Exploiting the appropriate inductive bias based on the knowledge of data is essential for achieving good performance in statistical machine learning. In practice, however, the domain knowledge of interest often provides information on the relationship of data attributes only distantly, which hinders direct utilization of such domain knowledge in popular regularization methods. In this paper, we propose the knowledge-based distant regularization framework, in which we utilize the distant information encoded in a knowledge graph for regularization of probabilistic model estimation. In particular, we propose to impose prior distributions on model parameters specified by knowledge graph embeddings. As an instance of the proposed framework, we present the factor analysis model with the knowledge-based distant regularization. We show the results of preliminary experiments on the improvement of the generalization capability of such model.

Bayesian learning is built on an assumption that the model space contains a true reflection of the data generating mechanism. This assumption is problematic, particularly in complex data environments. Here we present a Bayesian nonparametric approach to learning that makes use of statistical models, but does not assume that the model is true. Our approach has provably better properties than using a parametric model and admits a trivially parallelizable Monte Carlo sampling scheme that affords massive scalability on modern computer architectures. The model-based aspect of learning is particularly attractive for regularizing nonparametric inference when the sample size is small, and also for correcting approximate approaches such as variational Bayes (VB). We demonstrate the approach on a number of examples including VB classifiers and Bayesian random forests.

Probabilistic Bisection Algorithm performs root finding based on knowledge acquired from noisy oracle responses. We consider the generalized PBA setting (G-PBA) where the statistical distribution of the oracle is unknown and location-dependent, so that model inference and Bayesian knowledge updating must be performed simultaneously. To this end, we propose to leverage the spatial structure of a typical oracle by constructing a statistical surrogate for the underlying logistic regression step. We investigate several non-parametric surrogates, including Binomial Gaussian Processes (B-GP), Polynomial, Kernel, and Spline Logistic Regression. In parallel, we develop sampling policies that adaptively balance learning the oracle distribution and learning the root. One of our proposals mimics active learning with B-GPs and provides a novel look-ahead predictive variance formula. The resulting gains of our Spatial PBA algorithm relative to earlier G-PBA models are illustrated with synthetic examples and a challenging stochastic root finding problem from Bermudan option pricing.

We present a Bayesian view of counterfactual risk minimization (CRM), also known as offline policy optimization from logged bandit feedback. Using PAC-Bayesian analysis, we derive a new generalization bound for the truncated IPS estimator. We apply the bound to a class of Bayesian policies, which motivates a novel, potentially data-dependent, regularization technique for CRM.

The PC algorithm is a popular method for learning the structure of Gaussian Bayesian networks. It carries out statistical tests to determine absent edges in the network. It is hence governed by two parameters: (i) The type of test, and (ii) its significance level. These parameters are usually set to values recommended by an expert. Nevertheless, such an approach can suffer from human bias, leading to suboptimal reconstruction results. In this paper we consider a more principled approach for choosing these parameters in an automatic way. For this we optimize a reconstruction score evaluated on a set of different Gaussian Bayesian networks. This objective is expensive to evaluate and lacks a closed-form expression, which means that Bayesian optimization (BO) is a natural choice. BO methods use a model to guide the search and are hence able to exploit smoothness properties of the objective surface. We show that the parameters found by a BO method outperform those found by a random search strategy and the expert recommendation. Importantly, we have found that an often overlooked statistical test provides the best over-all reconstruction results.

Standard approaches to probabilistic reasoning require that one possesses an explicit model of the distribution in question. But, the empirical learning of models of probability distributions from partial observations is a problem for which efficient algorithms are generally not known. In this work we consider the use of bounded-degree fragments of the "sum-of-squares" logic as a probability logic. Prior work has shown that we can decide refutability for such fragments in polynomial-time. We propose to use such fragments to answer queries about whether a given probability distribution satisfies a given system of constraints and bounds on expected values. We show that in answering such queries, such constraints and bounds can be implicitly learned from partial observations in polynomial-time as well. It is known that this logic is capable of deriving many bounds that are useful in probabilistic analysis. We show here that it furthermore captures useful polynomial-time fragments of resolution. Thus, these fragments are also quite expressive.

