The paper proposes a new hybrid Bayesian network learning algorithm, termed Forward Early Dropping Hill Climbing (FEDHC), devised to work with either continuous or categorical variables. Further, the paper manifests that the only implementation of MMHC in the statistical software \textit{R}, is prohibitively expensive and a new implementation is offered. Further, specifically for the case of continuous data, a robust to outliers version of FEDHC, that can be adopted by other BN learning algorithms, is proposed. The FEDHC is tested via Monte Carlo simulations that distinctly show it is computationally efficient, and produces Bayesian networks of similar to, or of higher accuracy than MMHC and PCHC. Finally, an application of FEDHC, PCHC and MMHC algorithms to real data, from the field of economics, is demonstrated using the statistical software \textit{R}.
Bayesian Decision Trees are known for their probabilistic interpretability. However, their construction can sometimes be costly. In this article we present a general Bayesian Decision Tree algorithm applicable to both regression and classification problems. The algorithm does not apply Markov Chain Monte Carlo and does not require a pruning step. While it is possible to construct a weighted probability tree space we find that one particular tree, the greedy-modal tree (GMT), explains most of the information contained in the numerical examples. This approach seems to perform similarly to Random…
The last decade witnessed a growing interest in Bayesian learning. Yet, the technicality of the topic and the multitude of ingredients involved therein, besides the complexity of turning theory into practical implementations, limit the use of the Bayesian learning paradigm, preventing its widespread adoption across different fields and applications. This self-contained survey engages and introduces readers to the principles and algorithms of Bayesian Learning for Neural Networks. It provides an introduction to the topic from an accessible, practical-algorithmic perspective. Upon providing a ge…
This paper uses Gaussian mixture model instead of linear Gaussian model to fit the distribution of every node in Bayesian network. We will explain why and how we use Gaussian mixture models in Bayesian network. Meanwhile we propose a new method, called double iteration algorithm, to optimize the mixture model, the double iteration algorithm combines the expectation maximization algorithm and gradient descent algorithm, and it performs perfectly on the Bayesian network with mixture models. In experiments we test the Gaussian mixture model and the optimization algorithm on different graphs which…
K. N. Toosi University of Technology（K.N.托西理工大学）；Polytechnique Montréal（蒙特利尔理工学院）
This paper addresses the problem of learning a sparse structure Bayesian network from high-dimensional discrete data. Compared to continuous Bayesian networks, learning a discrete Bayesian network is a challenging problem due to the large parameter space. Although many approaches have been developed for learning continuous Bayesian networks, few approaches have been proposed for the discrete ones. In this paper, we address learning Bayesian networks as an optimization problem and propose a score function which guarantees the learnt structure to be a sparse directed acyclic graph. Besides, we i…
Queensland University of Technology（昆士兰科技大学）；University of Alberta（阿尔伯塔大学）
We study Bayesian optimal control of a general class of smoothly parameterized Markov decision problems. Since computing the optimal control is computationally expensive, we design an algorithm that trades off performance for computational efficiency. The algorithm is a lazy posterior sampling method that maintains a distribution over the unknown parameter. The algorithm changes its policy only when the variance of the distribution is reduced sufficiently. Importantly, we analyze the algorithm and show the precise nature of the performance vs. computation tradeoff. Finally, we show the effecti…
Univ Lyon, UJM-Saint-Etienne, CNRS, Institut d Optique Graduate School, Laboratoire Hubert Curien UMR 5516（里昂大学，圣艾蒂安大学，法国国家科学研究中心，光学研究所研究生院，于贝尔·居里安实验室）；Université Laval（拉瓦尔大学）
In the PAC-Bayesian literature, the C-Bound refers to an insightful relation between the risk of a majority vote classifier (under the zero-one loss) and the first two moments of its margin (i.e., the expected margin and the voters' diversity). Until now, learning algorithms developed in this framework minimize the empirical version of the C-Bound, instead of explicit PAC-Bayesian generalization bounds. In this paper, by directly optimizing PAC-Bayesian guarantees on the C-Bound, we derive self-bounding majority vote learning algorithms. Moreover, our algorithms based on gradient descent are s…
Ohio State University（俄亥俄州立大学）；UC Berkeley（加州大学伯克利分校）
