We develop variational Laplace for Bayesian neural networks (BNNs) which exploits a local approximation of the curvature of the likelihood to estimate the ELBO without the need for stochastic sampling of the neural-network weights. The Variational Laplace objective is simple to evaluate, as it is (in essence) the log-likelihood, plus weight-decay, plus a squared-gradient regularizer. Variational Laplace gave better test performance and expected calibration errors than maximum a-posteriori inference and standard sampling-based variational inference, despite using the same variational approximate posterior. Finally, we emphasise care needed in benchmarking standard VI as there is a risk of stopping before the variance parameters have converged. We show that early-stopping can be avoided by increasing the learning rate for the variance parameters.
We develop variational Laplace for Bayesian neural networks (BNNs) which exploits a local approximation of the curvature of the likelihood to estimate the ELBO without the need for stochastic sampling of the neural-network weights. The Variational Laplace objective is simple to evaluate, as it is (in essence) the log-likelihood, plus weight-decay, plus a squared-gradient regularizer. Variational Laplace gave better test performance and expected calibration errors than maximum a-posteriori inference and standard sampling-based variational inference, despite using the same variational approximat…
Addressing the Inconsistency in Bayesian Deep Learning via Generalized Laplace Approximation
School of Engineering Deakin University Melbourne, VIC, 3216, Australia（工程学院 德堪大学 澳大利亚墨尔本 VIC 3216）；Faculty of Mathematics（数学与计算机科学学院）；Department of Computing Technologies Swinburne University of Technology Hawthorn, VIC, 3122, Australia（计算技术系 斯winburne技术大学 澳大利亚霍斯顿 VIC 3122）
While learning the maximum likelihood value of parameters of an undirected graphical model is hard, modelling the posterior distribution over parameters given data is harder. Yet, undirected models are ubiquitous in computer vision and text modelling (e.g. conditional random fields). But where Bayesian approaches for directed models have been very successful, a proper Bayesian treatment of undirected models in still in its infant stages. We propose a new method for approximating the posterior of the parameters given data based on the Laplace approximation. This approximation requires the compu…
University of Cambridge（剑桥大学）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；University of Amsterdam（阿姆斯特丹大学）；Microsoft Research（微软研究院）；The Alan Turing Institute（艾伦·图灵研究所）
The Bayesian paradigm has the potential to solve core issues of deep neural networks such as poor calibration and data inefficiency. Alas, scaling Bayesian inference to large weight spaces often requires restrictive approximations. In this work, we show that it suffices to perform inference over a small subset of model weights in order to obtain accurate predictive posteriors. The other weights are kept as point estimates. This subnetwork inference framework enables us to use expressive, otherwise intractable, posterior approximations over such subsets. In particular, we implement subnetwork l…
University of Cambridge（剑桥大学）；MPI for Intelligent Systems（马克斯·普朗克智能系统研究所）；University of Tübingen（蒂宾根大学）；ETH Zurich（苏黎世联邦理工学院）；Max Planck ETH Center for Learning Systems（马克斯·普朗克-苏黎世联邦理工学院学习系统中心）；DeepMind（深度思维）
Bayesian formulations of deep learning have been shown to have compelling theoretical properties and offer practical functional benefits, such as improved predictive uncertainty quantification and model selection. The Laplace approximation (LA) is a classic, and arguably the simplest family of approximations for the intractable posteriors of deep neural networks. Yet, despite its simplicity, the LA is not as popular as alternatives like variational Bayes or deep ensembles. This may be due to assumptions that the LA is expensive due to the involved Hessian computation, that it is difficult to i…
Corrected Integrated Laplace Approximation for Bayesian Inference in Latent Gaussian Models
Manning College of Information and Computer Sciences University of Massachusetts Amherst（信息与计算机科学学院 马萨诸塞大学阿姆赫斯特分校）；Department of Statistics University of British Columbia（统计学系 不列颠哥伦比亚大学）
Variational approaches to approximate Bayesian inference provide very efficient means of performing parameter estimation and model selection. Among these, so-called variational-Laplace or VL schemes rely on Gaussian approximations to posterior densities on model parameters. In this note, we review the main variants of VL approaches, that follow from considering nonlinear models of continuous and/or categorical data. En passant, we also derive a few novel theoretical results that complete the portfolio of existing analyses of variational Bayesian approaches, including investigations of their as…
In this work we perform outlier detection using ensembles of neural networks obtained by variational approximation of the posterior in a Bayesian neural network setting. The variational parameters are obtained by sampling from the true posterior by gradient descent. We show our outlier detection results are comparable to those obtained using other efficient ensembling methods.
