations. We show that prediction accuracy, using the junction tree algorithm with US is comparable to that of Laplace smoothing. Moreover, in applications were sparsity of the data structures is utilized, US outperforms Laplace smoothing in terms of memory usage. Furthermore, we detail how to avoid redundant calculations that must otherwise be performed during the message passing scheme in the junction tree algorithm which we refer to as Unity Propagation (UP). Experimental results…
In this paper we present a synthesis of the work performed on two inference algorithms: the Pearl's belief propagation (BP) algorithm applied to Bayesian networks without loops (i.e. polytree) and the Loopy belief propagation (LBP) algorithm (inspired from the BP) which is applied to networks containing undirected cycles. It is known that the BP algorithm, applied to Bayesian networks with loops, gives incorrect numerical results i.e. incorrect posterior probabilities. Murphy and al. [7] find that the LBP algorithm converges on several networks and when this occurs, LBP gives a good approximat…
In Bayesian networks, exact belief propagation is achieved through message passing algorithms. These algorithms (ex: inward and outward) provide only a recursive definition of the corresponding messages. In contrast, when working on hidden Markov models and variants, one classically first defines explicitly these messages (forward and backward quantities), and then derive all results and algorithms. In this paper, we generalize the hidden Markov model approach by introducing an explicit definition of the messages in Bayesian networks, from which we derive all the relevant properties and result…
Continuous time Bayesian networks (CTBNs) describe structured stochastic processes with finitely many states that evolve over continuous time. A CTBN is a directed (possibly cyclic) dependency graph over a set of variables, each of which represents a finite state continuous time Markov process whose transition model is a function of its parents. As shown previously, exact inference in CTBNs is intractable. We address the problem of approximate inference, allowing for general queries conditioned on evidence over continuous time intervals and at discrete time points. We show how CTBNs can be par…
University of Cambridge（剑桥大学）；Universidad Autónoma de Madrid（马德里自治大学）；Harvard University（哈佛大学）
Deep Gaussian processes (DGPs) are multi-layer hierarchical generalisations of Gaussian processes (GPs) and are formally equivalent to neural networks with multiple, infinitely wide hidden layers. DGPs are nonparametric probabilistic models and as such are arguably more flexible, have a greater capacity to generalise, and provide better calibrated uncertainty estimates than alternative deep models. This paper develops a new approximate Bayesian learning scheme that enables DGPs to be applied to a range of medium to large scale regression problems for the first time. The new method uses an appr…
We show that a probabilistic version of the classical forward-stepwise variable inclusion procedure can serve as a general data-augmentation scheme for model space distributions in (generalized) linear models. This latent variable representation takes the form of a Markov process, thereby allowing information propagation algorithms to be applied for sampling from model space posteriors. In particular, we propose a sequential Monte Carlo method for achieving effective unbiased Bayesian model averaging in high-dimensional problems, utilizing proposal distributions constructed using local informa…
Probabilistic graphical models compactly represent joint distributions by decomposing them into factors over subsets of random variables. In Bayesian networks, the factors are conditional probability distributions. For many problems, common information exists among those factors. Adding similarity restrictions can be viewed as imposing prior knowledge for model regularization. With proper restrictions, learned models usually generalize better. In this work, we study methods that exploit such high-level similarities to regularize the learning process and apply them to the task of modeling the w…
University of Cambridge（剑桥大学）；Harvard University（哈佛大学）
Expectation propagation (EP) is a deterministic approximation algorithm that is often used to perform approximate Bayesian parameter learning. EP approximates the full intractable posterior distribution through a set of local approximations that are iteratively refined for each datapoint. EP can offer analytic and computational advantages over other approximations, such as Variational Inference (VI), and is the method of choice for a number of models. The local nature of EP appears to make it an ideal candidate for performing Bayesian learning on large models in large-scale dataset settings. H…
Recent developments show that Multiply Sectioned Bayesian Networks (MSBNs) can be used for diagnosis of natural systems as well as for model-based diagnosis of artificial systems. They can be applied to single-agent oriented reasoning systems as well as multi-agent distributed probabilistic reasoning systems. Belief propagation between a pair of subnets plays a central role in maintenance of global consistency in a MSBN. This paper studies the operation UpdateBelief, presented originally with MSBNs, for inter-subnet propagation. We analyze how the operation achieves its intended functionality,…
In this paper, we consider Hybrid Mixed Networks (HMN) which are Hybrid Bayesian Networks that allow discrete deterministic information to be modeled explicitly in the form of constraints. We present two approximate inference algorithms for HMNs that integrate and adjust well known algorithmic principles such as Generalized Belief Propagation, Rao-Blackwellised Importance Sampling and Constraint Propagation to address the complexity of modeling and reasoning in HMNs. We demonstrate the performance of our approximate inference algorithms on randomly generated HMNs.
