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 application in statistics is somewhat limited. This paper develops a variational Bayesian neural network estimation methodology and related statistical theory. The numerical algorithms and their implementational are discussed in detail. The theory for posterior consistency, a desirable property in nonparametric Bayesian statistics, is also developed. This theory provides an assessment of prediction accuracy and guidelines for characterizing the prior distributions and variational family. The loss of using a variational posterior over the true posterior has also been quantified. The development is motivated by an important biomedical engineering application, namely building predictive tools for the transition from mild cognitive impairment to Alzheimer's disease. The predictors are multi-modal and may involve complex interactive relations.

Recent work has attempted to directly approximate the `function-space' or predictive posterior distribution of Bayesian models, without approximating the posterior distribution over the parameters. This is appealing in e.g. Bayesian neural networks, where we only need the former, and the latter is hard to represent. In this work, we highlight some advantages and limitations of employing the Kullback-Leibler divergence in this setting. For example, we show that minimizing the KL divergence between a wide class of parametric distributions and the posterior induced by a (non-degenerate) Gaussian process prior leads to an ill-defined objective function. Then, we propose (featurized) Bayesian linear regression as a benchmark for `function-space' inference methods that directly measures approximation quality. We apply this methodology to assess aspects of the objective function and inference scheme considered in Sun, Zhang, Shi, and Grosse (2018), emphasizing the quality of approximation to Bayesian inference as opposed to predictive performance.

We introduce a new variational inference (VI) framework, called energetic variational inference (EVI). It minimizes the VI object function based on a prescribed energy-dissipation law. Using the EVI framework, we can derive many existing Particle-based Variational Inference (ParVI) methods, including the popular Stein Variational Gradient Descent (SVGD) approach. More importantly, many new ParVI schemes can be created under this framework. For illustration, we propose a new particle-based EVI scheme, which performs the particle-based approximation of the density first and then uses the approximated density in the variational procedure, or "Approximation-then-Variation" for short. Thanks to this order of approximation and variation, the new scheme can maintain the variational structure at the particle level and can significantly decrease the KL-divergence in each iteration. Numerical experiments show the proposed method outperforms some existing ParVI methods in terms of fidelity to the target distribution.

Query optimizers rely on accurate cardinality estimation (CardEst) to produce good execution plans. The core problem of CardEst is how to model the rich joint distribution of attributes in an accurate and compact manner. Despite decades of research, existing methods either over simplify the models only using independent factorization which leads to inaccurate estimates and sub optimal query plans, or over-complicate them by lossless conditional factorization without any independent assumption which results in slow probability computation. In this paper, we propose FLAT, a CardEst method that is simultaneously fast in probability computation, lightweight in model size and accurate in estimation quality. The key idea of FLAT is a novel unsupervised graphical model, called FSPN. It utilizes both independent and conditional factorization to adaptively model different levels of attributes correlations, and thus subsumes all existing CardEst models and dovetails their advantages. FLAT supports efficient online probability computation in near liner time on the underlying FSPN model, and provides effective offline model construction. It can estimate cardinality for both single table queries and multi-table join queries. Extensive experimental study demonstrates the superiority of FLAT over existing CardEst methods on well-known benchmarks: FLAT achieves 1 to 5 orders of magnitude better accuracy, 1 to 3 orders of magnitude faster probability computation speed (around 0.2ms) and 1 to 2 orders of magnitude lower storage cost (only tens of KB).

We study the problem of predicting and controlling the future state distribution of an autonomous agent. This problem, which can be viewed as a reframing of goal-conditioned reinforcement learning (RL), is centered around learning a conditional probability density function over future states. Instead of directly estimating this density function, we indirectly estimate this density function by training a classifier to predict whether an observation comes from the future. Via Bayes' rule, predictions from our classifier can be transformed into predictions over future states. Importantly, an off-policy variant of our algorithm allows us to predict the future state distribution of a new policy, without collecting new experience. This variant allows us to optimize functionals of a policy's future state distribution, such as the density of reaching a particular goal state. While conceptually similar to Q-learning, our work lays a principled foundation for goal-conditioned RL as density estimation, providing justification for goal-conditioned methods used in prior work. This foundation makes hypotheses about Q-learning, including the optimal goal-sampling ratio, which we confirm experimentally. Moreover, our proposed method is competitive with prior goal-conditioned RL methods.

In discriminative settings such as regression and classification there are two random variables at play, the inputs X and the targets Y. Here, we demonstrate that the Variational Information Bottleneck can be viewed as a compromise between fully empirical and fully Bayesian objectives, attempting to minimize the risks due to finite sampling of Y only. We argue that this approach provides some of the benefits of Bayes while requiring only some of the work.

Complex simulators have become a ubiquitous tool in many scientific disciplines, providing high-fidelity, implicit probabilistic models of natural and social phenomena. Unfortunately, they typically lack the tractability required for conventional statistical analysis. Approximate Bayesian computation (ABC) has emerged as a key method in simulation-based inference, wherein the true model likelihood and posterior are approximated using samples from the simulator. In this paper, we draw connections between ABC and generalized Bayesian inference (GBI). First, we re-interpret the accept/reject step in ABC as an implicitly defined error model. We then argue that these implicit error models will invariably be misspecified. While ABC posteriors are often treated as a necessary evil for approximating the standard Bayesian posterior, this allows us to re-interpret ABC as a potential robustification strategy. This leads us to suggest the use of GBI within ABC, a use case we explore empirically.

Semi-continuous data comes from a distribution that is a mixture of the point mass at zero and a continuous distribution with support on the positive real line. A clear example is the daily rainfall data. In this paper, we present a novel algorithm to estimate the density function for semi-continuous data using the principle of maximum entropy. Unlike existing methods in the literature, our algorithm needs only the sample values of the constraint functions in the entropy maximization problem and does not need the entire sample. Using simulations, we show that the estimate of the entropy produced by our algorithm has significantly less bias compared to existing methods. An application to the daily rainfall data is provided.

Predictions obtained by, e.g., artificial neural networks have a high accuracy but humans often perceive the models as black boxes. Insights about the decision making are mostly opaque for humans. Particularly understanding the decision making in highly sensitive areas such as healthcare or fifinance, is of paramount importance. The decision-making behind the black boxes requires it to be more transparent, accountable, and understandable for humans. This survey paper provides essential definitions, an overview of the different principles and methodologies of explainable Supervised Machine Learning (SML). We conduct a state-of-the-art survey that reviews past and recent explainable SML approaches and classifies them according to the introduced definitions. Finally, we illustrate principles by means of an explanatory case study and discuss important future directions.

Contemporary machine learning applications often involve classification tasks with many classes. Despite their extensive use, a precise understanding of the statistical properties and behavior of classification algorithms is still missing, especially in modern regimes where the number of classes is rather large. In this paper, we take a step in this direction by providing the first asymptotically precise analysis of linear multiclass classification. Our theoretical analysis allows us to precisely characterize how the test error varies over different training algorithms, data distributions, problem dimensions as well as number of classes, inter/intra class correlations and class priors. Specifically, our analysis reveals that the classification accuracy is highly distribution-dependent with different algorithms achieving optimal performance for different data distributions and/or training/features sizes. Unlike linear regression/binary classification, the test error in multiclass classification relies on intricate functions of the trained model (e.g., correlation between some of the trained weights) whose asymptotic behavior is difficult to characterize. This challenge is already present in simple classifiers, such as those minimizing a square loss. Our novel theoretical techniques allow us to overcome some of these challenges. The insights gained may pave the way for a precise understanding of other classification algorithms beyond those studied in this paper.