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；RIKEN Center for Advanced Intelligence Project（理化学研究所高级智能中心）；EPFL School of Computer and Communication Sciences（洛桑联邦理工学院计算机与通信科学学院）
In this thesis, we disentangle the generalized Gauss-Newton and approximate inference for Bayesian deep learning. The generalized Gauss-Newton method is an optimization method that is used in several popular Bayesian deep learning algorithms. Algorithms that combine the Gauss-Newton method with the Laplace and Gaussian variational approximation have recently led to state-of-the-art results in Bayesian deep learning. While the Laplace and Gaussian variational approximation have been studied extensively, their interplay with the Gauss-Newton method remains unclear. Recent criticism of priors and…
A Variational Approach to Bayesian Phylogenetic Inference
Peking University（北京大学）；Howard Hughes Medical Institute（霍华德·休斯医学研究所）；Fred Hutchinson Cancer Research Center（弗雷德·哈钦森癌症研究中心）；University of Washington（华盛顿大学）
Harvard T.H. Chan School of Public Health（哈佛大学陈曾熙公共卫生学院）；Harvard University（哈佛大学）
Variational inference enables approximate posterior inference of the highly over-parameterized neural networks that are popular in modern machine learning. Unfortunately, such posteriors are known to exhibit various pathological behaviors. We prove that as the number of hidden units in a single-layer Bayesian neural network tends to infinity, the function-space posterior mean under mean-field variational inference actually converges to zero, completely ignoring the data. This is in contrast to the true posterior, which converges to a Gaussian process. Our work provides insight into the over-re…
In this note we consider setups in which variational objectives for Bayesian neural networks can be computed in closed form. In particular we focus on single-layer networks in which the activation function is piecewise polynomial (e.g. ReLU). In this case we show that for a Normal likelihood and structured Normal variational distributions one can compute a variational lower bound in closed form. In addition we compute the predictive mean and variance in closed form. Finally, we also show how to compute approximate lower bounds for other likelihoods (e.g. softmax classification). In experiments…
VinAI Research（VinAI研究院）；University of Texas, Austin（德克萨斯大学奥斯汀分校）；Hanoi University of Science and Technology（河内科技大学）
Approximate inference in Bayesian deep networks exhibits a dilemma of how to yield high fidelity posterior approximations while maintaining computational efficiency and scalability. We tackle this challenge by introducing a novel variational structured approximation inspired by the Bayesian interpretation of Dropout regularization. Concretely, we focus on the inflexibility of the factorized structure in Dropout posterior and then propose an improved method called Variational Structured Dropout (VSD). VSD employs an orthogonal transformation to learn a structured representation on the variation…
University of Amsterdam（阿姆斯特丹大学）；TNO Intelligent Imaging（荷兰应用科学研究组织智能成像）；Canadian Institute For Advanced Research (CIFAR)（加拿大高级研究院）
We reinterpret multiplicative noise in neural networks as auxiliary random variables that augment the approximate posterior in a variational setting for Bayesian neural networks. We show that through this interpretation it is both efficient and straightforward to improve the approximation by employing normalizing flows while still allowing for local reparametrizations and a tractable lower bound. In experiments we show that with this new approximation we can significantly improve upon classical mean field for Bayesian neural networks on both predictive accuracy as well as predictive uncertaint…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；Tsinghua University（清华大学）
Variational Bayesian neural networks (BNNs) perform variational inference over weights, but it is difficult to specify meaningful priors and approximate posteriors in a high-dimensional weight space. We introduce functional variational Bayesian neural networks (fBNNs), which maximize an Evidence Lower BOund (ELBO) defined directly on stochastic processes, i.e. distributions over functions. We prove that the KL divergence between stochastic processes equals the supremum of marginal KL divergences over all finite sets of inputs. Based on this, we introduce a practical training objective which ap…
National University of Singapore（新加坡国立大学）；MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）；MIT（麻省理工学院）
This paper presents a novel variational inference framework for deriving a family of Bayesian sparse Gaussian process regression (SGPR) models whose approximations are variationally optimal with respect to the full-rank GPR model enriched with various corresponding correlation structures of the observation noises. Our variational Bayesian SGPR (VBSGPR) models jointly treat both the distributions of the inducing variables and hyperparameters as variational parameters, which enables the decomposability of the variational lower bound that in turn can be exploited for stochastic optimization. Such…
