The Poisson distribution arises naturally when dealing with data involving counts, and it has found many applications in inverse problems and imaging. In this work, we develop an approximate Bayesian inference technique based on expectation propagation for approximating the posterior distribution formed from the Poisson likelihood function and a Laplace type prior distribution, e.g., the anisotropic total variation prior. The approach iteratively yields a Gaussian approximation, and at each iteration, it updates the Gaussian approximation to one factor of the posterior distribution by moment matching. We derive explicit update formulas in terms of one-dimensional integrals, and also discuss stable and efficient quadrature rules for evaluating these integrals. The method is showcased on two-dimensional PET images.
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.
Variational methods have been recently considered for scaling the training process of Gaussian process classifiers to large datasets. As an alternative, we describe here how to train these classifiers efficiently using expectation propagation. The proposed method allows for handling datasets with millions of data instances. More precisely, it can be used for (i) training in a distributed fashion where the data instances are sent to different nodes in which the required computations are carried out, and for (ii) maximizing an estimate of the marginal likelihood using a stochastic approximation …
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…
A method for large scale Gaussian process classification has been recently proposed based on expectation propagation (EP). Such a method allows Gaussian process classifiers to be trained on very large datasets that were out of the reach of previous deployments of EP and has been shown to be competitive with related techniques based on stochastic variational inference. Nevertheless, the memory resources required scale linearly with the dataset size, unlike in variational methods. This is a severe limitation when the number of instances is very large. Here we show that this problem is avoided wh…
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 …
Aalto University（阿尔托大学）；Technical University of Denmark（丹麦技术大学）
We formulate approximate Bayesian inference in non-conjugate temporal and spatio-temporal Gaussian process models as a simple parameter update rule applied during Kalman smoothing. This viewpoint encompasses most inference schemes, including expectation propagation (EP), the classical (Extended, Unscented, etc.) Kalman smoothers, and variational inference. We provide a unifying perspective on these algorithms, showing how replacing the power EP moment matching step with linearisation recovers the classical smoothers. EP provides some benefits over the traditional methods via introduction of th…
We consider the inverse problem of reconstructing the posterior measure over the trajec- tories of a diffusion process from discrete time observations and continuous time constraints. We cast the problem in a Bayesian framework and derive approximations to the posterior distributions of single time marginals using variational approximate inference. We then show how the approximation can be extended to a wide class of discrete-state Markov jump pro- cesses by making use of the chemical Langevin equation. Our empirical results show that the proposed method is computationally efficient and provid…
This paper presents a new Expectation Propagation (EP) framework for image restoration using patch-based prior distributions. While Monte Carlo techniques are classically used to sample from intractable posterior distributions, they can suffer from scalability issues in high-dimensional inference problems such as image restoration. To address this issue, EP is used here to approximate the posterior distributions using products of multivariate Gaussian densities. Moreover, imposing structural constraints on the covariance matrices of these densities allows for greater scalability and distribute…
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…
This paper describes an expectation propagation (EP) method for multi-class classification with Gaussian processes that scales well to very large datasets. In such a method the estimate of the log-marginal-likelihood involves a sum across the data instances. This enables efficient training using stochastic gradients and mini-batches. When this type of training is used, the computational cost does not depend on the number of data instances $N$. Furthermore, extra assumptions in the approximate inference process make the memory cost independent of $N$. The consequence is that the proposed EP met…
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 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 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 consider probabilistic multinomial probit classification using Gaussian process (GP) priors. The challenges with the multiclass GP classification are the integration over the non-Gaussian posterior distribution, and the increase of the number of unknown latent variables as the number of target classes grows. Expectation propagation (EP) has proven to be a very accurate method for approximate inference but the existing EP approaches for the multinomial probit GP classification rely on numerical quadratures or independence assumptions between the latent values from different classes to facili…
We propose a novel approach for nonlinear regression using a two-layer neural network (NN) model structure with sparsity-favoring hierarchical priors on the network weights. We present an expectation propagation (EP) approach for approximate integration over the posterior distribution of the weights, the hierarchical scale parameters of the priors, and the residual scale. Using a factorized posterior approximation we derive a computationally efficient algorithm, whose complexity scales similarly to an ensemble of independent sparse linear models. The approach enables flexible definition of wei…
Gaussian processes (GPs) are flexible distributions over functions that enable high-level assumptions about unknown functions to be encoded in a parsimonious, flexible and general way. Although elegant, the application of GPs is limited by computational and analytical intractabilities that arise when data are sufficiently numerous or when employing non-Gaussian models. Consequently, a wealth of GP approximation schemes have been developed over the last 15 years to address these key limitations. Many of these schemes employ a small set of pseudo data points to summarise the actual data. In this…
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…
Bayesian inference is a popular method to build learning algorithms but it is hampered by the fact that its key object, the posterior probability distribution, is often uncomputable. Expectation Propagation (EP) (Minka (2001)) is a popular algorithm that solves this issue by computing a parametric approximation (e.g: Gaussian) to the density of the posterior. However, while it is known empirically to quickly compute fine approximations, EP is extremely poorly understood which prevents it from being adopted by a larger fraction of the community. The object of the present article is to shed intu…
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…
In this paper, I propose a new class of Zero-Inflated Poisson models into the family of Cluster Weighted Models (CWMs) called Zero-Inflated Poisson CWMs (ZIPCWM). ZIPCWM extends Poisson cluster weighted models and other mixture models. I propose an Expectation-Maximization (EM) algorithm via an iteratively reweighted least squares for the model. I theoretically and analytically investigate the identifiability of the proposed model through an extensive simulation study. Parameter recovery, classification assessment, and performance of different information criteria are investigated through broa…
In this work, we introduce a novel estimator of the predictive risk with Poisson data, when the loss function is the Kullback-Leibler divergence, in order to define a regularization parameter's choice rule for the Expectation Maximization (EM) algorithm. To this aim, we prove a Poisson counterpart of the Stein's Lemma for Gaussian variables, and from this result we derive the proposed estimator showing its analogies with the well-known Stein's Unbiased Risk Estimator valid for a quadratic loss. We prove that the proposed estimator is asymptotically unbiased with increasing number of measured c…