Bayesian models offer great flexibility for clustering applications---Bayesian nonparametrics can be used for modeling infinite mixtures, and hierarchical Bayesian models can be utilized for sharing clusters across multiple data sets. For the most part, such flexibility is lacking in classical clustering methods such as k-means. In this paper, we revisit the k-means clustering algorithm from a Bayesian nonparametric viewpoint. Inspired by the asymptotic connection between k-means and mixtures of Gaussians, we show that a Gibbs sampling algorithm for the Dirichlet process mixture approaches a h…
Chalmers University of Technology（查尔姆斯理工大学）；University of Oslo（奥斯陆大学）
Bayesian reinforcement learning (BRL) offers a decision-theoretic solution for reinforcement learning. While "model-based" BRL algorithms have focused either on maintaining a posterior distribution on models or value functions and combining this with approximate dynamic programming or tree search, previous Bayesian "model-free" value function distribution approaches implicitly make strong assumptions or approximations. We describe a novel Bayesian framework, Inferential Induction, for correctly inferring value function distributions from data, which leads to the development of a new class of B…
Using Bayes's theorem, we derive a unit-wise recurrence as well as a backward recursion similar to the forward-backward algorithm. The resulting Bayesian recurrent units can be integrated as recurrent neural networks within deep learning frameworks, while retaining a probabilistic interpretation from the direct correspondence with hidden Markov models. Whilst the contribution is mainly theoretical, experiments on speech recognition indicate that adding the derived units at the end of state-of-the-art recurrent architectures can improve the performance at a very low cost in terms of trainable p…
University of California, San Diego（加利福尼亚大学圣地亚哥分校）；The Ohio State University（俄亥俄州立大学）
In this paper, we present an algorithm for the sparse signal recovery problem that incorporates damped Gaussian generalized approximate message passing (GGAMP) into Expectation-Maximization (EM)-based sparse Bayesian learning (SBL). In particular, GGAMP is used to implement the E-step in SBL in place of matrix inversion, leveraging the fact that GGAMP is guaranteed to converge with appropriate damping. The resulting GGAMP-SBL algorithm is much more robust to arbitrary measurement matrix $\boldsymbol{A}$ than the standard damped GAMP algorithm while being much lower complexity than the standard…
This paper describes a new library for learning Bayesian networks from data containing discrete and continuous variables (mixed data). In addition to the classical learning methods on discretized data, this library proposes its algorithm that allows structural learning and parameters learning from mixed data without discretization since data discretization leads to information loss. This algorithm based on mixed MI score function for structural learning, and also linear regression and Gaussian distribution approximation for parameters learning. The library also offers two algorithms for enumer…
Information Fusion and Mining Laboratory（信息融合与挖掘实验室）
In this paper, we will provide an introduction to the derivative-free optimization algorithms which can be potentially applied to train deep learning models. Existing deep learning model training is mostly based on the back propagation algorithm, which updates the model variables layers by layers with the gradient descent algorithm or its variants. However, the objective functions of deep learning models to be optimized are usually non-convex and the gradient descent algorithms based on the first-order derivative can get stuck into the local optima very easily. To resolve such a problem, vario…
Universidad Politécnica de Madrid（马德里理工大学）；Universidad Autónoma de Madrid（马德里自治大学）；Departamento de Inteligencia Artificial（人工智能系）；Departamento de Ingeniería Informática（信息工程系）
The PC algorithm is a popular method for learning the structure of Gaussian Bayesian networks. It carries out statistical tests to determine absent edges in the network. It is hence governed by two parameters: (i) The type of test, and (ii) its significance level. These parameters are usually set to values recommended by an expert. Nevertheless, such an approach can suffer from human bias, leading to suboptimal reconstruction results. In this paper we consider a more principled approach for choosing these parameters in an automatic way. For this we optimize a reconstruction score evaluated on …
Finding an optimal parameter of a black-box function is important for searching stable material structures and finding optimal neural network structures, and Bayesian optimization algorithms are widely used for the purpose. However, most of existing Bayesian optimization algorithms can only handle vector data and cannot handle complex structured data. In this paper, we propose the topological Bayesian optimization, which can efficiently find an optimal solution from structured data using \emph{topological information}. More specifically, in order to apply Bayesian optimization to structured da…