Two main obstacles preventing the widespread adoption of variational Bayesian neural networks are the high parameter overhead that makes them infeasible on large networks, and the difficulty of implementation, which can be thought of as "programming overhead." MC dropout [Gal and Ghahramani, 2016] is popular because it sidesteps these obstacles. Nevertheless, dropout is often harmful to model performance when used in networks with batch normalization layers [Li et al., 2018], which are an indispensable part of modern neural networks. We construct a general variational family for ensemble-based…
We propose organisation conditions that yield a method for training SOM with adaptative neighborhood radius in a variational Bayesian framework. This method is validated on a non-stationary setting and compared in an high-dimensional setting with an other adaptative method.
A method for sequential inference of the fixed parameters of a dynamic latent Gaussian models is proposed and evaluated that is based on the iterated Laplace approximation. The method provides a useful trade-off between computational performance and the accuracy of the approximation to the true posterior distribution. Approximation corrections are shown to improve the accuracy of the approximation in simulation studies. A population-based approach is also shown to provide a more robust inference method.
We consider Bayesian variable selection in sparse high-dimensional regression, where the number of covariates $p$ may be large relative to the samples size $n$, but at most a moderate number $q$ of covariates are active. Specifically, we treat generalized linear models. For a single fixed sparse model with well-behaved prior distribution, classical theory proves that the Laplace approximation to the marginal likelihood of the model is accurate for sufficiently large sample size $n$. We extend this theory by giving results on uniform accuracy of the Laplace approximation across all models in a …
Self-Supervised Laplace Approximation for Bayesian Uncertainty Quantification
Rational Intelligence Lab, CISPA Helmholtz Center for Information Security Department of Statistics, LMU Munich（理性智能实验室，CISPA海德堡信息安全中心统计学系，慕尼黑大学）；Department of Statistics, LMU Munich（统计学系，慕尼黑大学）；Department of Computer Science, The University of Manchester（计算机科学系，曼彻斯特大学）
The integrated nested Laplace approximation (INLA) for Bayesian inference is an efficient approach to estimate the posterior marginal distributions of the parameters and latent effects of Bayesian hierarchical models that can be expressed as latent Gaussian Markov random fields (GMRF). The representation as a GMRF allows the associated software R-INLA to estimate the posterior marginals in a fraction of the time as typical Markov chain Monte Carlo algorithms. INLA can be extended by means of Bayesian model averaging (BMA) to increase the number of models that it can fit to conditional latent G…
Gaussian random field (GRF) models are widely used in spatial statistics to capture spatially correlated error. We investigate the results of replacing Gaussian processes with Laplace moving averages (LMAs) in spatial generalized linear mixed models (SGLMMs). We demonstrate that LMAs offer improved predictive power when the data exhibits localized spikes in the response. SGLMMs with LMAs are shown to maintain analogous parameter inference and similar computing to Gaussian SGLMMs. We propose a novel discrete space LMA model for irregular lattices and construct conjugate samplers for LMAs with g…
The Bayesian approach to solving inverse problems relies on the choice of a prior. This critical ingredient allows the formulation of expert knowledge or physical constraints in a probabilistic fashion and plays an important role for the success of the inference. Recently, Bayesian inverse problems were solved using generative models as highly informative priors. Generative models are a popular tool in machine learning to generate data whose properties closely resemble those of a given database. Typically, the generated distribution of data is embedded in a low-dimensional manifold. For the in…
We consider latent Gaussian fields for modelling spatial dependence in the context of both spatial point patterns and areal data, providing two different applications. The inhomogeneous Log-Gaussian Cox Process model is specified to describe a seismic sequence occurred in Greece, resorting to the Stochastic Partial Differential Equations. The Besag-York-Mollie model is fitted for disease mapping of the Covid-19 infection in the North of Italy. These models both belong to the class of Bayesian hierarchical models with latent Gaussian fields whose posterior is not available in closed form. There…
The asymmetric Laplace density (ALD) is used as a working likelihood for Bayesian quantile regression. Sriram et al.(2013) derived posterior consistency for Bayesian linear quantile regression based on the misspecified ALD. While their paper also argued for $\sqrt{n}-$consistency, Sriram and Ramamoorthi (2017) highlighted that the argument was only valid for a rate less than $\sqrt{n}$. So, the question of $\sqrt{n}-$rate has remained unaddressed, but is necessary to carry out meaningful Bayesian inference based on the ALD. In this paper, we derive results to obtain posterior consistency rates…