Long-lead forecasting for spatio-temporal problems can often entail complex nonlinear dynamics that are difficult to specify it a priori. Current statistical methodologies for modeling these processes are often overparameterized and thus, struggle from a computational perspective. One potential parsimonious solution to this problem is a method from the dynamical systems and engineering literature referred to as an echo state network (ESN). ESN models use so-called reservoir computing to efficiently estimate a dynamical neural network forecast, model referred to as a recurrent neural network (RNN). Moreover, so-called deep models have recently been shown to be successful at predicting high-dimensional complex nonlinear processes. These same traits can be used to characterize many spatio-temporal processes. Here we introduce a deep ensemble ESN (D-EESN) model. Through the use of an ensemble framework, this model is able to generate forecasts that are accompanied by uncertainty estimates. After introducing the D-EESN, we then develop a hierarchical Bayesian implementation. We use a general hierarchical Bayesian framework that accommodates non-Gaussian data types and multiple levels of uncertainties. The proposed methodology is first applied to a data set simulated from a novel non-Gaussian multiscale Lorenz-96 dynamical system simulation model and then to a long-lead United States (U.S.) soil moisture forecasting application.

We define a new class of Bayesian point estimators, which we refer to as risk-averse estimators. We then use this definition to formulate several axioms that we claim to be natural requirements for good inference procedures, and show that for two classes of estimation problems the axioms uniquely characterise an estimator. Namely, for estimation problems with a discrete hypothesis space, we show that the axioms lead to the MAP estimate, whereas for well-behaved, purely continuous estimation problems the axioms lead to the Wallace-Freeman estimate. Interestingly, this combined use of MAP and Wallace-Freeman estimation reflects the common practice in the Minimum Message Length (MML) community, but there these two estimators are used as approximations for the information-theoretic Strict MML estimator, whereas we derive them exactly, not as approximations, and do so with no use of encoding or information theory. Keywords: Bayes estimation, risk-averse, inference, axiomatic approach, MML, Wallace-Freeman, invariance 1. Introduction One of the fundamental statistical problems is point estimation. In a Bayesian setting, this can be described as follows. Let (x,θ) X Θ be a pair of random variables with a known joint distribution that assigns positive probability / probability density to any (x,θ) X Θ.

Learning to infer Bayesian posterior from a few-shot dataset is an important step towards robust meta-learning due to the model uncertainty inherent in the problem. In this paper, we propose a novel Bayesian model-agnostic meta-learning method. The proposed method combines scalable gradient-based meta-learning with nonparametric variational inference in a principled probabilistic framework. During fast adaptation, the method is capable of learning complex uncertainty structure beyond a point estimate or a simple Gaussian approximation. In addition, a robust Bayesian meta-update mechanism with a new meta-loss prevents overfitting during meta-update. Remaining an efficient gradient-based meta-learner, the method is also model-agnostic and simple to implement. Experiment results show the accuracy and robustness of the proposed method in various tasks: sinusoidal regression, image classification, active learning, and reinforcement learning.

Bayesian neural networks (BNNs) allow us to reason about uncertainty in a principled way. Stochastic Gradient Langevin Dynamics (SGLD) enables efficient BNN learning by drawing samples from the BNN posterior using mini-batches. However, SGLD and its extensions require storage of many copies of the model parameters, a potentially prohibitive cost, especially for large neural networks. We propose a framework, Adversarial Posterior Distillation, to distill the SGLD samples using a Generative Adversarial Network (GAN). At test-time, samples are generated by the GAN. We show that this distillation framework incurs no loss in performance on recent BNN applications including anomaly detection, active learning, and defense against adversarial attacks. By construction, our framework not only distills the Bayesian predictive distribution, but the posterior itself. This allows one to compute quantities such as the approximate model variance, which is useful in downstream tasks. To our knowledge, these are the first results applying MCMC-based BNNs to the aforementioned downstream applications.