Bayesian inference provides an attractive online-learning framework to analyze sequential data, and offers generalization guarantees which hold even with model mismatch and adversaries. Unfortunately, exact Bayesian inference is rarely feasible in practice and approximation methods are usually employed, but do such methods preserve the generalization properties of Bayesian inference ? In this paper, we show that this is indeed the case for some variational inference (VI) algorithms. We consider a few existing online, tempered VI algorithms, as well as a new algorithm, and derive their generalization bounds. Our theoretical result relies on the convexity of the variational objective, but we argue that the result should hold more generally and present empirical evidence in support of this. Our work in this paper presents theoretical justifications in favor of online algorithms relying on approximate Bayesian methods.
We advocate an optimization-centric view on and introduce a novel generalization of Bayesian inference. Our inspiration is the representation of Bayes' rule as infinite-dimensional optimization problem (Csiszar, 1975; Donsker and Varadhan; 1975, Zellner; 1988). First, we use it to prove an optimality result of standard Variational Inference (VI): Under the proposed view, the standard Evidence Lower Bound (ELBO) maximizing VI posterior is preferable to alternative approximations of the Bayesian posterior. Next, we argue for generalizing standard Bayesian inference. The need for this arises in situations of severe misalignment between reality and three assumptions underlying standard Bayesian inference: (1) Well-specified priors, (2) well-specified likelihoods, (3) the availability of infinite computing power. Our generalization addresses these shortcomings with three arguments and is called the Rule of Three (RoT). We derive it axiomatically and recover existing posteriors as special cases, including the Bayesian posterior and its approximation by standard VI. In contrast, approximations based on alternative ELBO-like objectives violate the axioms. Finally, we study a special case of the RoT that we call Generalized Variational Inference (GVI). GVI posteriors are a large and tractable family of belief distributions specified by three arguments: A loss, a divergence and a variational family. GVI posteriors have appealing properties, including consistency and an interpretation as approximate ELBO. The last part of the paper explores some attractive applications of GVI in popular machine learning models, including robustness and more appropriate marginals. After deriving black box inference schemes for GVI posteriors, their predictive performance is investigated on Bayesian Neural Networks and Deep Gaussian Processes, where GVI can comprehensively improve upon existing methods.
This paper is on Bayesian inference for parametric statistical models that are defined by a stochastic simulator which specifies how data is generated. Exact sampling is then possible but evaluating the likelihood function is typically prohibitively expensive. Approximate Bayesian Computation (ABC) is a framework to perform approximate inference in such situations. While basic ABC algorithms are widely applicable, they are notoriously slow and much research has focused on increasing their efficiency. Optimisation Monte Carlo (OMC) has recently been proposed as an efficient and embarrassingly parallel method that leverages optimisation to accelerate the inference. In this paper, we demonstrate an important previously unrecognised failure mode of OMC: It generates strongly overconfident approximations by collapsing regions of similar or near-constant likelihood into a single point. We propose an efficient, robust generalisation of OMC that corrects this. It makes fewer assumptions, retains the main benefits of OMC, and can be performed either as post-processing to OMC or as a stand-alone computation. We demonstrate the effectiveness of the proposed Robust OMC on toy examples and tasks in inverse-graphics where we perform Bayesian inference with a complex image renderer.
Thermal preferences vary from person to person and may change over time. The main objective of this paper is to sequentially pose intelligent queries to occupants in order to optimally learn the indoor air temperature values which maximize their satisfaction. Our central hypothesis is that an occupant's preference relation over indoor air temperature can be described using a scalar function of these temperatures, which we call the "occupant's thermal utility function". Information about an occupant's preference over these temperatures is available to us through their response to thermal preference queries : "prefer warmer," "prefer cooler" and "satisfied" which we interpret as statements about the derivative of their utility function, i.e. the utility function is "increasing", "decreasing" and "constant" respectively. We model this hidden utility function using a Gaussian process prior with built-in unimodality constraint, i.e., the utility function has a unique maximum, and we train this model using Bayesian inference. This permits an expected improvement based selection of next preference query to pose to the occupant, which takes into account both exploration (sampling from areas of high uncertainty) and exploitation (sampling from areas which are likely to offer an improvement over current best observation). We use this framework to sequentially design experiments and illustrate its benefits by showing that it requires drastically fewer observations to learn the maximally preferred temperature values as compared to other methods. This framework is an important step towards the development of intelligent HVAC systems which would be able to respond to occupants' personalized thermal comfort needs. In order to encourage the use of our PE framework and ensure reproducibility in results, we publish an implementation of our work named GPPrefElicit as an open-source package in Python.