University of New Mexico（新墨西哥大学）；Google, Inc（谷歌公司）
Bayesian network structure learning algorithms with limited data are being used in domains such as systems biology and neuroscience to gain insight into the underlying processes that produce observed data. Learning reliable networks from limited data is difficult, therefore transfer learning can improve the robustness of learned networks by leveraging data from related tasks. Existing transfer learning algorithms for Bayesian network structure learning give a single maximum a posteriori estimate of network models. Yet, many other models may be equally likely, and so a more informative result i…
As an automatic method of determining model complexity using the training data alone, Bayesian linear regression provides us a principled way to select hyperparameters. But one often needs approximation inference if distribution assumption is beyond Gaussian distribution. In this paper, we propose a Bayesian linear regression model with Student-t assumptions (BLRS), which can be inferred exactly. In this framework, both conjugate prior and expectation maximization (EM) algorithm are generalized. Meanwhile, we prove that the maximum likelihood solution is equivalent to the standard Bayesian lin…
Inria Lille - Nord Europe（法国国家信息与自动化研究所里尔-北欧中心）；University College London（伦敦大学学院）
Generalised Bayesian learning algorithms are increasingly popular in machine learning, due to their PAC generalisation properties and flexibility. The present paper aims at providing a self-contained survey on the resulting PAC-Bayes framework and some of its main theoretical and algorithmic developments.
We study a decentralized cooperative multi-agent multi-armed bandit problem with $K$ arms and $N$ agents connected over a network. In our model, each arm's reward distribution is same for all agents, and rewards are drawn independently across agents and over time steps. In each round, agents choose an arm to play and subsequently send a message to their neighbors. The goal is to minimize cumulative regret averaged over the entire network. We propose a decentralized Bayesian multi-armed bandit framework that extends single-agent Bayesian bandit algorithms to the decentralized setting. Specifica…
Bayesian optimization has shown to be a fundamental global optimization algorithm in many applications: ranging from automatic machine learning, robotics, reinforcement learning, experimental design, simulations, etc. The most popular and effective Bayesian optimization relies on a surrogate model in the form of a Gaussian process due to its flexibility to represent a prior over function. However, many algorithms and setups relies on the stationarity assumption of the Gaussian process. In this paper, we present a novel nonstationary strategy for Bayesian optimization that is able to outperform…
This paper extends the work in [Suzuki, 1996] and presents an efficient depth-first branch-and-bound algorithm for learning Bayesian network structures, based on the minimum description length (MDL) principle, for a given (consistent) variable ordering. The algorithm exhaustively searches through all network structures and guarantees to find the network with the best MDL score. Preliminary experiments show that the algorithm is efficient, and that the time complexity grows slowly with the sample size. The algorithm is useful for empirically studying both the performance of suboptimal heuristic…
Princeton University（普林斯顿大学）；University of Pennsylvania（宾夕法尼亚大学）
We consider the correlated multiarmed bandit (MAB) problem in which the rewards associated with each arm are modeled by a multivariate Gaussian random variable, and we investigate the influence of the assumptions in the Bayesian prior on the performance of the upper credible limit (UCL) algorithm and a new correlated UCL algorithm. We rigorously characterize the influence of accuracy, confidence, and correlation scale in the prior on the decision-making performance of the algorithms. Our results show how priors and correlation structure can be leveraged to improve performance.
Bayesian Reinforcement Learning (RL) is capable of not only incorporating domain knowledge, but also solving the exploration-exploitation dilemma in a natural way. As Bayesian RL is intractable except for special cases, previous work has proposed several approximation methods. However, these methods are usually too sensitive to parameter values, and finding an acceptable parameter setting is practically impossible in many applications. In this paper, we propose a new algorithm that greedily approximates Bayesian RL to achieve robustness in parameter space. We show that for a desired learning b…
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.
