Variational inference has been widely used in machine learning literature to fit various Bayesian models. In network analysis, this method has been successfully applied to solve the community detection problems. Although these results are promising, their theoretical support is only for relatively dense networks, an assumption that may not hold for real networks. In addition, it has been shown recently that the variational loss surface has many saddle points, which may severely affect its performance, especially when applied to sparse networks. This paper proposes a simple way to improve the variational inference method by hard thresholding the posterior of the community assignment after each iteration. Using a random initialization that correlates with the true community assignment, we show that the proposed method converges and can accurately recover the true community labels, even when the average node degree of the network is bounded. Extensive numerical study further confirms the advantage of the proposed method over the classical variational inference and another state-of-the-art algorithm.
Konstanz University of Applied Sciences（康斯坦茨应用科学大学）；Zurich University of Applied Sciences（苏黎世应用科学大学）；University of Zurich（苏黎世大学）
The main challenge in Bayesian models is to determine the posterior for the model parameters. Already, in models with only one or few parameters, the analytical posterior can only be determined in special settings. In Bayesian neural networks, variational inference is widely used to approximate difficult-to-compute posteriors by variational distributions. Usually, Gaussians are used as variational distributions (Gaussian-VI) which limits the quality of the approximation due to their limited flexibility. Transformation models on the other hand are flexible enough to fit any distribution. Here w…
Heidelberg Institute for Theoretical Studies gGmbH（海德堡理论研究所）
The Laplace approximation has been one of the workhorses of Bayesian inference. It often delivers good approximations in practice despite the fact that it does not strictly take into account where the volume of posterior density lies. Variational approaches avoid this issue by explicitly minimising the Kullback-Leibler divergence DKL between a postulated posterior and the true (unnormalised) logarithmic posterior. However, they rely on a closed form DKL in order to update the variational parameters. To address this, stochastic versions of variational inference have been devised that approximat…
Variational inference methods often focus on the problem of efficient model optimization, with little emphasis on the choice of the approximating posterior. In this paper, we review and implement the various methods that enable us to develop a rich family of approximating posteriors. We show that one particular method employing transformations on distributions results in developing very rich and complex posterior approximation. We analyze its performance on the MNIST dataset by implementing with a Variational Autoencoder and demonstrate its effectiveness in learning better posterior distributi…
University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；Mount Holyoke College（曼荷莲学院）
Recent work in variational inference (VI) uses ideas from Monte Carlo estimation to tighten the lower bounds on the log-likelihood that are used as objectives. However, there is no systematic understanding of how optimizing different objectives relates to approximating the posterior distribution. Developing such a connection is important if the ideas are to be applied to inference-i.e., applications that require an approximate posterior and not just an approximation of the log-likelihood. Given a VI objective defined by a Monte Carlo estimator of the likelihood, we use a "divide and couple" pr…
University of Michigan（密歇根大学）；Columbia University（哥伦比亚大学）
Variational autoencoders model high-dimensional data by positing low-dimensional latent variables that are mapped through a flexible distribution parametrized by a neural network. Unfortunately, variational autoencoders often suffer from posterior collapse: the posterior of the latent variables is equal to its prior, rendering the variational autoencoder useless as a means to produce meaningful representations. Existing approaches to posterior collapse often attribute it to the use of neural networks or optimization issues due to variational approximation. In this paper, we consider posterior …
Massachusetts Institute of Technology（麻省理工学院）；Duke University（杜克大学）
Variational inference (VI) provides fast approximations of a Bayesian posterior in part because it formulates posterior approximation as an optimization problem: to find the closest distribution to the exact posterior over some family of distributions. For practical reasons, the family of distributions in VI is usually constrained so that it does not include the exact posterior, even as a limit point. Thus, no matter how long VI is run, the resulting approximation will not approach the exact posterior. We propose to instead consider a more flexible approximating family consisting of all possib…
