Showing papers on "Bayesian probability published in 2018"

Posterior Summarization in Bayesian Phylogenetics Using Tracer 1.7.

Abstract: Bayesian inference of phylogeny using Markov chain Monte Carlo (MCMC) plays a central role in understanding evolutionary history from molecular sequence data. Visualizing and analyzing the MCMC-generated samples from the posterior distribution is a key step in any non-trivial Bayesian inference. We present the software package Tracer (version 1.7) for visualizing and analyzing the MCMC trace files generated through Bayesian phylogenetic inference. Tracer provides kernel density estimation, multivariate visualization, demographic trajectory reconstruction, conditional posterior distribution summary, and more. Tracer is open-source and available at http://beast.community/tracer.

New tools for automated high-resolution cryo-EM structure determination in RELION-3.

Abstract: Here, we describe the third major release of RELION. CPU-based vector acceleration has been added in addition to GPU support, which provides flexibility in use of resources and avoids memory limitations. Reference-free autopicking with Laplacian-of-Gaussian filtering and execution of jobs from python allows non-interactive processing during acquisition, including 2D-classification, de novo model generation and 3D-classification. Per-particle refinement of CTF parameters and correction of estimated beam tilt provides higher resolution reconstructions when particles are at different heights in the ice, and/or coma-free alignment has not been optimal. Ewald sphere curvature correction improves resolution for large particles. We illustrate these developments with publicly available data sets: together with a Bayesian approach to beam-induced motion correction it leads to resolution improvements of 0.2-0.7 A compared to previous RELION versions.

Bayesian phylogenetic and phylodynamic data integration using BEAST 1.10

Abstract: The Bayesian Evolutionary Analysis by Sampling Trees (BEAST) software package has become a primary tool for Bayesian phylogenetic and phylodynamic inference from genetic sequence data. BEAST unifies molecular phylogenetic reconstruction with complex discrete and continuous trait evolution, divergence-time dating, and coalescent demographic models in an efficient statistical inference engine using Markov chain Monte Carlo integration. A convenient, cross-platform, graphical user interface allows the flexible construction of complex evolutionary analyses.

Bayesian inference for psychology. Part II: Example applications with JASP

Abstract: Bayesian hypothesis testing presents an attractive alternative to p value hypothesis testing. Part I of this series outlined several advantages of Bayesian hypothesis testing, including the ability to quantify evidence and the ability to monitor and update this evidence as data come in, without the need to know the intention with which the data were collected. Despite these and other practical advantages, Bayesian hypothesis tests are still reported relatively rarely. An important impediment to the widespread adoption of Bayesian tests is arguably the lack of user-friendly software for the run-of-the-mill statistical problems that confront psychologists for the analysis of almost every experiment: the t-test, ANOVA, correlation, regression, and contingency tables. In Part II of this series we introduce JASP (http://www.jasp-stats.org), an open-source, cross-platform, user-friendly graphical software package that allows users to carry out Bayesian hypothesis tests for standard statistical problems. JASP is based in part on the Bayesian analyses implemented in Morey and Rouder’s BayesFactor package for R. Armed with JASP, the practical advantages of Bayesian hypothesis testing are only a mouse click away.

A Tutorial on Bayesian Optimization

Abstract: Bayesian optimization is an approach to optimizing objective functions that take a long time (minutes or hours) to evaluate. It is best-suited for optimization over continuous domains of less than 20 dimensions, and tolerates stochastic noise in function evaluations. It builds a surrogate for the objective and quantifies the uncertainty in that surrogate using a Bayesian machine learning technique, Gaussian process regression, and then uses an acquisition function defined from this surrogate to decide where to sample. In this tutorial, we describe how Bayesian optimization works, including Gaussian process regression and three common acquisition functions: expected improvement, entropy search, and knowledge gradient. We then discuss more advanced techniques, including running multiple function evaluations in parallel, multi-fidelity and multi-information source optimization, expensive-to-evaluate constraints, random environmental conditions, multi-task Bayesian optimization, and the inclusion of derivative information. We conclude with a discussion of Bayesian optimization software and future research directions in the field. Within our tutorial material we provide a generalization of expected improvement to noisy evaluations, beyond the noise-free setting where it is more commonly applied. This generalization is justified by a formal decision-theoretic argument, standing in contrast to previous ad hoc modifications.