Hong Kong University of Science & Technology（香港科技大学）
This paper explores the role of independence of causal influence (ICI) in Bayesian network inference. ICI allows one to factorize a conditional probability table into smaller pieces. We describe a method for exploiting the factorization in clique tree propagation (CTP) - the state-of-the-art exact inference algorithm for Bayesian networks. We also present empirical results showing that the resulting algorithm is significantly more efficient than the combination of CTP and previous techniques for exploiting ICI.
University of Oxford（牛津大学）；University of Amsterdam（阿姆斯特丹大学）
We investigate the benefit of treating all the parameters in a Bayesian neural network stochastically and find compelling theoretical and empirical evidence that this standard construction may be unnecessary. To this end, we prove that expressive predictive distributions require only small amounts of stochasticity. In particular, partially stochastic networks with only $n$ stochastic biases are universal probabilistic predictors for $n$-dimensional predictive problems. In empirical investigations, we find no systematic benefit of full stochasticity across four different inference modalities an…
We present SPUX - a modular framework for Bayesian inference enabling uncertainty quantification and propagation in linear and nonlinear, deterministic and stochastic models, and supporting Bayesian model selection. SPUX can be coupled to any serial or parallel application written in any programming language, (e.g. including Python, R, Julia, C/C++, Fortran, Java, or a binary executable), scales effortlessly from serial runs on a personal computer to parallel high performance computing clusters, and aims to provide a platform particularly suited to support and foster reproducibility in computa…
Expectation-Propagation for Bayesian Empirical Likelihood Inference
College of Computing and Data Science, Nanyang Technological University（南洋理工大学计算与数据科学学院）；School of Mathematics and Statistics, The University of Melbourne（墨尔本大学数学与统计学院）
A common divide-and-conquer approach for Bayesian computation with big data is to partition the data, perform local inference for each piece separately, and combine the results to obtain a global posterior approximation. While being conceptually and computationally appealing, this method involves the problematic need to also split the prior for the local inferences; these weakened priors may not provide enough regularization for each separate computation, thus eliminating one of the key advantages of Bayesian methods. To resolve this dilemma while still retaining the generalizability of the un…
Calibrated Sampling-Free Uncertainty Estimation in Bayesian Deep Learning
TU Darmstadt & hessian.AI, Darmstadt, Germany（达姆施塔特工业大学 & hessian.AI，德国达姆施塔特）；RIKEN Center for Advanced Intelligence Project, Tokyo, Japan（日本理化学研究所革新智能研究中心，日本东京）
We explain how effective automatic probability density function estimates can be constructed using contemporary Bayesian inference engines such as those based on no-U-turn sampling and expectation propagation. Extensive simulation studies demonstrate that the proposed density estimates have excellent comparative performance and scale well to very large sample sizes due to a binning strategy. Moreover, the approach is fully Bayesian and all estimates are accompanied by pointwise credible intervals. An accompanying package in the R language facilitates easy use of the new density estimates.
We explore the application of a nonlinear MCMC technique first introduced in [1] to problems in Bayesian machine learning. We provide a convergence guarantee in total variation that uses novel results for long-time convergence and large-particle ("propagation of chaos") convergence. We apply this nonlinear MCMC technique to sampling problems including a Bayesian neural network on CIFAR10.