Broad College of Business, Michigan State University（密歇根州立大学布罗德商学院）；the University of Texas at Austin（德克萨斯大学奥斯汀分校）；McCombs School of Business, the University of Texas at Austin（德克萨斯大学奥斯汀分校麦库姆斯商学院）
Leveraging well-established MCMC strategies, we propose MCMC-interactive variational inference (MIVI) to not only estimate the posterior in a time constrained manner, but also facilitate the design of MCMC transitions. Constructing a variational distribution followed by a short Markov chain that has parameters to learn, MIVI takes advantage of the complementary properties of variational inference and MCMC to encourage mutual improvement. On one hand, with the variational distribution locating high posterior density regions, the Markov chain is optimized within the variational inference framewo…
We propose a simple and general variant of the standard reparameterized gradient estimator for the variational evidence lower bound. Specifically, we remove a part of the total derivative with respect to the variational parameters that corresponds to the score function. Removing this term produces an unbiased gradient estimator whose variance approaches zero as the approximate posterior approaches the exact posterior. We analyze the behavior of this gradient estimator theoretically and empirically, and generalize it to more complex variational distributions such as mixtures and importance-weig…
Particle-based variational inference offers a flexible way of approximating complex posterior distributions with a set of particles. In this paper we introduce a new particle-based variational inference method based on the theory of semi-discrete optimal transport. Instead of minimizing the KL divergence between the posterior and the variational approximation, we minimize a semi-discrete optimal transport divergence. The solution of the resulting optimal transport problem provides both a particle approximation and a set of optimal transportation densities that map each particle to a segment of…
Tsinghua University（清华大学）；University of California, Riverside（加州大学河滨分校）
Neyman-Scott processes (NSPs) have been applied across a range of fields to model points or temporal events with a hierarchy of clusters. Markov chain Monte Carlo (MCMC) is typically used for posterior sampling in the model. However, MCMC's mixing time can cause the resulting inference to be slow, and thereby slow down model learning and prediction. We develop the first variational inference (VI) algorithm for NSPs, and give two examples of suitable variational posterior point process distributions. Our method minimizes the inclusive Kullback-Leibler (KL) divergence for VI to obtain the variat…
Variational inference has become one of the most widely used methods in latent variable modeling. In its basic form, variational inference employs a fully factorized variational distribution and minimizes its KL divergence to the posterior. As the minimization can only be carried out approximately, this approximation induces a bias. In this paper, we revisit perturbation theory as a powerful way of improving the variational approximation. Perturbation theory relies on a form of Taylor expansion of the log marginal likelihood, vaguely in terms of the log ratio of the true posterior and its vari…
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）；National Research University Higher School of Economics（国立研究型高等经济学院）；Samsung AI Center（三星人工智能中心）
One of the core problems in variational inference is a choice of approximate posterior distribution. It is crucial to trade-off between efficient inference with simple families as mean-field models and accuracy of inference. We propose a variant of a greedy approximation of the posterior distribution with tractable base learners. Using Max-Entropy approach, we obtain a well-defined optimization problem. We demonstrate the ability of the method to capture complex multimodal posterior via continual learning setting for neural networks.
University of Cambridge（剑桥大学）；HHMI Janelia Research Campus（HHMI珍妮亚研究园区）
Each training step for a variational autoencoder (VAE) requires us to sample from the approximate posterior, so we usually choose simple (e.g. factorised) approximate posteriors in which sampling is an efficient computation that fully exploits GPU parallelism. However, such simple approximate posteriors are often insufficient, as they eliminate statistical dependencies in the posterior. While it is possible to use normalizing flow approximate posteriors for continuous latents, some problems have discrete latents and strong statistical dependencies. The most natural approach to model these depe…
This paper establishes the asymptotic consistency of the {\it loss-calibrated variational Bayes} (LCVB) method. LCVB was proposed in~\cite{LaSiGh2011} as a method for approximately computing Bayesian posteriors in a `loss aware' manner. This methodology is also highly relevant in general data-driven decision-making contexts. Here, we not only establish the asymptotic consistency of the calibrated approximate posterior, but also the asymptotic consistency of decision rules. We also establish the asymptotic consistency of decision rules obtained from a `naive' variational Bayesian procedure.