Fitting cross-classified multilevel models with binary response is challenging. In this setting a promising method is Bayesian inference through Integrated Nested Laplace Approximations (INLA), which performs well in several latent variable models. Therefore we devise a systematic simulation study to assess the performance of INLA with cross-classified logistic data under different scenarios defined by the magnitude of the random effects variances, the number of observations, the number of clusters, and the degree of cross-classification. In the simulations INLA is systematically compared with…
Microsoft Research（微软研究院）；University of California（加利福尼亚大学）
We discuss Bayesian methods for learning Bayesian networks when data sets are incomplete. In particular, we examine asymptotic approximations for the marginal likelihood of incomplete data given a Bayesian network. We consider the Laplace approximation and the less accurate but more efficient BIC/MDL approximation. We also consider approximations proposed by Draper (1993) and Cheeseman and Stutz (1995). These approximations are as efficient as BIC/MDL, but their accuracy has not been studied in any depth. We compare the accuracy of these approximations under the assumption that the Laplace app…
School of Mathematics and Statistics（数学与统计学院）；University of Melbourne（墨尔本大学）
In this work, we advocate for the importance of singular learning theory (SLT) as it pertains to the theory and practice of variational inference in Bayesian neural networks (BNNs). To begin, using SLT, we lay to rest some of the confusion surrounding discrepancies between downstream predictive performance measured via e.g., the test log predictive density, and the variational objective. Next, we use the SLT-corrected asymptotic form for singular posterior distributions to inform the design of the variational family itself. Specifically, we build upon the idealized variational family introduce…
We develop an optimization algorithm suitable for Bayesian learning in complex models. Our approach relies on natural gradient updates within a general black-box framework for efficient training with limited model-specific derivations. It applies within the class of exponential-family variational posterior distributions, for which we extensively discuss the Gaussian case for which the updates have a rather simple form. Our Quasi Black-box Variational Inference (QBVI) framework is readily applicable to a wide class of Bayesian inference problems and is of simple implementation as the updates of…
We show how to use a variational approximation to the logistic function to perform approximate inference in Bayesian networks containing discrete nodes with continuous parents. Essentially, we convert the logistic function to a Gaussian, which facilitates exact inference, and then iteratively adjust the variational parameters to improve the quality of the approximation. We demonstrate experimentally that this approximation is faster and potentially more accurate than sampling. We also introduce a simple new technique for handling evidence, which allows us to handle arbitrary distributions on o…
Various computational challenges arise when applying Bayesian inference approaches to complex hierarchical models. Sampling-based inference methods, such as Markov Chain Monte Carlo strategies, are renowned for providing accurate results but with high computational costs and slow or questionable convergence. On the contrary, approximate methods like the Integrated Nested Laplace Approximation (INLA) construct a deterministic approximation to the univariate posteriors through nested Laplace Approximations. This method enables fast inference performance in Latent Gaussian Models, which encode a …
Federated Bayesian learning offers a principled framework for the definition of collaborative training algorithms that are able to quantify epistemic uncertainty and to produce trustworthy decisions. Upon the completion of collaborative training, an agent may decide to exercise her legal "right to be forgotten", which calls for her contribution to the jointly trained model to be deleted and discarded. This paper studies federated learning and unlearning in a decentralized network within a Bayesian framework. It specifically develops federated variational inference (VI) solutions based on the d…
The goal of phylodynamics, an area on the intersection of phylogenetics and population genetics, is to reconstruct population size dynamics from genetic data. Recently, a series of nonparametric Bayesian methods have been proposed for such demographic reconstructions. These methods rely on prior specifications based on Gaussian processes and proceed by approximating the posterior distribution of population size trajectories via Markov chain Monte Carlo (MCMC) methods. In this paper, we adapt an integrated nested Laplace approximation (INLA), a recently proposed approximate Bayesian inference f…
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.
