In this paper, we consider high-dimensional Gaussian graphical models where the true underlying graph is decomposable. A hierarchical $G$-Wishart prior is proposed to conduct a Bayesian inference for the precision matrix and its graph structure. Although the posterior asymptotics using the $G$-Wishart prior has received increasing attention in recent years, most of results assume moderate high-dimensional settings, where the number of variables $p$ is smaller than the sample size $n$. However, this assumption might not hold in many real applications such as genomics, speech recognition and climatology. Motivated by this gap, we investigate asymptotic properties of posteriors under the high-dimensional setting where $p$ can be much larger than $n$. The pairwise Bayes factor consistency, posterior ratio consistency and graph selection consistency are obtained in this high-dimensional setting. Furthermore, the posterior convergence rate for precision matrices under the matrix $\ell_1$-norm is derived, which turns out to coincide with the minimax convergence rate for sparse precision matrices. A simulation study confirms that the proposed Bayesian procedure outperforms competitors.
Gaussian graphical models are a popular tool to learn the dependence structure in the form of a graph among variables of interest. Bayesian methods have gained in popularity in the last two decades due to their ability to simultaneously learn the covariance and the graph and characterize uncertainty in the selection. For scalability of the Markov chain Monte Carlo algorithms, decomposability is commonly imposed on the graph space. A wide variety of graphical conjugate priors are proposed jointly on the covariance matrix and the graph with improved algorithms to search along the space of decomp…
A Bayesian approach is used to estimate the covariance matrix of Gaussian data. Ideas from Gaussian graphical models and model selection are used to construct a prior for the covariance matrix that is a mixture over all decomposable graphs. For this prior the probability of each graph size is specified by the user and graphs of equal size are assigned equal probability. Most previous approaches assume that all graphs are equally probable. We show empirically that the prior that assigns equal probability over graph sizes outperforms the prior that assigns equal probability over all graphs, both…
Bayesian inference for graphical models has received much attention in the literature in recent years. It is well known that when the graph G is decomposable, Bayesian inference is significantly more tractable than in the general non-decomposable setting. Penalized likelihood inference on the other hand has made tremendous gains in the past few years in terms of scalability and tractability. Bayesian inference, however, has not had the same level of success, though a scalable Bayesian approach has its respective strengths, especially in terms of quantifying uncertainty. To address this gap, we…
In this paper we propose a class of prior distributions on decomposable graphs, allowing for improved modeling flexibility. While existing methods solely penalize the number of edges, the proposed work empowers practitioners to control clustering, level of separation, and other features of the graph. Emphasis is placed on a particular prior distribution which derives its motivation from the class of product partition models; the properties of this prior relative to existing priors is examined through theory and simulation. We then demonstrate the use of graphical models in the field of agricul…
This paper considers the problem of defining distributions over graphical structures. We propose an extension of the hyper Markov properties of Dawid and Lauritzen [Ann. Statist. 21 (1993) 1272-1317], which we term structural Markov properties, for both undirected decomposable and directed acyclic graphs, which requires that the structure of distinct components of the graph be conditionally independent given the existence of a separating component. This allows the analysis and comparison of multiple graphical structures, while being able to take advantage of the common conditional independence…
We introduce efficient Markov chain Monte Carlo methods for inference and model determination in multivariate and matrix-variate Gaussian graphical models. Our framework is based on the G-Wishart prior for the precision matrix associated with graphs that can be decomposable or non-decomposable. We extend our sampling algorithms to a novel class of conditionally autoregressive models for sparse estimation in multivariate lattice data, with a special emphasis on the analysis of spatial data. These models embed a great deal of flexibility in estimating both the correlation structure across outcom…
Carnegie Mellon University（卡内基梅隆大学）；Massachusetts Institute of Technology（麻省理工学院）
In this paper we present decomposable priors, a family of priors over structure and parameters of tree belief nets for which Bayesian learning with complete observations is tractable, in the sense that the posterior is also decomposable and can be completely determined analytically in polynomial time. This follows from two main results: First, we show that factored distributions over spanning trees in a graph can be integrated in closed form. Second, we examine priors over tree parameters and show that a set of assumptions similar to (Heckerman and al. 1995) constrain the tree parameter priors…
In this study, we present a multi-class graphical Bayesian predictive classifier that incorporates the uncertainty in the model selection into the standard Bayesian formalism. For each class, the dependence structure underlying the observed features is represented by a set of decomposable Gaussian graphical models. Emphasis is then placed on the Bayesian model averaging which takes full account of the class-specific model uncertainty by averaging over the posterior graph model probabilities. An explicit evaluation of the model probabilities is well known to be infeasible. To address this issue…
We discuss Bayesian inference for parameters selected using the data. First, we provide a critical analysis of the existing positions in the literature regarding the correct Bayesian approach under selection. Second, we propose two types of non-informative priors for selection models. These priors may be employed to produce a posterior distribution in the absence of prior information as well as to provide well-calibrated frequentist inference for the selected parameter. We test the proposed priors empirically in several scenarios.