School of Mathematical and Data Science（数学与数据科学学院）；West Virginia University（西弗吉尼亚大学）；Department of Chemical and Biomedical Engineering（化学与生物医学工程系）；The Medical College of Wisconsin（威斯康星医学院）；Department of Biostatistics and Bioinformatics（生物统计学与生物信息学系）
We present K-Means Batch Bayesian Optimization (KMBBO), a novel batch sampling algorithm for Bayesian Optimization (BO). KMBBO uses unsupervised learning to efficiently estimate peaks of the model acquisition function. We show in empirical experiments that our method outperforms the current state-of-the-art batch allocation algorithms on a variety of test problems including tuning of algorithm hyper-parameters and a challenging drug discovery problem. In order to accommodate the real-world problem of high dimensional data, we propose a modification to KMBBO by combining it with compressed sens…
Université de Lyon（里昂大学）；CNRS（法国国家科学研究中心）；Université Lyon 1（里昂第一大学）
We present a novel hybrid algorithm for Bayesian network structure learning, called Hybrid HPC (H2PC). It first reconstructs the skeleton of a Bayesian network and then performs a Bayesian-scoring greedy hill-climbing search to orient the edges. It is based on a subroutine called HPC, that combines ideas from incremental and divide-and-conquer constraint-based methods to learn the parents and children of a target variable. We conduct an experimental comparison of H2PC against Max-Min Hill-Climbing (MMHC), which is currently the most powerful state-of-the-art algorithm for Bayesian network stru…
Coherent uncertainty quantification is a key strength of Bayesian methods. But modern algorithms for approximate Bayesian posterior inference often sacrifice accurate posterior uncertainty estimation in the pursuit of scalability. This work shows that previous Bayesian coreset construction algorithms---which build a small, weighted subset of the data that approximates the full dataset---are no exception. We demonstrate that these algorithms scale the coreset log-likelihood suboptimally, resulting in underestimated posterior uncertainty. To address this shortcoming, we develop greedy iterative …
University of Milan-Bicocca（米兰比可卡大学）；Istituto Dalle Molle di Studi sull’Intelligenza Artificiale (IDSIA)（达勒莫勒人工智能研究所（IDSIA））
Dynamic Bayesian networks have been well explored in the literature as discrete-time models: however, their continuous-time extensions have seen comparatively little attention. In this paper, we propose the first constraint-based algorithm for learning the structure of continuous-time Bayesian networks. We discuss the different statistical tests and the underlying hypotheses used by our proposal to establish conditional independence. Furthermore, we analyze and discuss the computational complexity of the best and worst cases for the proposed algorithm. Finally, we validate its performance usin…
A statistical algorithm for categorizing different types of matches and fraud in image databases is presented. The approach is based on a generative model of a graph representing images and connections between pairs of identities, trained using properties of a matching algorithm between images.
We present new algorithms for learning Bayesian networks from data with missing values using a data augmentation approach. An exact Bayesian network learning algorithm is obtained by recasting the problem into a standard Bayesian network learning problem without missing data. To the best of our knowledge, this is the first exact algorithm for this problem. As expected, the exact algorithm does not scale to large domains. We build on the exact method to create an approximate algorithm using a hill-climbing technique. This algorithm scales to large domains so long as a suitable standard structur…
Distributed inference/estimation in Bayesian framework in the context of sensor networks has recently received much attention due to its broad applicability. The variational Bayesian (VB) algorithm is a technique for approximating intractable integrals arising in Bayesian inference. In this paper, we propose two novel distributed VB algorithms for general Bayesian inference problem, which can be applied to a very general class of conjugate-exponential models. In the first approach, the global natural parameters at each node are optimized using a stochastic natural gradient that utilizes the Ri…
National University of Defense Technology（国防科技大学）；Samsung R&D Institute America - Dallas（三星美国研究院（达拉斯））
The performance of sparse signal recovery from noise corrupted, underdetermined measurements can be improved if both sparsity and correlation structure of signals are exploited. One typical correlation structure is the intra-block correlation in block sparse signals. To exploit this structure, a framework, called block sparse Bayesian learning (BSBL), has been proposed recently. Algorithms derived from this framework showed superior performance but they are not very fast, which limits their applications. This work derives an efficient algorithm from this framework, using a marginalized likelih…