The Bayesian New Statistics: Hypothesis testing, estimation, meta-analysis, and power analysis from a Bayesian perspective

Abstract: In the practice of data analysis, there is a conceptual distinction between hypothesis testing, on the one hand, and estimation with quantified uncertainty on the other. Among frequentists in psychology, a shift of emphasis from hypothesis testing to estimation has been dubbed "the New Statistics" (Cumming 2014). A second conceptual distinction is between frequentist methods and Bayesian methods. Our main goal in this article is to explain how Bayesian methods achieve the goals of the New Statistics better than frequentist methods. The article reviews frequentist and Bayesian approaches to hypothesis testing and to estimation with confidence or credible intervals. The article also describes Bayesian approaches to meta-analysis, randomized controlled trials, and power analysis.

Rejecting or Accepting Parameter Values in Bayesian Estimation

Abstract: This article explains a decision rule that uses Bayesian posterior distributions as the basis for accepting or rejecting null values of parameters. This decision rule focuses on the range of plausible values indicated by the highest density interval of the posterior distribution and the relation between this range and a region of practical equivalence (ROPE) around the null value. The article also discusses considerations for setting the limits of a ROPE and emphasizes that analogous considerations apply to setting the decision thresholds for p values and Bayes factors.

Bayesian Model-Agnostic Meta-Learning

Abstract: Due to the inherent model uncertainty, learning to infer Bayesian posterior from a few-shot dataset is an important step towards robust meta-learning. In this paper, we propose a novel Bayesian model-agnostic meta-learning method. The proposed method combines efficient gradient-based meta-learning with nonparametric variational inference in a principled probabilistic framework. Unlike previous methods, during fast adaptation, the method is capable of learning complex uncertainty structure beyond a simple Gaussian approximation, and during meta-update, a novel Bayesian mechanism prevents meta-level overfitting. Remaining a gradient-based method, it is also the first Bayesian model-agnostic meta-learning method applicable to various tasks including reinforcement learning. Experiment results show the accuracy and robustness of the proposed method in sinusoidal regression, image classification, active learning, and reinforcement learning.

Discussion of "Using Stacking to Average Bayesian Predictive Distributions" by Yao et. al

Abstract: Main article included invited and contributed discussions: https://dx.doi.org/10.1214/17-BA1091 (Bayesian Analysis 13:3 (2018), pages 917-1007).

Bayesian alternatives for common null-hypothesis significance tests in psychiatry: a non-technical guide using JASP

Abstract: Despite its popularity as an inferential framework, classical null hypothesis significance testing (NHST) has several restrictions. Bayesian analysis can be used to complement NHST, however, this approach has been underutilized largely due to a dearth of accessible software options. JASP is a recently developed open-source statistical package that facilitates both Bayesian and NHST analysis using a graphical interface. This article provides an applied introduction to Bayesian inference with Bayes factors using JASP. We use JASP to compare and contrast Bayesian alternatives for several common classical null hypothesis significance tests: correlations, frequency distributions, t-tests, ANCOVAs, and ANOVAs. These examples are also used to illustrate the strengths and limitations of both NHST and Bayesian hypothesis testing. A comparison of NHST and Bayesian inferential frameworks demonstrates that Bayes factors can complement p-values by providing additional information for hypothesis testing. Namely, Bayes factors can quantify relative evidence for both alternative and null hypotheses. Moreover, the magnitude of this evidence can be presented as an easy-to-interpret odds ratio. While Bayesian analysis is by no means a new method, this type of statistical inference has been largely inaccessible for most psychiatry researchers. JASP provides a straightforward means of performing reproducible Bayesian hypothesis tests using a graphical “point and click” environment that will be familiar to researchers conversant with other graphical statistical packages, such as SPSS.

Four reasons to prefer Bayesian analyses over significance testing

Abstract: Inference using significance testing and Bayes factors is compared and contrasted in five case studies based on real research. The first study illustrates that the methods will often agree, both in motivating researchers to conclude that H1 is supported better than H0, and the other way round, that H0 is better supported than H1. The next four, however, show that the methods will also often disagree. In these cases, the aim of the paper will be to motivate the sensible evidential conclusion, and then see which approach matches those intuitions. Specifically, it is shown that a high-powered non-significant result is consistent with no evidence for H0 over H1 worth mentioning, which a Bayes factor can show, and, conversely, that a low-powered non-significant result is consistent with substantial evidence for H0 over H1, again indicated by Bayesian analyses. The fourth study illustrates that a high-powered significant result may not amount to any evidence for H1 over H0, matching the Bayesian conclusion. Finally, the fifth study illustrates that different theories can be evidentially supported to different degrees by the same data; a fact that P-values cannot reflect but Bayes factors can. It is argued that appropriate conclusions match the Bayesian inferences, but not those based on significance testing, where they disagree.