A common concern with Bayesian methodology in scientific contexts is that inferences can be heavily influenced by subjective biases. As presented here, there are two types of bias for some quantity of interest: bias against and bias in favor. Based upon the principle of evidence, it is shown how to measure and control these biases for both hypothesis assessment and estimation problems. Optimality results are established for the principle of evidence as the basis of the approach to these problems. A close relationship is established between measuring bias in Bayesian inferences and frequentist …
We formulate Bayesian updates in Markov processes by means of path integral techniques and derive the imaginary-time Schrödinger equation with likelihood to direct the inference incorporated as a potential for the posterior probability distribution
We introduce new estimation methods for a sub-class of the Gaussian scale mixture models for wavelet trees by Wainwright, Simoncelli & Willsky that rely on modern results for composite likelihoods and approximate Bayesian inference. Our methodology is illustrated for denoising and edge detection problems in two-dimensional images.
In Bayesian learning of Gaussian graphical model structure, it is common to restrict attention to certain classes of graphs and approximate the posterior distribution by repeatedly moving from one graph to another, using MCMC or methods such as stochastic shotgun search (SSS). I give two corrected versions of an algorithm for non-decomposable graphs and discuss random graph distributions, in particular as prior distributions. The main topic of the thesis is Bayesian structure-learning with forests or trees. Restricting attention to these graphs can be justified using theorems on random graphs.…
Although there is much recent work developing flexible variational methods for Bayesian computation, Gaussian approximations with structured covariance matrices are often preferred computationally in high-dimensional settings. This paper considers approximate inference methods for complex latent variable models where the posterior is close to Gaussian, but with some skewness in the posterior marginals. We consider skew decomposable graphical models (SDGMs), which are based on the closed skew normal family of distributions, as variational approximations. These approximations can reflect the tru…
Bayesian, frequentist and fiducial (BFF) inferences are much more congruous than they have been perceived historically in the scientific community (cf., Reid and Cox 2015; Kass 2011; Efron 1998). Most practitioners are probably more familiar with the two dominant statistical inferential paradigms, Bayesian inference and frequentist inference. The third, lesser known fiducial inference paradigm was pioneered by R.A. Fisher in an attempt to define an inversion procedure for inference as an alternative to Bayes' theorem. Although each paradigm has its own strengths and limitations subject to thei…
We introduce the concepts of Bayesian lens, characterizing the bidirectional structure of exact Bayesian inference, and statistical game, formalizing the optimization objectives of approximate inference problems. We prove that Bayesian inversions compose according to the compositional lens pattern, and exemplify statistical games with a number of classic statistical concepts, from maximum likelihood estimation to generalized variational Bayesian methods. This paper is the first in a series laying the foundations for a compositional account of the theory of active inference, and we therefore pa…
The missionary zeal of many Bayesians of old has been matched, in the other direction, by a view among some theoreticians that Bayesian methods are absurd-not merely misguided but obviously wrong in principle. We consider several examples, beginning with Feller's classic text on probability theory and continuing with more recent cases such as the perceived Bayesian nature of the so-called doomsday argument. We analyze in this note the intellectual background behind various misconceptions about Bayesian statistics, without aiming at a complete historical coverage of the reasons for this dismiss…
This survey reviews the existing literature on the most relevant Bayesian inference methods for univariate and multivariate GARCH models. The advantages and drawbacks of each procedure are outlined as well as the advantages of the Bayesian approach versus classical procedures. The paper makes emphasis on recent Bayesian non-parametric approaches for GARCH models that avoid imposing arbitrary parametric distributional assumptions. These novel approaches implicitly assume infinite mixture of Gaussian distributions on the standardized returns which have been shown to be more flexible and describe…
Fitting models to data using Bayesian inference is quite common, but when each point in parameter space gives a curve, fitting the curve to a data set requires new nuisance parameters, which specify the metric embedding the one-dimensional curve into the higher-dimensional space occupied by the data. A generic formalism for curve fitting in the context of Bayesian inference is developed which shows how the aforementioned metric arises. The result is a natural generalization of previous works, and is compared to oft-used frequentist approaches and similar Bayesian techniques.