Inspired by applications in sports where the skill of players or teams competing against each other varies over time, we propose a probabilistic model of pairwise-comparison outcomes that can capture a wide range of time dynamics. We achieve this by replacing the static parameters of a class of popular pairwise-comparison models by continuous-time Gaussian processes; the covariance function of these processes enables expressive dynamics. We develop an efficient inference algorithm that computes an approximate Bayesian posterior distribution. Despite the flexbility of our model, our inference algorithm requires only a few linear-time iterations over the data and can take advantage of modern multiprocessor computer architectures. We apply our model to several historical databases of sports outcomes and find that our approach outperforms competing approaches in terms of predictive performance, scales to millions of observations, and generates compelling visualizations that help in understanding and interpreting the data.
In image processing tasks, spatial priors are essential for robust computations, regularization, algorithmic design and Bayesian inference. In this paper, we introduce weighted mean curvature (WMC) as a novel image prior and present an efficient computation scheme for its discretization in practical image processing applications. We first demonstrate the favorable properties of WMC, such as sampling invariance, scale invariance, and contrast invariance with Gaussian noise model; and we show the relation of WMC to area regularization. We further propose an efficient computation scheme for discretized WMC, which is demonstrated herein to process over 33.2 giga-pixels/second on GPU. This scheme yields itself to a convolutional neural network representation. Finally, WMC is evaluated on synthetic and real images, showing its superiority quantitatively to total-variation and mean curvature.
Bayesian neural networks (BNNs) have recently regained a significant amount of attention in the deep learning community due to the development of scalable approximate Bayesian inference techniques. There are several advantages of using Bayesian approach: Parameter and prediction uncertainty become easily available, facilitating rigid statistical analysis. Furthermore, prior knowledge can be incorporated. However so far there have been no scalable techniques capable of combining both model (structural) and parameter uncertainty. In this paper we introduce the concept of model uncertainty in BNNs and hence make inference in the joint space of models and parameters. Moreover, we suggest an adaptation of a scalable variational inference approach with reparametrization of marginal inclusion probabilities to incorporate the model space constraints. Finally, we show that incorporating model uncertainty via Bayesian model averaging and Bayesian model selection allows to drastically sparsify the structure of BNNs.
Markov chain Monte Carlo (MCMC) algorithms are widely used to sample from complicated distributions, especially to sample from the posterior distribution in Bayesian inference. However, MCMC is not directly applicable when facing the doubly intractable problem. In this paper, we discussed and compared two existing solutions -- Pseudo-marginal Monte Carlo and Exchange Algorithm. This paper also proposes a novel algorithm: Multi-armed Bandit MCMC (MABMC), which chooses between two (or more) randomized acceptance ratios in each step. MABMC could be applied directly to incorporate Pseudo-marginal Monte Carlo and Exchange algorithm, with higher average acceptance probability.
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 approximates the functional ELBO using finite measurement sets and the spectral Stein gradient estimator. With fBNNs, we can specify priors entailing rich structures, including Gaussian processes and implicit stochastic processes. Empirically, we find fBNNs extrapolate well using various structured priors, provide reliable uncertainty estimates, and scale to large datasets.