Spatial modeling with R-INLA: A review

Abstract: Coming up with Bayesian models for spatial data is easy, but performing inference with them can be challenging. Writing fast inference code for a complex spatial model with realistically-sized datasets from scratch is time-consuming, and if changes are made to the model, there is little guarantee that the code performs well. The key advantages of R-INLA are the ease with which complex models can be created and modified, without the need to write complex code, and the speed at which inference can be done even for spatial problems with hundreds of thousands of observations. R-INLA handles latent Gaussian models, where fixed effects, structured and unstructured Gaussian random effects are combined linearly in a linear predictor, and the elements of the linear predictor are observed through one or more likelihoods. The structured random effects can be both standard areal model such as the Besag and the BYM models, and geostatistical models from a subset of the Matern Gaussian random fields. In this review, we discuss the large success of spatial modeling with R-INLA and the types of spatial models that can be fitted, we give an overview of recent developments for areal models, and we give an overview of the stochastic partial differential equation (SPDE) approach and some of the ways it can be extended beyond the assumptions of isotropy and separability. In particular, we describe how slight changes to the SPDE approach leads to straight-forward approaches for nonstationary spatial models and nonseparable space–time models. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Bayesian Methods and Theory Statistical Models > Bayesian Models Data: Types and Structure > Massive Data.

Accurate Uncertainties for Deep Learning Using Calibrated Regression.

Abstract: Methods for reasoning under uncertainty are a key building block of accurate and reliable machine learning systems. Bayesian methods provide a general framework to quantify uncertainty. However, because of model misspecification and the use of approximate inference, Bayesian uncertainty estimates are often inaccurate -- for example, a 90% credible interval may not contain the true outcome 90% of the time. Here, we propose a simple procedure for calibrating any regression algorithm; when applied to Bayesian and probabilistic models, it is guaranteed to produce calibrated uncertainty estimates given enough data. Our procedure is inspired by Platt scaling and extends previous work on classification. We evaluate this approach on Bayesian linear regression, feedforward, and recurrent neural networks, and find that it consistently outputs well-calibrated credible intervals while improving performance on time series forecasting and model-based reinforcement learning tasks.

Adversarial Personalized Ranking for Recommendation

Abstract: Item recommendation is a personalized ranking task. To this end, many recommender systems optimize models with pairwise ranking objectives, such as the Bayesian Personalized Ranking (BPR). Using matrix Factorization (MF) --- the most widely used model in recommendation --- as a demonstration, we show that optimizing it with BPR leads to a recommender model that is not robust. In particular, we find that the resultant model is highly vulnerable to adversarial perturbations on its model parameters, which implies the possibly large error in generalization. To enhance the robustness of a recommender model and thus improve its generalization performance, we propose a new optimization framework, namely Adversarial Personalized Ranking (APR). In short, our APR enhances the pairwise ranking method BPR by performing adversarial training. It can be interpreted as playing a minimax game, where the minimization of the BPR objective function meanwhile defends an adversary, which adds adversarial perturbations on model parameters to maximize the BPR objective function. To illustrate how it works, we implement APR on MF by adding adversarial perturbations on the embedding vectors of users and items. Extensive experiments on three public real-world datasets demonstrate the effectiveness of APR --- by optimizing MF with APR, it outperforms BPR with a relative improvement of 11.2% on average and achieves state-of-the-art performance for item recommendation. Our implementation is available at: this https URL.

An introduction to Bayesian inference in gravitational-wave astronomy: parameter estimation, model selection, and hierarchical models

Abstract: This is an introduction to Bayesian inference with a focus on hierarchical models and hyper-parameters. We write primarily for an audience of Bayesian novices, but we hope to provide useful insights for seasoned veterans as well. Examples are drawn from gravitational-wave astronomy, though we endeavor for the presentation to be understandable to a broader audience. We begin with a review of the fundamentals: likelihoods, priors, and posteriors. Next, we discuss Bayesian evidence, Bayes factors, odds ratios, and model selection. From there, we describe how posteriors are estimated using samplers such as Markov Chain Monte Carlo algorithms and nested sampling. Finally, we generalize the formalism to discuss hyper-parameters and hierarchical models. We include extensive appendices discussing the creation of credible intervals, Gaussian noise, explicit marginalization, posterior predictive distributions, and selection effects.