We propose a general formalism of iterated random functions with semigroup property, under which exact and approximate Bayesian posterior updates can be viewed as specific instances. A convergence theory for iterated random functions is presented. As an application of the general theory we analyze convergence behaviors of exact and approximate message-passing algorithms that arise in a sequential change point detection problem formulated via a latent variable directed graphical model. The sequential inference algorithm and its supporting theory are illustrated by simulated examples.
The University of Melbourne（墨尔本大学）；Univ-Lille-3（里尔第三大学）；Chalmers University of Technology（查尔姆斯理工大学）
We study how to communicate findings of Bayesian inference to third parties, while preserving the strong guarantee of differential privacy. Our main contributions are four different algorithms for private Bayesian inference on proba-bilistic graphical models. These include two mechanisms for adding noise to the Bayesian updates, either directly to the posterior parameters, or to their Fourier transform so as to preserve update consistency. We also utilise a recently introduced posterior sampling mechanism, for which we prove bounds for the specific but general case of discrete Bayesian network…
The aim of this paper is to introduce new statistical criterions for estimation, suitable for inference in models with common continuous support. This proposal is in the direct line of a renewed interest for divergence based inference tools imbedding the most classical ones, such as maximum likelihood, Chi-square or Kullback Leibler. General pseudodistances with decomposable structure are considered, they allowing to define minimum pseudodistance estimators, without using nonparametric density estimators. A special class of pseudodistances indexed by α>0, leading for α\downarrow0 to the Kulbac…
Istituto Dalle Molle di studi sull’Intelligenza Artificiale (IDSIA)（达莱莫勒人工智能研究所）
For decomposable score-based structure learning of Bayesian networks, existing approaches first compute a collection of candidate parent sets for each variable and then optimize over this collection by choosing one parent set for each variable without creating directed cycles while maximizing the total score. We target the task of constructing the collection of candidate parent sets when the score of choice is the Bayesian Information Criterion (BIC). We provide new non-trivial results that can be used to prune the search space of candidate parent sets of each node. We analyze how these new re…
We address the problem of providing inference from a Bayesian perspective for parameters selected after viewing the data. We present a Bayesian framework for providing inference for selected parameters, based on the observation that providing Bayesian inference for selected parameters is a truncated data problem. We show that if the prior for the parameter is non-informative, or if the parameter is a "fixed" unknown constant, then it is necessary to adjust the Bayesian inference for selection. Our second contribution is the introduction of Bayesian False Discovery Rate controlling methodology,…
Bayesian nonparametric (BNP) models provide elegant methods for discovering underlying latent features within a data set, but inference in such models can be slow. We exploit the fact that completely random measures, which commonly used models like the Dirichlet process and the beta-Bernoulli process can be expressed as, are decomposable into independent sub-measures. We use this decomposition to partition the latent measure into a finite measure containing only instantiated components, and an infinite measure containing all other components. We then select different inference algorithms for t…
Graphical models express conditional independence relationships among variables. Although methods for vector-valued data are well established, functional data graphical models remain underdeveloped. We introduce a notion of conditional independence between random functions, and construct a framework for Bayesian inference of undirected, decomposable graphs in the multivariate functional data context. This framework is based on extending Markov distributions and hyper Markov laws from random variables to random processes, providing a principled alternative to naive application of multivariate m…
BayesPy is an open-source Python software package for performing variational Bayesian inference. It is based on the variational message passing framework and supports conjugate exponential family models. By removing the tedious task of implementing the variational Bayesian update equations, the user can construct models faster and in a less error-prone way. Simple syntax, flexible model construction and efficient inference make BayesPy suitable for both average and expert Bayesian users. It also supports some advanced methods such as stochastic and collapsed variational inference.