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.
The marginal likelihood is a well established model selection criterion in Bayesian statistics. It also allows to efficiently calculate the marginal posterior model probabilities that can be used for Bayesian model averaging of quantities of interest. For many complex models, including latent modeling approaches, marginal likelihoods are however difficult to compute. One recent promising approach for approximating the marginal likelihood is Integrated Nested Laplace Approximation (INLA), design for models with latent Gaussian structures. In this study we compare the approximations obtained wit…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）
This paper focuses on Bayesian Optimization in combinatorial spaces. In many applications in the natural science. Broad applications include the study of molecules, proteins, DNA, device structures and quantum circuit designs, a on optimization over combinatorial categorical spaces is needed to find optimal or pareto-optimal solutions. However, only a limited amount of methods have been proposed to tackle this problem. Many of them depend on employing Gaussian Process for combinatorial Bayesian Optimizations. Gaussian Processes suffer from scalability issues for large data sizes as their scali…
Natural image statistics exhibit hierarchical dependencies across multiple scales. Representing such prior knowledge in non-factorial latent tree models can boost performance of image denoising, inpainting, deconvolution or reconstruction substantially, beyond standard factorial "sparse" methodology. We derive a large scale approximate Bayesian inference algorithm for linear models with non-factorial (latent tree-structured) scale mixture priors. Experimental results on a range of denoising and inpainting problems demonstrate substantially improved performance compared to MAP estimation or to …
Generalized additive models (GAMs) are a well-established statistical tool for modeling complex nonlinear relationships between covariates and a response assumed to have a conditional distribution in the exponential family. In this article, P-splines and the Laplace approximation are coupled for flexible and fast approximate Bayesian inference in GAMs. The proposed Laplace-P-spline model contributes to the development of a new methodology to explore the posterior penalty space by considering a deterministic grid-based strategy or a Markov chain sampler, depending on the number of smooth additi…
The motivations for using variational inference (VI) in neural networks differ significantly from those in latent variable models. This has a counter-intuitive consequence; more expressive variational approximations can provide significantly worse predictions as compared to those with less expressive families. In this work we make two contributions. First, we identify a cause of this performance gap, variational over-pruning. Second, we introduce a theoretically grounded explanation for this phenomenon. Our perspective sheds light on several related published results and provides intuition int…
There is a growing demand for performing larger-scale Bayesian inference tasks, arising from greater data availability and higher-dimensional model parameter spaces. In this work we present parallelization strategies for the methodology of integrated nested Laplace approximations (INLA), a popular framework for performing approximate Bayesian inference on the class of Latent Gaussian models. Our approach makes use of nested OpenMP parallelism, a parallel line search procedure using robust regression in INLA's optimization phase and the state-of-the-art sparse linear solver PARDISO. We leverage…
NTT DATA Mathematical Systems Inc.（NTT数据数学系统公司）；Tokyo Institute of Technology（东京工业大学）
Non-negative matrix factorization (NMF) is a knowledge discovery method that is used in many fields. Variational inference and Gibbs sampling methods for it are also wellknown. However, the variational approximation error has not been clarified yet, because NMF is not statistically regular and the prior distribution used in variational Bayesian NMF (VBNMF) has zero or divergence points. In this paper, using algebraic geometrical methods, we theoretically analyze the difference in negative log evidence (a.k.a. free energy) between VBNMF and Bayesian NMF, i.e., the Kullback-Leibler divergence be…