Dropout Sampling for Robust Object Detection in Open-Set Conditions

Abstract: Dropout Variational Inference, or Dropout Sampling, has been recently proposed as an approximation technique for Bayesian Deep Learning and evaluated for image classification and regression tasks. This paper investigates the utility of Dropout Sampling for object detection for the first time. We demonstrate how label uncertainty can be extracted from a state-of-the-art object detection system via Dropout Sampling. We evaluate this approach on a large synthetic dataset of 30,000 images, and a real-world dataset captured by a mobile robot in a versatile campus environment. We show that this uncertainty can be utilized to increase object detection performance under the open-set conditions that are typically encountered in robotic vision. A Dropout Sampling network is shown to achieve a 12.3 % increase in recall (for the same precision score as a standard network) and a 15.1 % increase in precision (for the same recall score as the standard network).

Deep Bayesian Bandits Showdown: An Empirical Comparison of Bayesian Deep Networks for Thompson Sampling

Abstract: Recent advances in deep reinforcement learning have made significant strides in performance on applications such as Go and Atari games. However, developing practical methods to balance exploration and exploitation in complex domains remains largely unsolved. Thompson Sampling and its extension to reinforcement learning provide an elegant approach to exploration that only requires access to posterior samples of the model. At the same time, advances in approximate Bayesian methods have made posterior approximation for flexible neural network models practical. Thus, it is attractive to consider approximate Bayesian neural networks in a Thompson Sampling framework. To understand the impact of using an approximate posterior on Thompson Sampling, we benchmark well-established and recently developed methods for approximate posterior sampling combined with Thompson Sampling over a series of contextual bandit problems. We found that many approaches that have been successful in the supervised learning setting underperformed in the sequential decision-making scenario. In particular, we highlight the challenge of adapting slowly converging uncertainty estimates to the online setting.

Bayesian Naïve Bayes classifiers to text classification

Abstract: Text classification is the task of assigning predefined categories to natural language documents, and it can provide conceptual views of document collections. The Naive Bayes NB classifier is a family of simple probabilistic classifiers based on a common assumption that all features are independent of each other, given the category variable, and it is often used as the baseline in text classification. However, classical NB classifiers with multinomial, Bernoulli and Gaussian event models are not fully Bayesian. This study proposes three Bayesian counterparts, where it turns out that classical NB classifier with Bernoulli event model is equivalent to Bayesian counterpart. Finally, experimental results on 20 newsgroups and WebKB data sets show that the performance of Bayesian NB classifier with multinomial event model is similar to that of classical counterpart, but Bayesian NB classifier with Gaussian event model is obviously better than classical counterpart.

Parallelised Bayesian Optimisation via Thompson Sampling

Abstract: We design and analyse variations of the classical Thompson sampling (TS) procedure for Bayesian optimisation (BO) in settings where function evaluations are expensive but can be performed in parallel. Our theoretical analysis shows that a direct application of the sequential Thompson sampling algorithm in either synchronous or asynchronous parallel settings yields a surprisingly powerful result: making n evaluations distributed among M workers is essentially equivalent to performing n evaluations in sequence. Further, by modelling the time taken to complete a function evaluation, we show that, under a time constraint, asynchronous parallel TS achieves asymptotically lower regret than both the synchronous and sequential versions. These results are complemented by an experimental analysis, showing that asynchronous TS outperforms a suite of existing parallel BO algorithms in simulations and in an application involving tuning hyper-parameters of a convolutional neural network. In addition to these, the proposed procedure is conceptually much simpler than existing work for parallel BO.

Validating Bayesian Inference Algorithms with Simulation-Based Calibration

Abstract: Verifying the correctness of Bayesian computation is challenging. This is especially true for complex models that are common in practice, as these require sophisticated model implementations and algorithms. In this paper we introduce \emph{simulation-based calibration} (SBC), a general procedure for validating inferences from Bayesian algorithms capable of generating posterior samples. This procedure not only identifies inaccurate computation and inconsistencies in model implementations but also provides graphical summaries that can indicate the nature of the problems that arise. We argue that SBC is a critical part of a robust Bayesian workflow, as well as being a useful tool for those developing computational algorithms and statistical software.

Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau

Abstract: Modern imaging methods rely strongly on Bayesian inference techniques to solve challenging imaging problems. Currently, the predominant Bayesian computation approach is convex optimization, which scales very efficiently to high-dimensional image models and delivers accurate point estimation results. However, in order to perform more complex analyses, for example, image uncertainty quantification or model selection, it is necessary to use more computationally intensive Bayesian computation techniques such as Markov chain Monte Carlo methods. This paper presents a new and highly efficient Markov chain Monte Carlo methodology to perform Bayesian computation for high-dimensional models that are log-concave and nonsmooth, a class of models that is central in imaging sciences. The methodology is based on a regularized unadjusted Langevin algorithm that exploits tools from convex analysis, namely, Moreau--Yoshida envelopes and proximal operators, to construct Markov chains with favorable convergence properties. ...

Mixture Models With a Prior on the Number of Components

Abstract: A natural Bayesian approach for mixture models with an unknown number of components is to take the usual finite mixture model with symmetric Dirichlet weights, and put a prior on the number of components—that is, to use a mixture of finite mixtures (MFM). The most commonly used method of inference for MFMs is reversible jump Markov chain Monte Carlo, but it can be nontrivial to design good reversible jump moves, especially in high-dimensional spaces. Meanwhile, there are samplers for Dirichlet process mixture (DPM) models that are relatively simple and are easily adapted to new applications. It turns out that, in fact, many of the essential properties of DPMs are also exhibited by MFMs—an exchangeable partition distribution, restaurant process, random measure representation, and stick-breaking representation—and crucially, the MFM analogues are simple enough that they can be used much like the corresponding DPM properties. Consequently, many of the powerful methods developed for inference in DPMs ...

Deep Bayesian Bandits Showdown: An Empirical Comparison of Bayesian Deep Networks for Thompson Sampling

Abstract: Recent advances in deep reinforcement learning have made significant strides in performance on applications such as Go and Atari games. However, developing practical methods to balance exploration and exploitation in complex domains remains largely unsolved. Thompson Sampling and its extension to reinforcement learning provide an elegant approach to exploration that only requires access to posterior samples of the model. At the same time, advances in approximate Bayesian methods have made posterior approximation for flexible neural network models practical. Thus, it is attractive to consider approximate Bayesian neural networks in a Thompson Sampling framework. To understand the impact of using an approximate posterior on Thompson Sampling, we benchmark well-established and recently developed methods for approximate posterior sampling combined with Thompson Sampling over a series of contextual bandit problems. We found that many approaches that have been successful in the supervised learning setting underperformed in the sequential decision-making scenario. In particular, we highlight the challenge of adapting slowly converging uncertainty estimates to the online setting.

A guide to Bayesian model checking for ecologists

Abstract: Checking that models adequately represent data is an essential component of applied statistical inference. Ecologists increasingly use hierarchical Bayesian statistical models in their research. The appeal of this modeling paradigm is undeniable, as researchers can build and fit models that embody complex ecological processes while simultaneously controlling observation error. However, ecologists tend to be less focused on checking model assumptions and assessing potential lack-of-fit when applying Bayesian methods than when applying more traditional modes of inference such as maximum likelihood. There are also multiple ways of assessing the fit of Bayesian models, each of which has strengths and weaknesses. For instance, Bayesian p-values are relatively easy to compute, but are well known to be conservative, producing p-values biased toward 0.5. Alternatively, lesser known approaches to model checking, such as prior predictive checks, cross-validation probability integral transforms, and pivot discrepancy measures may produce more accurate characterizations of goodness-of-fit but are not as well known to ecologists. In addition, a suite of visual and targeted diagnostics can be used to examine violations of different model assumptions and lack-of-fit at different levels of the modeling hierarchy, and to check for residual temporal or spatial autocorrelation. In this review, we synthesize existing literature to guide ecologists through the many available options for Bayesian model checking. We illustrate methods and procedures with several ecological case studies, including i) analysis of simulated spatio-temporal count data, (ii) N-mixture models for estimating abundance and detection probability of sea otters from an aircraft, and (iii) hidden Markov modeling to describe attendance patterns of California sea lion mothers on a rookery. We find that commonly used procedures based on posterior predictive p-values detect extreme model inadequacy, but often do not detect more subtle cases of lack of fit. Tests based on cross-validation and pivot discrepancy measures (including the ``sampled predictive p-value'') appear to be better suited to model checking and to have better overall statistical performance. We conclude that model checking is an essential component of scientific discovery and learning that should accompany most Bayesian analyses presented in the literature.

