In this paper we study properties of the Laplace approximation of the posterior distribution arising in nonlinear Bayesian inverse problems. Our work is motivated by Schillings et al. (2020), where it is shown that in such a setting the Laplace approximation error in Hellinger distance converges to zero in the order of the noise level. Here, we prove novel error estimates for a given noise level that also quantify the effect due to the nonlinearity of the forward mapping and the dimension of the problem. In particular, we are interested in settings in which a linear forward mapping is perturbed by a small nonlinear mapping. Our results indicate that in this case, the Laplace approximation error is of the size of the perturbation. The paper provides insight into Bayesian inference in nonlinear inverse problems, where linearization of the forward mapping has suitable approximation properties.
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 …
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…
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…
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.
Spatially inhomogeneous functions, which may be smooth in some regions and rough in other regions, are modelled naturally in a Bayesian manner using so-called Besov priors which are given by random wavelet expansions with Laplace-distributed coefficients. This paper studies theoretical guarantees for such prior measures - specifically, we examine their frequentist posterior contraction rates in the setting of non-linear inverse problems with Gaussian white noise. Our results are first derived under a general local Lipschitz assumption on the forward map. We then verify the assumption for two n…
University of Sussex（萨塞克斯大学）；University of Bristol（布里斯托尔大学）
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…
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…
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…
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…
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…
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…
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 …
We study nonparametric Bayesian inference with location mixtures of the Laplace density and a Dirichlet process prior on the mixing distribution. We derive a contraction rate of the corresponding posterior distribution, both for the mixing distribution relative to the Wasserstein metric and for the mixed density relative to the Hellinger and $L_q$ metrics.
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…
This paper develops meshless methods for probabilistically describing discretisation error in the numerical solution of partial differential equations. This construction enables the solution of Bayesian inverse problems while accounting for the impact of the discretisation of the forward problem. In particular, this drives statistical inferences to be more conservative in the presence of significant solver error. Theoretical results are presented describing rates of convergence for the posteriors in both the forward and inverse problems. This method is tested on a challenging inverse problem w…
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…
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 Bayesian inference in inverse regression problems where the objective is to infer about unobserved covariates from observed responses and covariates. We establish posterior consistency of such unobserved covariates in Bayesian inverse regression problemsunder appropriate priors in a leave-one-out cross-validation setup. We relate this to posterior consistency of inverse reference distributions (Bhattacharya (2013)) for assessing model adequacy. We illustrate our theory and methods with various examples of Bayesian inverse regression, along with adequate simulation experiments.
The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e. a MAP estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse problem, seen as minimisation of the Onsager-Machlup functional of the posterior measure. An open problem in the stability analysis of inverse problems is to establish a relationship between the convergence properties of solutions obtained by the variational approach and by the Bayesian approach. To address this problem, we propose a general convergence theory fo…
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…
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…
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…
In the literature surrounding Bayesian penalized regression, the two primary choices of prior distribution on the regression coefficients are zero-mean Gaussian and Laplace. While both have been compared numerically and theoretically, there remains little guidance on which to use in real-life situations. We propose two viable solutions to this problem in the form of prior distributions which combine and compromise between Laplace and Gaussian priors, respectively. Through cross-validation the prior which optimizes prediction performance is automatically selected. We then demonstrate the improv…
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…
The Bayesian approach is effective for inverse problems. The posterior density distribution provides useful information of the unknowns. However, for problems with non-unique solutions, the classical estimators such as the maximum a posterior (MAP) and conditional mean (CM) are not enough. We introduce two new estimators, the local maximum a posterior (LMAP) and local conditional mean (LCM). Their applications are demonstrated by three inverse problems: an inverse spectral problem, an inverse source problem, and an inverse medium problem.
We present non-asymptotic two-sided bounds to the log-marginal likelihood in Bayesian inference. The classical Laplace approximation is recovered as the leading term. Our derivation permits model misspecification and allows the parameter dimension to grow with the sample size. We do not make any assumptions about the asymptotic shape of the posterior, and instead require certain regularity conditions on the likelihood ratio and that the posterior to be sufficiently concentrated.
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…
When using R package tmbstan for Bayesian inference, the built-in feature Laplace approximation to the marginal likelihood with random effects integrated out can be switched on and off. There exists no guideline on whether Laplace approximation should be used to achieve better efficiency especially when the statistical model for estimating selection is complicated. To answer this question, we conducted simulation studies under different scenarios with a state-space model employing a VAR(1) state equation. We found that turning on Laplace approximation in tmbstan would probably lower the comput…
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 key operation in Bayesian inference, is to compute high-dimensional integrals. An old approximate technique is the Laplace method or approximation, which dates back to Pierre- Simon Laplace (1774). This simple idea approximates the integrand with a second order Taylor expansion around the mode and computes the integral analytically. By developing a nested version of this classical idea, combined with modern numerical techniques for sparse matrices, we obtain the approach of Integrated Nested Laplace Approximations (INLA) to do approximate Bayesian inference for latent Gaussian models (LGMs…
We prove a Bernstein-von Mises theorem for a general class of high dimensional nonlinear Bayesian inverse problems in the vanishing noise limit. We propose a sufficient condition on the growth rate of the number of unknown parameters under which the posterior distribution is asymptotically normal. This growth condition is expressed explicitly in terms of the model dimension, the degree of ill-posedness of the inverse problem and the noise parameter. The theoretical results are applied to a Bayesian estimation of the medium parameter in an elliptic problem.
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）
Penalized regression methods, such as $L_1$ regularization, are routinely used in high-dimensional applications, and there is a rich literature on optimality properties under sparsity assumptions. In the Bayesian paradigm, sparsity is routinely induced through two-component mixture priors having a probability mass at zero, but such priors encounter daunting computational problems in high dimensions. This has motivated an amazing variety of continuous shrinkage priors, which can be expressed as global-local scale mixtures of Gaussians, facilitating computation. In sharp contrast to the correspo…
Measurement error (ME) and missing values in covariates are often unavoidable in disciplines that deal with data, and both problems have separately received considerable attention during the past decades. However, while most researchers are familiar with methods for treating missing data, accounting for ME in covariates of regression models is less common. In addition, ME and missing data are typically treated as two separate problems, despite practical and theoretical similarities. Here, we exploit the fact that missing data in a continuous covariate is an extreme case of classical ME, allowi…
It is common practice to use Laplace approximations to compute marginal likelihoods in Bayesian versions of generalised linear models (GLM). Marginal likelihoods combined with model priors are then used in different search algorithms to compute the posterior marginal probabilities of models and individual covariates. This allows performing Bayesian model selection and model averaging. For large sample sizes, even the Laplace approximation becomes computationally challenging because the optimisation routine involved needs to evaluate the likelihood on the full set of data in multiple iterations…