Machine learning provides algorithms that can learn from data and make inferences or predictions on data. Bayesian networks are a class of graphical models that allow to represent a collection of random variables and their condititional dependencies by directed acyclic graphs. In this paper, an inference algorithm for the hidden random variables of a Bayesian network is given by using the tropicalization of the marginal distribution of the observed variables. By restricting the topological structure to graded networks, an inference algorithm for graded Bayesian networks will be established tha…
Unsupervised image segmentation aims at clustering the set of pixels of an image into spatially homogeneous regions. We introduce here a class of Bayesian nonparametric models to address this problem. These models are based on a combination of a Potts-like spatial smoothness component and a prior on partitions which is used to control both the number and size of clusters. This class of models is flexible enough to include the standard Potts model and the more recent Potts-Dirichlet Process model \cite{Orbanz2008}. More importantly, any prior on partitions can be introduced to control the globa…
In this paper, we present a Bayesian channel estimation algorithm for multicarrier receivers based on pilot symbol observations. The inherent sparse nature of wireless multipath channels is exploited by modeling the prior distribution of multipath components' gains with a hierarchical representation of the Bessel K probability density function; a highly efficient, fast iterative Bayesian inference method is then applied to the proposed model. The resulting estimator outperforms other state-of-the-art Bayesian and non-Bayesian estimators, either by yielding lower mean squared estimation error o…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Brown University（布朗大学）；University of Chicago（芝加哥大学）
A popular approach to semi-supervised learning proceeds by endowing the input data with a graph structure in order to extract geometric information and incorporate it into a Bayesian framework. We introduce new theory that gives appropriate scalings of graph parameters that provably lead to a well-defined limiting posterior as the size of the unlabeled data set grows. Furthermore, we show that these consistency results have profound algorithmic implications. When consistency holds, carefully designed graph-based Markov chain Monte Carlo algorithms are proved to have a uniform spectral gap, ind…
We present new multi-test Bayesian optimization models and algorithms for use in large scale material screening applications. Our screening problems are designed around two tests, one expensive and one cheap. This paper differs from other recent work on multi-test Bayesian optimization through use of a flexible model that allows for complex, non-linear relationships between the cheap and expensive test scores. This additional modeling flexibility is essential in the material screening applications which we describe. We demonstrate the power of our new algorithms on a family of synthetic toy pr…
In this paper, we consider Bayesian image denoising based on a Gaussian Markov random field (GMRF) model, for which we propose an new algorithm. Our method can solve Bayesian image denoising problems, including hyperparameter estimation, in $O(n)$-time, where $n$ is the number of pixels in a given image. From the perspective of the order of the computational time, this is a state-of-the-art algorithm for the present problem setting. Moreover, the results of our numerical experiments we show our method is in fact effective in practice.
This paper is about index policies for minimizing (frequentist) regret in a stochastic multi-armed bandit model, inspired by a Bayesian view on the problem. Our main contribution is to prove that the Bayes-UCB algorithm, which relies on quantiles of posterior distributions, is asymptotically optimal when the reward distributions belong to a one-dimensional exponential family, for a large class of prior distributions. We also show that the Bayesian literature gives new insight on what kind of exploration rates could be used in frequentist, UCB-type algorithms. Indeed, approximations of the Baye…
We use the minorization-maximization principle (Lange, Hunter and Yang 2000) to establish the monotonicity of a multiplicative algorithm for computing Bayesian D-optimal designs. This proves a conjecture of Dette, Pepelyshev and Zhigljavsky (2008).