The notion of Craig interpolant, used as a form of explanation in automated reasoning, is adapted from logical inference to statistical inference and used to explain inferences made by neural networks. The method produces explanations that are at the same time concise, understandable and precise.
This survey covers state-of-the-art Bayesian techniques for the estimation of mixtures. It complements the earlier Marin, Mengersen and Robert (2005) by studying new types of distributions, the multinomial, latent class and t distributions. It also exhibits closed form solutions for Bayesian inference in some discrete setups. Lastly, it sheds a new light on the computation of Bayes factors via the approximation of Chib (1995).
We consider the problem of estimating high-dimensional covariance matrices of a particular structure, which is a summation of low rank and sparse matrices. This covariance structure has a wide range of applications including factor analysis and random effects models. We propose a Bayesian method of estimating the covariance matrices by representing the covariance model in the form of a factor model with unknown number of latent factors. We introduce binary indicators for factor selection and rank estimation for the low rank component combined with a Bayesian lasso method for the sparse compone…
We propose a quasi-Bayesian method to conduct inference for the production frontier. This approach combines multiple first-stage extreme quantile estimates by the quasi-Bayesian method to produce the point estimate and confidence interval for the production frontier. We show the asymptotic properties of the proposed estimator and the validity of the inference procedure. The finite sample performance of our method is illustrated through simulations and an empirical application.
We present a new probabilistic method for detecting, deblending, and cataloging astronomical sources called the Bayesian Light Source Separator (BLISS). BLISS is based on deep generative models, which embed neural networks within a Bayesian model. For posterior inference, BLISS uses a new form of variational inference known as Forward Amortized Variational Inference. The BLISS inference routine is fast, requiring a single forward pass of the encoder networks on a GPU once the encoder networks are trained. BLISS can perform fully Bayesian inference on megapixel images in seconds, and produces h…
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…
Typical Bayesian inference requires parameter identification via likelihood parameterization, which has invited criticism for being less flexible than the Frequentist framework and subject to misspecification. Though misspecification may be avoided by functional parameter inference under a nonparametric model space, there does not exist a flexible Bayesian semiparametric model that would allow full control over the marginal prior over any general functional parameter. We present the technique of $θ$-augmentation which helps us manipulate nonparametric models into semiparametric ones that direc…
Inspired by the pioneering work of Rubin (1978), we employ the potential outcomes framework to develop a finite-population Bayesian causal inference framework for randomized controlled $2^K$ factorial designs with binary outcomes, which are common in medical research. As demonstrated by simulated and empirical examples, the proposed framework corrects the well-known variance over-estimation issue of the classic "Neymanian" inference framework, under various settings.
Modern deep neural network models suffer from adversarial examples, i.e. confidently misclassified points in the input space. It has been shown that Bayesian neural networks are a promising approach for detecting adversarial points, but careful analysis is problematic due to the complexity of these models. Recently Gilmer et al. (2018) introduced adversarial spheres, a toy set-up that simplifies both practical and theoretical analysis of the problem. In this work, we use the adversarial sphere set-up to understand the properties of approximate Bayesian inference methods for a linear model in a…
We revisit and generalize the concept of composite likelihood as a method to make a probabilistic inference by aggregation of multiple Bayesian agents, thereby defining a class of predictive models which we call composite Bayesian. This perspective gives insight to choose the weights associated with composite likelihood, either a priori or via learning; in the latter case, they may be tuned so as to minimize prediction cross-entropy, yielding an easy-to-solve convex problem. We argue that composite Bayesian inference is a middle way between generative and discriminative models that trades off …
Many modern data analysis problems involve inferences from streaming data. However, streaming data is not easily amenable to the standard probabilistic modeling approaches, which assume that we condition on finite data. We develop population variational Bayes, a new approach for using Bayesian modeling to analyze streams of data. It approximates a new type of distribution, the population posterior, which combines the notion of a population distribution of the data with Bayesian inference in a probabilistic model. We study our method with latent Dirichlet allocation and Dirichlet process mixtur…
This note is commenting on Ronald Gallant's (2015) reflections on the construction of Bayesian prior distributions from moment conditions. The main conclusion is that the paper does not deliver a working principle that could justify inference based on such priors.