Deep Gaussian processes (DGPs) are multi-layer hierarchical generalisations of Gaussian processes (GPs) and are formally equivalent to neural networks with multiple, infinitely wide hidden layers. DGPs are probabilistic and non-parametric and as such are arguably more flexible, have a greater capacity to generalise, and provide better calibrated uncertainty estimates than alternative deep models. The focus of this paper is scalable approximate Bayesian learning of these networks. The paper develops a novel and efficient extension of probabilistic backpropagation, a state-of-the-art method for …
Precision achieved by stochastic sampling algorithms for Bayesian networks typically deteriorates in face of extremely unlikely evidence. To address this problem, we propose the Evidence Pre-propagation Importance Sampling algorithm (EPIS-BN), an importance sampling algorithm that computes an approximate importance function by the heuristic methods: loopy belief Propagation and e-cutoff. We tested the performance of e-cutoff on three large real Bayesian networks: ANDES, CPCS, and PATHFINDER. We observed that on each of these networks the EPIS-BN algorithm gives us a considerable improvement ov…
I propose a novel approach for nonlinear Logistic regression using a two-layer neural network (NN) model structure with hierarchical priors on the network weights. I present a hybrid of expectation propagation called Variational Expectation Propagation approach (VEP) for approximate integration over the posterior distribution of the weights, the hierarchical scale parameters of the priors and zeta. Using a factorized posterior approximation I derive a computationally efficient algorithm, whose complexity scales similarly to an ensemble of independent sparse logistic models. The approach can be…
In the presence of modeling errors, the mainstream Bayesian methods seldom give a realistic account of uncertainties as they commonly underestimate the inherent variability of parameters. This problem is not due to any misconception in the Bayesian framework since it is absolutely robust with respect to the modeling assumptions and the observed data. Rather, this issue has deep roots in users' inability to develop an appropriate class of probabilistic models. This paper bridges this significant gap, introducing a novel Bayesian hierarchical setting, which breaks time-history vibrational respon…
Rowan University（罗文大学）；University of Sheffield（谢菲尔德大学）
Learning in uncertain, noisy, or adversarial environments is a challenging task for deep neural networks (DNNs). We propose a new theoretically grounded and efficient approach for robust learning that builds upon Bayesian estimation and Variational Inference. We formulate the problem of density propagation through layers of a DNN and solve it using an Ensemble Density Propagation (EnDP) scheme. The EnDP approach allows us to propagate moments of the variational probability distribution across the layers of a Bayesian DNN, enabling the estimation of the mean and covariance of the predictive dis…
University of Stuttgart（斯图加特大学）；Fraunhofer Institute for Manufacturing Engineering and Automation IPA（弗劳恩霍夫制造工程与自动化研究所）
Artificial neural networks (NNs) have become the de facto standard in machine learning. They allow learning highly nonlinear transformations in a plethora of applications. However, NNs usually only provide point estimates without systematically quantifying corresponding uncertainties. In this paper a novel approach towards fully Bayesian NNs is proposed, where training and predictions of a perceptron are performed within the Bayesian inference framework in closed-form. The weights and the predictions of the perceptron are considered Gaussian random variables. Analytical expressions for predict…
This paper considers the computational power of constant size, dynamic Bayesian networks. Although discrete dynamic Bayesian networks are no more powerful than hidden Markov models, dynamic Bayesian networks with continuous random variables and discrete children of continuous parents are capable of performing Turing-complete computation. With modified versions of existing algorithms for belief propagation, such a simulation can be carried out in real time. This result suggests that dynamic Bayesian networks may be more powerful than previously considered. Relationships to causal models and rec…
Politecnico di Torino（都灵理工大学）；Italian Institute for Genomic Medicine（意大利基因组医学研究所）；INFN Sezione di Torino（意大利国家核物理研究所都灵分部）；Amazon Alexa（亚马逊Alexa）
Efficient feature selection from high-dimensional datasets is a very important challenge in many data-driven fields of science and engineering. We introduce a statistical mechanics inspired strategy that addresses the problem of sparse feature selection in the context of binary classification by leveraging a computational scheme known as expectation propagation (EP). The algorithm is used in order to train a continuous-weights perceptron learning a classification rule from a set of (possibly partly mislabeled) examples provided by a teacher perceptron with diluted continuous weights. We test t…
Record linkage (de-duplication or entity resolution) is the process of merging noisy databases to remove duplicate entities. While record linkage removes duplicate entities from such databases, the downstream task is any inferential, predictive, or post-linkage task on the linked data. One goal of the downstream task is obtaining a larger reference data set, allowing one to perform more accurate statistical analyses. In addition, there is inherent record linkage uncertainty passed to the downstream task. Motivated by the above, we propose a generalized Bayesian record linkage method and consid…
Instituto de Investigacion en Ingenieria de Aragon（阿拉贡工程研究院）；University of Zaragoza（萨拉戈萨大学）；Centro Universitario de la Defensa（国防大学中心）