How can we perform efficient inference and learning in directed probabilistic models, in the presence of continuous latent variables with intractable posterior distributions, and large datasets? We introduce a stochastic variational inference and learning algorithm that scales to large datasets and, under some mild differentiability conditions, even works in the intractable case. Our contributions are two-fold. First, we show that a reparameterization of the variational lower bound yields a lower bound estimator that can be straightforwardly optimized using standard stochastic gradient methods…
Xylem Inc.（赛莱默公司）；The University of Chicago（芝加哥大学）
Variational autoencoder is a powerful deep generative model with variational inference. The practice of modeling latent variables in the VAE's original formulation as normal distributions with a diagonal covariance matrix limits the flexibility to match the true posterior distribution. We propose a new transformation, dyadic transformation (DT), that can model a multivariate normal distribution. DT is a single-stage transformation with low computational requirements. We demonstrate empirically on MNIST dataset that DT enhances the posterior flexibility and attains competitive results compared …
Fast and Robust Likelihood-Guided Diffusion Posterior Sampling with Amortized Variational Inference
Institute of Foundation Models, MBZUAI（基础模型研究所，MBZUAI）；Huawei Lagrange Mathematics and Computing Research Center, Paris, France（华为拉格朗日数学与计算研究中心，法国巴黎）；Mohamed Bin Zayed University of AI, Chair of the ML Department, EPITA, Laboratoire Recherche de l'EPITA（穆罕默德·本·扎耶德人工智能大学，机器学习系主任，EPITA，EPITA研究实验室）
Bayesian neural network models (BNN) have re-surged in recent years due to the advancement of scalable computations and its utility in solving complex prediction problems in a wide variety of applications. Despite the popularity and usefulness of BNN, the conventional Markov Chain Monte Carlo based implementation suffers from high computational cost, limiting the use of this powerful technique in large scale studies. The variational Bayes inference has become a viable alternative to circumvent some of the computational issues. Although the approach is popular in machine learning, its applicati…
The choice of approximate posterior distribution is one of the core problems in variational inference. Most applications of variational inference employ simple families of posterior approximations in order to allow for efficient inference, focusing on mean-field or other simple structured approximations. This restriction has a significant impact on the quality of inferences made using variational methods. We introduce a new approach for specifying flexible, arbitrarily complex and scalable approximate posterior distributions. Our approximations are distributions constructed through a normalizi…
Tsinghua University（清华大学）；University of Toronto（多伦多大学）
Recent progress in variational inference has paid much attention to the flexibility of variational posteriors. One promising direction is to use implicit distributions, i.e., distributions without tractable densities as the variational posterior. However, existing methods on implicit posteriors still face challenges of noisy estimation and computational infeasibility when applied to models with high-dimensional latent variables. In this paper, we present a new approach named Kernel Implicit Variational Inference that addresses these challenges. As far as we know, for the first time implicit va…
Despite the popularism of Bayesian neural networks in recent years, its use is somewhat limited in complex and big data situations due to the computational cost associated with full posterior evaluations. Variational Bayes (VB) provides a useful alternative to circumvent the computational cost and time complexity associated with the generation of samples from the true posterior using Markov Chain Monte Carlo (MCMC) techniques. The efficacy of the VB methods is well established in machine learning literature. However, its potential broader impact is hindered due to a lack of theoretical validit…
In high-dimensions, the prior tails can have a significant effect on both posterior computation and asymptotic concentration rates. To achieve optimal rates while keeping the posterior computations relatively simple, an empirical Bayes approach has recently been proposed, featuring thin-tailed conjugate priors with data-driven centers. While conjugate priors ease some of the computational burden, Markov chain Monte Carlo methods are still needed, which can be expensive when dimension is high. In this paper, we develop a variational approximation to the empirical Bayes posterior that is fast to…
Variational Bayesian (VB) methods produce posterior inference in a time frame considerably smaller than traditional Markov Chain Monte Carlo approaches. Although the VB posterior is an approximation, it has been shown to produce good parameter estimates and predicted values when a rich classes of approximating distributions are considered. In this paper we propose Updating VB (UVB), a recursive algorithm used to update a sequence of VB posterior approximations in an online setting, with the computation of each posterior update requiring only the data observed since the previous update. An exte…
Ising models originated in statistical physics and are widely used in modeling spatial data and computer vision problems. However, statistical inference of this model remains challenging due to intractable nature of the normalizing constant in the likelihood. Here, we use a pseudo-likelihood instead to study the Bayesian estimation of two-parameter, inverse temperature, and magnetization, Ising model with a fully specified coupling matrix. We develop a computationally efficient variational Bayes procedure for model estimation. Under the Gaussian mean-field variational family, we derive posteri…
