Information theory provides a mathematical foundation to measure uncertainty in belief. Belief is represented by a probability distribution that captures our understanding of an outcome's plausibility. Information measures based on Shannon's concept of entropy include realization information, Kullback-Leibler divergence, Lindley's information in experiment, cross entropy, and mutual information. We derive a general theory of information from first principles that accounts for evolving belief and recovers all of these measures. Rather than simply gauging uncertainty, information is understood in this theory to measure change in belief. We may then regard entropy as the information we expect to gain upon realization of a discrete latent random variable. This theory of information is compatible with the Bayesian paradigm in which rational belief is updated as evidence becomes available. Furthermore, this theory admits novel measures of information with well-defined properties, which we explore in both analysis and experiment. This view of information illuminates the study of machine learning by allowing us to quantify information captured by a predictive model and distinguish it from residual information contained in training data. We gain related insights regarding feature selection, anomaly detection, and novel Bayesian approaches.

We consider the problem of identifying universal low-dimensional features from high-dimensional data for inference tasks in settings involving learning. For such problems, we introduce natural notions of universality and we show a local equivalence among them. Our analysis is naturally expressed via information geometry, and represents a conceptually and computationally useful analysis. The development reveals the complementary roles of the singular value decomposition, Hirschfeld-Gebelein-R\'enyi maximal correlation, the canonical correlation and principle component analyses of Hotelling and Pearson, Tishby's information bottleneck, Wyner's common information, Ky Fan $k$-norms, and Brieman and Friedman's alternating conditional expectations algorithm. We further illustrate how this framework facilitates understanding and optimizing aspects of learning systems, including multinomial logistic (softmax) regression and the associated neural network architecture, matrix factorization methods for collaborative filtering and other applications, rank-constrained multivariate linear regression, and forms of semi-supervised learning.

Often the development of novel materials is not amenable to high-throughput or purely computational screening methods. Instead, materials must be synthesized one at a time in a process that does not generate significant amounts of data. One way this method can be improved is by ensuring that each experiment provides the best improvement in both material properties and predictive modeling accuracy. In this work, we study the effectiveness of active learning, which optimizes the order of experiments, and meta learning, which transfers knowledge from one context to another, to reduce the number of experiments necessary to build a predictive model. We present a novel multi-task benchmark database of peptides designed to advance active, few-shot, and meta-learning methods for experimental design. Each task is binary classification of peptides represented as a sequence string. We show results of standard active learning and meta-learning methods across these datasets to assess their ability to improve predictive models with the fewest number of experiments. We find the ensemble query by committee active learning method to be effective. The meta-learning method Reptile was found to improve accuracy. The robustness of these conclusions were tested across multiple model choices.

Due to limited computational power, performing uncertainty quantification analyses with complex computational models can be a challenging task. This is exacerbated in the context of stochastic simulators, the response of which to a given set of input parameters, rather than being a deterministic value, is a random variable with unknown probability density function (PDF). Of interest in this paper is the construction of a surrogate that can accurately predict this response PDF for any input parameters. We suggest using a flexible distribution family -- the generalized lambda distribution -- to approximate the response PDF. The associated distribution parameters are cast as functions of input parameters and represented by sparse polynomial chaos expansions. To build such a surrogate model, we propose an approach based on a local inference of the response PDF at each point of the experimental design based on replicated model evaluations. Two versions of this framework are proposed and compared on analytical examples and case studies.

It is a particularly well-suited approach to better understand the underlying structure of data when scientific understanding of the data is at an early stage. BN modelling is designed to sort out directly from indirectly related variables and offers a far richer modelling framework than classical approaches in epidemiology like, e.g., regression techniques or extensions thereof. In contrast to structural equation modelling (Hair, Black, Babin, Anderson, Tatham et al. 1998), which requires expert knowledge to design the model, the Additive Bayesian Network (ABN) method is a data-driven approach (Lewis and Ward 2013; Kratzer, Pittavino, Lewis, and Furrer 2019b). It does not rely on expert knowledge, but it can possiarXiv:1911.09006v1

Sparse convex clustering is to cluster observations and conduct variable selection simultaneously in the framework of convex clustering. Although the weighted $L_1$ norm as the regularization term is usually employed in the sparse convex clustering, this increases the dependence on the data and reduces the estimation accuracy if the sample size is not sufficient. To tackle these problems, this paper proposes a Bayesian sparse convex clustering via the idea of Bayesian lasso and global-local shrinkage priors. We introduce Gibbs sampling algorithms for our method using scale mixtures of normals. The effectiveness of the proposed methods is shown in simulation studies and a real data analysis.

This paper studies the theoretical predictive properties of classes of forecast combination methods. The study is motivated by the recently developed Bayesian framework for synthesizing predictive densities: Bayesian predictive synthesis. A novel strategy based on continuous time stochastic processes is proposed and developed, where the combined predictive error processes are expressed as stochastic differential equations, evaluated using Ito's lemma. We show that a subclass of synthesis functions under Bayesian predictive synthesis, which we categorize as non-linear synthesis, entails an extra term that "corrects" the bias from misspecification and dependence in the predictive error process, effectively improving forecasts. Theoretical properties are examined and shown that this subclass improves the expected squared forecast error over any and all linear combination, averaging, and ensemble of forecasts, under mild conditions. We discuss the conditions for which this subclass outperforms others, and its implications for developing forecast combination methods. A finite sample simulation study is presented to illustrate our results.

Flow-based generative models leverage invertible generator functions to fit a distribution to the training data using maximum likelihood. Despite their use in several application domains, robustness of these models to adversarial attacks has hardly been explored. In this paper, we study adversarial robustness of flow-based generative models both theoretically (for some simple models) and empirically (for more complex ones). First, we consider a linear flow-based generative model and compute optimal sample-specific and universal adversarial perturbations that maximally decrease the likelihood scores. Using this result, we study the robustness of the well-known adversarial training procedure, where we characterize the fundamental trade-off between model robustness and accuracy. Next, we empirically study the robustness of two prominent deep, non-linear, flow-based generative models, namely GLOW and RealNVP. We design two types of adversarial attacks; one that minimizes the likelihood scores of in-distribution samples, while the other that maximizes the likelihood scores of out-of-distribution ones. We find that GLOW and RealNVP are extremely sensitive to both types of attacks. Finally, using a hybrid adversarial training procedure, we significantly boost the robustness of these generative models.

We study the problem of learning permutation invariant representations that can capture "flexible" notions of containment. We formalize this problem via a measure theoretic definition of multisets, and obtain a theoretically-motivated learning model. We propose training this model on a novel task: predicting the size of the symmetric difference (or intersection) between pairs of multisets. We demonstrate that our model not only performs very well on predicting containment relations (and more effectively predicts the sizes of symmetric differences and intersections than DeepSets-based approaches with unconstrained object representations), but that it also learns meaningful representations.

How to Make AdaBoost.M1 Work for Weak Base Classifiers by Changing Only One Line of the Code (ECML 2002)