Arizona State University（亚利桑那州立大学）；Boston University（波士顿大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
We overview some results on distributed learning with focus on a family of recently proposed algorithms known as non-Bayesian social learning. We consider different approaches to the distributed learning problem and its algorithmic solutions for the case of finitely many hypotheses. The original centralized problem is discussed at first, and then followed by a generalization to the distributed setting. The results on convergence and convergence rate are presented for both asymptotic and finite time regimes. Various extensions are discussed such as those dealing with directed time-varying netwo…
In literature there are several studies on the performance of Bayesian network structure learning algorithms. The focus of these studies is almost always the heuristics the learning algorithms are based on, i.e. the maximisation algorithms (in score-based algorithms) or the techniques for learning the dependencies of each variable (in constraint-based algorithms). In this paper we investigate how the use of permutation tests instead of parametric ones affects the performance of Bayesian network structure learning from discrete data. Shrinkage tests are also covered to provide a broad overview …
In this rejoinder we summarize the comments, questions and remarks on the paper "A novel algorithmic approach to Bayesian Logic Regression" from the discussants. We then respond to those comments, questions and remarks, provide several extensions of the original model and give a tutorial on our R-package EMJMCMC (http://aliaksah.github.io/EMJMCMC2016/)
University of Illinois at Chicago（伊利诺伊大学芝加哥分校）；Shanghai University of Finance and Economics（上海财经大学）；Institute of Advanced Study（高等研究院）；Shanghai Jiao Tong University（上海交通大学）
We study the problem of learning Bayesian networks where an $ε$-fraction of the samples are adversarially corrupted. We focus on the fully-observable case where the underlying graph structure is known. In this work, we present the first nearly-linear time algorithm for this problem with a dimension-independent error guarantee. Previous robust algorithms with comparable error guarantees are slower by at least a factor of $(d/ε)$, where $d$ is the number of variables in the Bayesian network and $ε$ is the fraction of corrupted samples. Our algorithm and analysis are considerably simpler than tho…
In many real-world problems, we want to infer some property of an expensive black-box function $f$, given a budget of $T$ function evaluations. One example is budget constrained global optimization of $f$, for which Bayesian optimization is a popular method. Other properties of interest include local optima, level sets, integrals, or graph-structured information induced by $f$. Often, we can find an algorithm $\mathcal{A}$ to compute the desired property, but it may require far more than $T$ queries to execute. Given such an $\mathcal{A}$, and a prior distribution over $f$, we refer to the pro…
Explosive growth in data and availability of cheap computing resources have sparked increasing interest in Big learning, an emerging subfield that studies scalable machine learning algorithms, systems, and applications with Big Data. Bayesian methods represent one important class of statistic methods for machine learning, with substantial recent developments on adaptive, flexible and scalable Bayesian learning. This article provides a survey of the recent advances in Big learning with Bayesian methods, termed Big Bayesian Learning, including nonparametric Bayesian methods for adaptively inferr…
Bayesian algorithmic perfumery: A Hierarchical Relevance Vector Machine for the Estimation of Personalized Fragrance Preferences based on Three Sensory Layers and Jungian Personality Archetypes
Brain tumor segmentation from Magnetic Resonance Images (MRIs) is an important task to measure tumor responses to treatments. However, automatic segmentation is very challenging. This paper presents an automatic brain tumor segmentation method based on a Normalized Gaussian Bayesian classification and a new 3D Fluid Vector Flow (FVF) algorithm. In our method, a Normalized Gaussian Mixture Model (NGMM) is proposed and used to model the healthy brain tissues. Gaussian Bayesian Classifier is exploited to acquire a Gaussian Bayesian Brain Map (GBBM) from the test brain MR images. GBBM is further p…
Linear dynamical systems are canonical models for learning-based control of plants with uncertain dynamics. The setting consists of a stochastic differential equation that captures the state evolution of the plant understudy, while the true dynamics matrices are unknown and need to be learned from the observed data of state trajectory. An important issue is to ensure that the system is stabilized and destabilizing control actions due to model uncertainties are precluded as soon as possible. A reliable stabilization procedure for this purpose that can effectively learn from unstable data to sta…
Verifying the correctness of Bayesian computation is challenging. This is especially true for complex models that are common in practice, as these require sophisticated model implementations and algorithms. In this paper we introduce \emph{simulation-based calibration} (SBC), a general procedure for validating inferences from Bayesian algorithms capable of generating posterior samples. This procedure not only identifies inaccurate computation and inconsistencies in model implementations but also provides graphical summaries that can indicate the nature of the problems that arise. We argue that…
In this paper we consider the problem of model choice for a set of insurance loss ratios. We use a reversible jump algorithm for our model discrimination and show how the vanilla reversible jump algorithm can be improved on using recent methodological advances in reversible jump computation.