Current black-box variational inference (BBVI) methods require the user to make numerous design choices -- such as the selection of variational objective and approximating family -- yet there is little principled guidance on how to do so. We develop a conceptual framework and set of experimental tools to understand the effects of these choices, which we leverage to propose best practices for maximizing posterior approximation accuracy. Our approach is based on studying the pre-asymptotic tail behavior of the density ratios between the joint distribution and the variational approximation, then …
Boston University（波士顿大学）；University of Luxembourg（卢森堡大学）；University of British Columbia（不列颠哥伦比亚大学）；MIT（麻省理工学院）
Variational inference has become an increasingly attractive fast alternative to Markov chain Monte Carlo methods for approximate Bayesian inference. However, a major obstacle to the widespread use of variational methods is the lack of post-hoc accuracy measures that are both theoretically justified and computationally efficient. In this paper, we provide rigorous bounds on the error of posterior mean and uncertainty estimates that arise from full-distribution approximations, as in variational inference. Our bounds are widely applicable, as they require only that the approximating and exact pos…
Variational Bayes (VB) is a scalable alternative to Markov chain Monte Carlo (MCMC) for Bayesian posterior inference. Though popular, VB comes with few theoretical guarantees, most of which focus on well-specified models. However, models are rarely well-specified in practice. In this work, we study VB under model misspecification. We prove the VB posterior is asymptotically normal and centers at the value that minimizes the Kullback-Leibler (KL) divergence to the true data-generating distribution. Moreover, the VB posterior mean centers at the same value and is also asymptotically normal. Thes…
In this paper, we develop an unsupervised generative clustering framework that combines the Variational Information Bottleneck and the Gaussian Mixture Model. Specifically, in our approach, we use the Variational Information Bottleneck method and model the latent space as a mixture of Gaussians. We derive a bound on the cost function of our model that generalizes the Evidence Lower Bound (ELBO) and provide a variational inference type algorithm that allows computing it. In the algorithm, the coders' mappings are parametrized using neural networks, and the bound is approximated by Monte Carlo s…
Parameter estimation for model-based clustering using a finite mixture of normal inverse Gaussian (NIG) distributions is achieved through variational Bayes approximations. Univariate NIG mixtures and multivariate NIG mixtures are considered. The use of variational Bayes approximations here is a substantial departure from the traditional EM approach and alleviates some of the associated computational complexities and uncertainties. Our variational algorithm is applied to simulated and real data. The paper concludes with discussion and suggestions for future work.
Gaussian process (GP) models form a core part of probabilistic machine learning. Considerable research effort has been made into attacking three issues with GP models: how to compute efficiently when the number of data is large; how to approximate the posterior when the likelihood is not Gaussian and how to estimate covariance function parameter posteriors. This paper simultaneously addresses these, using a variational approximation to the posterior which is sparse in support of the function but otherwise free-form. The result is a Hybrid Monte-Carlo sampling scheme which allows for a non-Gaus…
Columbia University（哥伦比亚大学）；Princeton University（普林斯顿大学）
Variational inference (VI) is widely used as an efficient alternative to Markov chain Monte Carlo. It posits a family of approximating distributions $q$ and finds the closest member to the exact posterior $p$. Closeness is usually measured via a divergence $D(q || p)$ from $q$ to $p$. While successful, this approach also has problems. Notably, it typically leads to underestimation of the posterior variance. In this paper we propose CHIVI, a black-box variational inference algorithm that minimizes $D_χ(p || q)$, the $χ$-divergence from $p$ to $q$. CHIVI minimizes an upper bound of the model evi…
This paper investigates Frequentist consistency properties of the posterior distributions constructed via Generalized Variational Inference (GVI). A number of generic and novel strategies are given for proving consistency, relying on the theory of $Γ$-convergence. Specifically, this paper shows that under minimal regularity conditions, the sequence of GVI posteriors is consistent and collapses to a point mass at the population-optimal parameter value as the number of observations goes to infinity. The results extend to the latent variable case without additional assumptions and hold under miss…
Mean-field Variational Bayes (MFVB) is an approximate Bayesian posterior inference technique that is increasingly popular due to its fast runtimes on large-scale datasets. However, even when MFVB provides accurate posterior means for certain parameters, it often mis-estimates variances and covariances. Furthermore, prior robustness measures have remained undeveloped for MFVB. By deriving a simple formula for the effect of infinitesimal model perturbations on MFVB posterior means, we provide both improved covariance estimates and local robustness measures for MFVB, thus greatly expanding the pr…