Mental simulation is a critical cognitive function for goal-directed behavior because it is essential for assessing actions and their consequences. When a self-generated or externally specified goal is given, a sequence of actions that is most likely to attain that goal is selected among other candidates via mental simulation. Therefore, better mental simulation leads to better goal-directed action planning. However, developing a mental simulation model is challenging because it requires knowledge of self and the environment. The current paper studies how adequate goal-directed action plans of robots can be mentally generated by dynamically organizing top-down visual attention and visual working memory. For this purpose, we propose a neural network model based on variational Bayes predictive coding, where goal-directed action planning is formulated by Bayesian inference of latent intentional space. Our experimental results showed that cognitively meaningful competencies, such as autonomous top-down attention to the robot end effector (its hand) as well as dynamic organization of occlusion-free visual working memory, emerged. Furthermore, our analysis of comparative experiments indicated that introduction of visual working memory and the inference mechanism using variational Bayes predictive coding significantly improve the performance in planning adequate goal-directed actions.
A core problem in statistics and probabilistic machine learning is to compute probability distributions and expectations. This is the fundamental problem of Bayesian statistics and machine learning, which frames all inference as expectations with respect to the posterior distribution. The key challenge is to approximate these intractable expectations. In this tutorial, we review sequential Monte Carlo (SMC), a random-sampling-based class of methods for approximate inference. First, we explain the basics of SMC, discuss practical issues, and review theoretical results. We then examine two of the main user design choices: the proposal distributions and the so called intermediate target distributions. We review recent results on how variational inference and amortization can be used to learn efficient proposals and target distributions. Next, we discuss the SMC estimate of the normalizing constant, how this can be used for pseudo-marginal inference and inference evaluation. Throughout the tutorial we illustrate the use of SMC on various models commonly used in machine learning, such as stochastic recurrent neural networks, probabilistic graphical models, and probabilistic programs.
While MCMC methods have become a main work-horse for Bayesian inference, scaling them to large distributed datasets is still a challenge. Embarrassingly parallel MCMC strategies take a divide-and-conquer stance to achieve this by writing the target posterior as a product of subposteriors, running MCMC for each of them in parallel and subsequently combining the results. The challenge then lies in devising efficient aggregation strategies. Current strategies trade-off between approximation quality, and costs of communication and computation. In this work, we introduce a novel method that addresses these issues simultaneously. Our key insight is to introduce a deep invertible transformation to approximate each of the subposteriors. These approximations can be made accurate even for complex distributions and serve as intermediate representations, keeping the total communication cost limited. Moreover, they enable us to sample from the product of the subposteriors using an efficient and stable importance sampling scheme. We demonstrate the approach outperforms available state-of-the-art methods in a range of challenging scenarios, including high-dimensional and heterogeneous subposteriors.
Energy-efficient simultaneous localization and mapping (SLAM) is crucial for mobile robots exploring unknown environments. The mammalian brain solves SLAM via a network of specialized neurons, exhibiting asynchronous computations and event-based communications, with very low energy consumption. We propose a brain-inspired spiking neural network (SNN) architecture that solves the unidimensional SLAM by introducing spike-based reference frame transformation, visual likelihood computation, and Bayesian inference. We integrated our neuromorphic algorithm to Intel's Loihi neuromorphic processor, a non-Von Neumann hardware that mimics the brain's computing paradigms. We performed comparative analyses for accuracy and energy-efficiency between our neuromorphic approach and the GMapping algorithm, which is widely used in small environments. Our Loihi-based SNN architecture consumes 100 times less energy than GMapping run on a CPU while having comparable accuracy in head direction localization and map-generation. These results pave the way for scaling our approach towards active-SLAM alternative solutions for Loihi-controlled autonomous robots.
Scalable communication is of utmost importance for reliable dissemination of time-sensitive information in cooperative vehicular ad-hoc networks (VANETs), which is, in turn, an essential prerequisite for the proper operation of the critical cooperative safety applications. The model-based communication (MBC) is a recently-explored scalability solution proposed in the literature, which has shown a promising potential to reduce the channel congestion to a great extent. In this work, based on the MBC notion, a technology-agnostic hybrid model selection policy for Vehicle-to-Everything (V2X) communication is proposed which benefits from the characteristics of the non-parametric Bayesian inference techniques, specifically Gaussian Processes. The results show the effectiveness of the proposed communication architecture on both reducing the required message exchange rate and increasing the remote agent tracking precision.