Flipout: Efficient Pseudo-Independent Weight Perturbations on Mini-Batches

Abstract: Stochastic neural net weights are used in a variety of contexts, including regularization, Bayesian neural nets, exploration in reinforcement learning, and evolution strategies. Unfortunately, due to the large number of weights, all the examples in a mini-batch typically share the same weight perturbation, thereby limiting the variance reduction effect of large mini-batches. We introduce flipout, an efficient method for decorrelating the gradients within a mini-batch by implicitly sampling pseudo-independent weight perturbations for each example. Empirically, flipout achieves the ideal linear variance reduction for fully connected networks, convolutional networks, and RNNs. We find significant speedups in training neural networks with multiplicative Gaussian perturbations. We show that flipout is effective at regularizing LSTMs, and outperforms previous methods. Flipout also enables us to vectorize evolution strategies: in our experiments, a single GPU with flipout can handle the same throughput as at least 40 CPU cores using existing methods, equivalent to a factor-of-4 cost reduction on Amazon Web Services.

Decomposition of Uncertainty in Bayesian Deep Learning for Efficient and Risk-sensitive Learning

Abstract: Bayesian neural networks with latent variables are scalable and flexible probabilistic models: They account for uncertainty in the estimation of the network weights and, by making use of latent variables, can capture complex noise patterns in the : data. Using these models we show how to per- ∗ form and utilize a decomposition of uncertainty in • aleatoric and epistemic components for decision making purposes. This allows us to successfully identify informative points for active learning of functions with heteroscedastic and bimodal noise. ' t Using the decomposition we further define a novel risk-sensitive criterion for reinforcement learning to identify policies that balance expected cost, model-bias and noise aversion.

The Anatomy of Inference: Generative Models and Brain Structure.

Abstract: To infer the causes of its sensations, the brain must call on a generative (predictive) model. This necessitates passing local messages between populations of neurons to update beliefs about hidden variables in the world beyond its sensory samples. It also entails inferences about how we will act. Active inference is a principled framework that frames perception and action as approximate Bayesian inference. This has been successful in accounting for a wide range of physiological and behavioural phenomena. Recently, a process theory has emerged that attempts to relate inferences to their neurobiological substrates. In this paper, we review and develop the anatomical aspects of this process theory. We argue that the form of the generative models required for inference constrains the way in which brain regions connect to one another. Specifically, neuronal populations representing beliefs about a variable must receive input from populations representing the Markov blanket of that variable. We illustrate this idea in four different domains: perception, planning, attention, and movement. In doing so, we attempt to show how appealing to generative models enables us to account for anatomical brain architectures. Ultimately, committing to an anatomical theory of inference ensures we can form empirical hypotheses that can be tested using neuroimaging, neuropsychological, and electrophysiological experiments.

BAMLSS: Bayesian Additive Models for Location, Scale and Shape (and Beyond)

Abstract: Bayesian analysis provides a convenient setting for the estimation of complex generalized additive regression models (GAMs). Since computational power has tremendously increased in the past decade,...

Bayesian Uncertainty Estimation for Batch Normalized Deep Networks

Abstract: We show that training a deep network using batch normalization is equivalent to approximate inference in Bayesian models. We further demonstrate that this finding allows us to make meaningful estim ...

Noisy Natural Gradient as Variational Inference

Abstract: Variational Bayesian neural nets combine the flexibility of deep learning with Bayesian uncertainty estimation. Unfortunately, there is a tradeoff between cheap but simple variational families (e.g.~fully factorized) or expensive and complicated inference procedures. We show that natural gradient ascent with adaptive weight noise implicitly fits a variational posterior to maximize the evidence lower bound (ELBO). This insight allows us to train full-covariance, fully factorized, or matrix-variate Gaussian variational posteriors using noisy versions of natural gradient, Adam, and K-FAC, respectively, making it possible to scale up to modern-size ConvNets. On standard regression benchmarks, our noisy K-FAC algorithm makes better predictions and matches Hamiltonian Monte Carlo's predictive variances better than existing methods. Its improved uncertainty estimates lead to more efficient exploration in active learning, and intrinsic motivation for reinforcement learning.