We study high-dimensional Bayesian linear regression with product priors. Using the nascent theory of non-linear large deviations (Chatterjee and Dembo,2016), we derive sufficient conditions for the leading-order correctness of the naive mean-field approximation to the log-normalizing constant of the posterior distribution. Subsequently, assuming a true linear model for the observed data, we derive a limiting infinite dimensional variational formula for the log normalizing constant of the posterior. Furthermore, we establish that under an additional "separation" condition, the variational prob…
We propose a new marginal data density estimator (MDDE) that uses the variational Bayes posterior density as a weighting density of the reciprocal importance sampling (RIS) MDDE. This computationally convenient estimator is based on variational Bayes posterior densities that are available for many models and requires simulated draws only from the posterior distribution. It provides accurate estimates with a moderate number of posterior draws, has a finite variance, and provides a minimum variance candidate for the class of RIS MDDEs. Its reciprocal is consistent, asymptotically normally distri…
In Bayesian analysis, the posterior follows from the data and a choice of a prior and a likelihood. One hopes that the posterior is robust to reasonable variation in the choice of prior and likelihood, since this choice is made by the modeler and is necessarily somewhat subjective. Despite the fundamental importance of the problem and a considerable body of literature, the tools of robust Bayes are not commonly used in practice. This is in large part due to the difficulty of calculating robustness measures from MCMC draws. Although methods for computing robustness measures from MCMC draws exis…
Bayesian hierarchical models are increasing popular in economics. When using hierarchical models, it is useful not only to calculate posterior expectations, but also to measure the robustness of these expectations to reasonable alternative prior choices. We use variational Bayes and linear response methods to provide fast, accurate posterior means and robustness measures with an application to measuring the effectiveness of microcredit in the developing world.
Bayesian analyses combine information represented by different terms in a joint Bayesian model. When one or more of the terms is misspecified, it can be helpful to restrict the use of information from suspect model components to modify posterior inference. This is called "cutting feedback", and both the specification and computation of the posterior for such "cut models" is challenging. In this paper, we define cut posterior distributions as solutions to constrained optimization problems, and propose optimization-based variational methods for their computation. These methods are faster than ex…
University of Oxford（牛津大学）；Delft University of Technology（代尔夫特理工大学）
Multimodal distributions of some physics based model parameters are often encountered in engineering due to different situations such as a change in some environmental conditions, and the presence of some types of damage and nonlinearity. In statistical model updating, for locally identifiable parameters, it can be anticipated that multi-modal posterior distributions would be found. The full characterization of these multi-modal distributions is important as methodologies for structural condition monitoring in structures are frequently based in the comparison of the damaged and healthy models …
Functional data analysis deals with data recorded densely over time (or any other continuum) with one or more observed curves per subject. Conceptually, functional data are continuously defined, but in practice, they are usually observed at discrete points. Among different kinds of functional data analyses, clustering analysis aims to determine underlying groups of curves in the dataset when there is no information on the group membership of each individual curve. In this work, we propose a new model-based approach for clustering and smoothing functional data simultaneously via variational inf…
Variational Bayes (VB) has shown itself to be a powerful approximation method in many application areas. This paper describes some diagnostics methods which can assess how well the VB approximates the true posterior, particularly with regards to its covariance structure. The methods proposed also allow us to generate simple corrections when the approximation error is large. It looks at joint, marginal and conditional aspects of the approximate posterior and shows how to apply these techniques in both simulated and real data examples.
In recent years, large-scale Bayesian learning draws a great deal of attention. However, in big-data era, the amount of data we face is growing much faster than our ability to deal with it. Fortunately, it is observed that large-scale datasets usually own rich internal structure and is somewhat redundant. In this paper, we attempt to simplify the Bayesian posterior via exploiting this structure. Specifically, we restrict our interest to the so-called well-clustered datasets and construct an \emph{approximate posterior} according to the clustering information. Fortunately, the clustering struct…
Gatsby Computational Neuroscience Unit（盖茨比计算神经科学单元）；University College London（伦敦大学学院）
Sparse variational approximations allow for principled and scalable inference in Gaussian Process (GP) models. In settings where several GPs are part of the generative model, theses GPs are a posteriori coupled. For many applications such as regression where predictive accuracy is the quantity of interest, this coupling is not crucial. Howewer if one is interested in posterior uncertainty, it cannot be ignored. A key element of variational inference schemes is the choice of the approximate posterior parameterization. When the number of latent variables is large, mean field (MF) methods provide…