Bayesian optimization has become a popular method for high-throughput computing, like the design of computer experiments or hyperparameter tuning of expensive models, where sample efficiency is mandatory. In these applications, distributed and scalable architectures are a necessity. However, Bayesian optimization is mostly sequential. Even parallel variants require certain computations between samples, limiting the parallelization bandwidth. Thompson sampling has been previously applied for distributed Bayesian optimization. But, when compared with other acquisition functions in the sequential…
EURECOM（欧洲通信研究院）；University of California, Irvine（加州大学欧文分校）
Autoencoders and their variants are among the most widely used models in representation learning and generative modeling. However, autoencoder-based models usually assume that the learned representations are i.i.d. and fail to capture the correlations between the data samples. To address this issue, we propose a novel Sparse Gaussian Process Bayesian Autoencoder (SGPBAE) model in which we impose fully Bayesian sparse Gaussian Process priors on the latent space of a Bayesian Autoencoder. We perform posterior estimation for this model via stochastic gradient Hamiltonian Monte Carlo. We evaluate …
Carnegie Mellon University（卡内基梅隆大学）；University of California, Berkeley（加州大学伯克利分校）
Compiling Bayesian networks (BNs) to junction trees and performing belief propagation over them is among the most prominent approaches to computing posteriors in BNs. However, belief propagation over junction tree is known to be computationally intensive in the general case. Its complexity may increase dramatically with the connectivity and state space cardinality of Bayesian network nodes. In this paper, we address this computational challenge using GPU parallelization. We develop data structures and algorithms that extend existing junction tree techniques, and specifically develop a novel ap…
Kavli Institute for Cosmology（卡弗里宇宙学研究所）；University of Cambridge（剑桥大学）；University of Sussex（萨塞克斯大学）
The true posterior distribution of a Bayesian neural network is massively multimodal. Whilst most of these modes are functionally equivalent, we demonstrate that there remains a level of real multimodality that manifests in even the simplest neural network setups. It is only by fully marginalising over all posterior modes, using appropriate Bayesian sampling tools, that we can capture the split personalities of the network. The ability of a network trained in this manner to reason between multiple candidate solutions dramatically improves the generalisability of the model, a feature we contend…
Real engineering and scientific applications often involve one or more qualitative inputs. Standard Gaussian processes (GPs), however, cannot directly accommodate qualitative inputs. The recently introduced latent variable Gaussian process (LVGP) overcomes this issue by first mapping each qualitative factor to underlying latent variables (LVs), and then uses any standard GP covariance function over these LVs. The LVs are estimated similarly to the other GP hyperparameters through maximum likelihood estimation, and then plugged into the prediction expressions. However, this plug-in approach wil…
National Institute of Informatics（国立信息学研究所）；Tokyo Institute of Technology（东京工业大学）
The stochastic block model (SBM) is a generative model revealing macroscopic structures in graphs. Bayesian methods are used for (i) cluster assignment inference and (ii) model selection for the number of clusters. In this paper, we study the behavior of Bayesian inference in the SBM in the large sample limit. Combining variational approximation and Laplace's method, a consistent criterion of the fully marginalized log-likelihood is established. Based on that, we derive a tractable algorithm that solves tasks (i) and (ii) concurrently, obviating the need for an outer loop to check all model ca…
KTH Royal Institute of Technology（KTH皇家理工学院）；Huawei Technologies Sweden AB（华为技术瑞典有限公司）
Belief propagation (BP) can do exact inference in loop-free graphs, but its performance could be poor in graphs with loops, and the understanding of its solution is limited. This work gives an interpretable belief propagation rule that is actually minimization of a localized $α$-divergence. We term this algorithm as $α$ belief propagation ($α$-BP). The performance of $α$-BP is tested in MAP (maximum a posterior) inference problems, where $α$-BP can outperform (loopy) BP by a significant margin even in fully-connected graphs.
University of Cambridge（剑桥大学）；The Alan Turing Institute（艾伦·图灵研究所）
Learning in Gaussian Process models occurs through the adaptation of hyperparameters of the mean and the covariance function. The classical approach entails maximizing the marginal likelihood yielding fixed point estimates (an approach called \textit{Type II maximum likelihood} or ML-II). An alternative learning procedure is to infer the posterior over hyperparameters in a hierarchical specification of GPs we call \textit{Fully Bayesian Gaussian Process Regression} (GPR). This work considers two approximation schemes for the intractable hyperparameter posterior: 1) Hamiltonian Monte Carlo (HMC…
University of Pennsylvania（宾夕法尼亚大学）；Medical College of Wisconsin（威斯康星医学院）；Arizona State University（亚利桑那州立大学）
Bayesian Additive Regression Trees (BART) is a fully Bayesian approach to modeling with ensembles of trees. BART can uncover complex regression functions with high dimensional regressors in a fairly automatic way and provide Bayesian quantification of the uncertainty through the posterior. However, BART assumes IID normal errors. This strong parametric assumption can lead to misleading inference and uncertainty quantification. In this paper, we use the classic Dirichlet process mixture (DPM) mechanism to nonparametrically model the error distribution. A key strength of BART is that default pri…