Variational Bayes (VB) is a recent approximate method for Bayesian inference. It has the merit of being a fast and scalable alternative to Markov Chain Monte Carlo (MCMC) but its approximation error is often unknown. In this paper, we derive the approximation error of VB in terms of mean, mode, variance, predictive density and KL divergence for the linear Gaussian multi-equation regression. Our results indicate that VB approximates the posterior mean perfectly. Factors affecting the magnitude of underestimation in posterior variance and mode are revealed. Importantly, We demonstrate that VB estimates predictive densities accurately.
Existing model-based reinforcement learning methods often study perception modeling and decision making separately. We introduce joint Perception and Control as Inference (PCI), a general framework to combine perception and control for partially observable environments through Bayesian inference. Based on the fact that object-level inductive biases are critical in human perceptual learning and reasoning, we propose Object-based Perception Control (OPC), an instantiation of PCI which manages to facilitate control using automatic discovered object-based representations. We develop an unsupervised end-to-end solution and analyze the convergence of the perception model update. Experiments in a high-dimensional pixel environment demonstrate the learning effectiveness of our object-based perception control approach. Specifically, we show that OPC achieves good perceptual grouping quality and outperforms several strong baselines in accumulated rewards.
Approximate Bayesian computation (ABC) is a set of techniques for Bayesian inference when the likelihood is intractable but sampling from the model is possible. This work presents a simple yet effective ABC algorithm based on the combination of two classical ABC approaches --- regression ABC and sequential ABC. The key idea is that rather than learning the posterior directly, we first target another auxiliary distribution that can be learned accurately by existing methods, through which we then subsequently learn the desired posterior with the help of a Gaussian copula. During this process, the complexity of the model changes adaptively according to the data at hand. Experiments on a synthetic dataset as well as three real-world inference tasks demonstrates that the proposed method is fast, accurate, and easy to use.
Dynamic paired comparison models, such as Elo and Glicko, are frequently used for sports prediction and ranking players or teams. We present an alternative dynamic paired comparison model which uses a Gaussian Process (GP) as a prior for the time dynamics rather than the Markovian dynamics usually assumed. In addition, we show that the GP model can easily incorporate covariates. We derive an efficient approximate Bayesian inference procedure based on the Laplace Approximation and sparse linear algebra. We select hyperparameters by maximising their marginal likelihood using Bayesian Optimisation, comparing the results against random search. Finally, we fit and evaluate the model on the 2018 season of ATP tennis matches, where it performs competitively, outperforming Elo and Glicko on log loss, particularly when surface covariates are included.
Robust MDPs (RMDPs) can be used to compute policies with provable worst-case guarantees in reinforcement learning. The quality and robustness of an RMDP solution are determined by the ambiguity set---the set of plausible transition probabilities---which is usually constructed as a multi-dimensional confidence region. Existing methods construct ambiguity sets as confidence regions using concentration inequalities which leads to overly conservative solutions. This paper proposes a new paradigm that can achieve better solutions with the same robustness guarantees without using confidence regions as ambiguity sets. To incorporate prior knowledge, our algorithms optimize the size and position of ambiguity sets using Bayesian inference. Our theoretical analysis shows the safety of the proposed method, and the empirical results demonstrate its practical promise.
Online detection of instantaneous changes in the generative process of a data sequence generally focuses on retrospective inference of such change points without considering their future occurrences. We extend the Bayesian Online Change Point Detection algorithm to also infer the number of time steps until the next change point (i.e., the residual time). This enables to handle observation models which depend on the total segment duration, which is useful to model data sequences with temporal scaling. The resulting inference algorithm for segment detection can be deployed in an online fashion, and we illustrate applications to synthetic and to two medical real-world data sets.