This paper provides a review of Approximate Bayesian Computation (ABC) methods for carrying out Bayesian posterior inference, through the lens of density estimation. We describe several recent algorithms and make connection with traditional approaches. We show advantages and limitations of models based on parametric approaches and we then draw attention to developments in machine learning, which we believe have the potential to make ABC scalable to higher dimensions and may be the future direction for research in this area.
In this paper, a scale mixture of Normal distributions model is developed for classification and clustering of data having outliers and missing values. The classification method, based on a mixture model, focuses on the introduction of latent variables that gives us the possibility to handle sensitivity of model to outliers and to allow a less restrictive modelling of missing data. Inference is processed through a Variational Bayesian Approximation and a Bayesian treatment is adopted for model learning, supervised classification and clustering.
University of Cambridge（剑桥大学）；IT University of Copenhagen（哥本哈根信息技术大学）
In this paper, we study the trade-offs of different inference approaches for Bayesian matrix factorisation methods, which are commonly used for predicting missing values, and for finding patterns in the data. In particular, we consider Bayesian nonnegative variants of matrix factorisation and tri-factorisation, and compare non-probabilistic inference, Gibbs sampling, variational Bayesian inference, and a maximum-a-posteriori approach. The variational approach is new for the Bayesian nonnegative models. We compare their convergence, and robustness to noise and sparsity of the data, on both synt…
In this article we consider Bayesian inference for partially observed Andersson-Madigan-Perlman (AMP) Gaussian chain graph (CG) models. Such models are of particular interest in applications such as biological networks and financial time series. The model itself features a variety of constraints which make both prior modeling and computational inference challenging. We develop a framework for the aforementioned challenges, using a sequential Monte Carlo (SMC) method for statistical inference. Our approach is illustrated on both simulated data as well as real case studies from university gradua…
Datasets are growing not just in size but in complexity, creating a demand for rich models and quantification of uncertainty. Bayesian methods are an excellent fit for this demand, but scaling Bayesian inference is a challenge. In response to this challenge, there has been considerable recent work based on varying assumptions about model structure, underlying computational resources, and the importance of asymptotic correctness. As a result, there is a zoo of ideas with few clear overarching principles. In this paper, we seek to identify unifying principles, patterns, and intuitions for scalin…
Balliol College（贝利奥尔学院）；University of Oxford（牛津大学）
Bayesian inference has theoretical attractions as a principled framework for reasoning about beliefs. However, the motivations of Bayesian inference which claim it to be the only 'rational' kind of reasoning do not apply in practice. They create a binary split in which all approximate inference is equally 'irrational'. Instead, we should ask ourselves how to define a spectrum of more- and less-rational reasoning that explains why we might prefer one Bayesian approximation to another. I explore approximate inference in Bayesian neural networks and consider the unintended interactions between th…
Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
We present an adaptive approach to the construction of Gaussian process surrogates for Bayesian inference with expensive-to-evaluate forward models. Our method relies on the fully Bayesian approach to training Gaussian process models and utilizes the expected improvement idea from Bayesian global optimization. We adaptively construct training designs by maximizing the expected improvement in fit of the Gaussian process model to the noisy observational data. Numerical experiments on model problems with synthetic data demonstrate the effectiveness of the obtained adaptive designs compared to the…
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…
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.
For many decades, statisticians have made attempts to prepare the Bayesian omelette without breaking the Bayesian eggs; that is, to obtain probabilistic likelihood-based inferences without relying on informative prior distributions. A recent example is Murray Aitkin's recent book, {\em Statistical Inference}, which presents an approach to statistical hypothesis testing based on comparisons of posterior distributions of likelihoods under competing models. Aitkin develops and illustrates his method using some simple examples of inference from iid data and two-way tests of independence. We analyz…
