Black-box machine learning learning methods are now routinely used in high-risk settings, like medical diagnostics, which demand uncertainty quantification to avoid consequential model failures. Distribution-free uncertainty quantification (distribution-free UQ) is a user-friendly paradigm for creating statistically rigorous confidence intervals/sets for such predictions. Critically, the intervals/sets are valid without distributional assumptions or model assumptions, with explicit guarantees with finitely many datapoints. Moreover, they adapt to the difficulty of the input; when the input example is difficult, the uncertainty intervals/sets are large, signaling that the model might be wrong. Without much work, one can use distribution-free methods on any underlying algorithm, such as a neural network, to produce confidence sets guaranteed to contain the ground truth with a user-specified probability, such as 90%. Indeed, the methods are easy-to-understand and general, applying to many modern prediction problems arising in the fields of computer vision, natural language processing, deep reinforcement learning, and so on. This hands-on introduction is aimed at a reader interested in the practical implementation of distribution-free UQ, including conformal prediction and related methods, who is not necessarily a statistician. We will include many explanatory illustrations, examples, and code samples in Python, with PyTorch syntax. The goal is to provide the reader a working understanding of distribution-free UQ, allowing them to put confidence intervals on their algorithms, with one self-contained document.

Bayesian neural networks provide a direct and natural way to extend standard deep neural networks to support probabilistic deep learning through the use of probabilistic layers that, traditionally, encode weight (and bias) uncertainty. In particular, hybrid Bayesian neural networks utilize standard deterministic layers together with few probabilistic layers judicially positioned in the networks for uncertainty estimation. A major aspect and benefit of Bayesian inference is that priors, in principle, provide the means to encode prior knowledge for use in inference and prediction. However, it is difficult to specify priors on weights since the weights have no intuitive interpretation. Further, the relationships of priors on weights to the functions computed by networks are difficult to characterize. In contrast, functions are intuitive to interpret and are direct since they map inputs to outputs. Therefore, it is natural to specify priors on functions to encode prior knowledge, and to use them in inference and prediction based on functions. To support this, we propose hybrid Bayesian neural networks with functional probabilistic layers that encode function (and activation) uncertainty. We discuss their foundations in functional Bayesian inference, functional variational inference, sparse Gaussian processes, and sparse variational Gaussian processes. We further perform few proof-of-concept experiments using GPflus, a new library that provides Gaussian process layers and supports their use with deterministic Keras layers to form hybrid neural network and Gaussian process models.

This paper presents a deep Inverse Reinforcement Learning (IRL) framework that can learn an a priori unknown number of nonlinear reward functions from unlabeled experts' demonstrations. For this purpose, we employ the tools from Dirichlet processes and propose an adaptive approach to simultaneously account for both complex and unknown number of reward functions. Using the conditional maximum entropy principle, we model the experts' multi-intention behaviors as a mixture of latent intention distributions and derive two algorithms to estimate the parameters of the deep reward network along with the number of experts' intentions from unlabeled demonstrations. The proposed algorithms are evaluated on three benchmarks, two of which have been specifically extended in this study for multi-intention IRL, and compared with well-known baselines. We demonstrate through several experiments the advantages of our algorithms over the existing approaches and the benefits of online inferring, rather than fixing beforehand, the number of expert's intentions.

Regression and interpolation with Gaussian processes, or kriging, is a popular statistical tool for non-parametric function estimation, originating from geostatistics and time series analysis, and later adopted in many other areas such as machine learning and the design and analysis of computer experiments (see, e.g., Stein, 1999; Santner et al., 2003; Rasmussen and Williams, 2006, and references therein). It is widely used for constructing fast approximations of time-consuming computer models, with applications to calibration and validation (Kennedy and O'Hagan, 2001; Bayarri et al., 2007), engineering design (Jones et al., 1998; Forrester et al., 2008), Bayesian inference (Calderhead et al., 2009; Wilkinson, 2014), and the optimization of machine learning algorithms (Bergstra et al., 2011)--to name but a few. A Gaussian process (GP) prior is characterized by its mean and covariance functions. They are usually chosen within parametric families (for instance, constant or linear mean functions, and Matérn covariance functions), which transfers the problem of choosing the mean and covariance functions to that of selecting parameters. The selection is most often carried out by optimization of a criterion that measures the goodness of fit of the predictive distributions, and a variety of such criteria--the likelihood function, the leave-one-out (LOO) squared-predictionerror criterion (hereafter denoted by LOO-SPE), and others--is available from the literature.

Hierarchical Bayesian methods enable information sharing across multiple related regression problems. While standard practice is to model regression parameters (effects) as (1) exchangeable across datasets and (2) correlated to differing degrees across covariates, we show that this approach exhibits poor statistical performance when the number of covariates exceeds the number of datasets. For instance, in statistical genetics, we might regress dozens of traits (defining datasets) for thousands of individuals (responses) on up to millions of genetic variants (covariates). When an analyst has more covariates than datasets, we argue that it is often more natural to instead model effects as (1) exchangeable across covariates and (2) correlated to differing degrees across datasets. To this end, we propose a hierarchical model expressing our alternative perspective. We devise an empirical Bayes estimator for learning the degree of correlation between datasets. We develop theory that demonstrates that our method outperforms the classic approach when the number of covariates dominates the number of datasets, and corroborate this result empirically on several high-dimensional multiple regression and classification problems.

Dempster-Shafer Theory (DST) generalizes Bayesian probability theory, offering useful additional information, but suffers from a high computational burden. A lot of work has been done to reduce the complexity of computations used in information fusion with Dempster's rule. Yet, few research had been conducted to reduce the complexity of computations for the conjunctive and disjunctive decompositions of evidence, which are at the core of other important methods of information fusion. In this paper, we propose a method designed to exploit the actual evidence (information) contained in these decompositions in order to compute them. It is based on a new notion that we call focal point, derived from the notion of focal set. With it, we are able to reduce these computations up to a linear complexity in the number of focal sets in some cases. In a broader perspective, our formulas have the potential to be tractable when the size of the frame of discernment exceeds a few dozen possible states, contrary to the existing litterature. This article extends (and translates) our work published at the french conference GRETSI in 2019.

Robots learning from observations in the real world using inverse reinforcement learning (IRL) may encounter objects or agents in the environment, other than the expert, that cause nuisance observations during the demonstration. These confounding elements are typically removed in fully-controlled environments such as virtual simulations or lab settings. When complete removal is impossible the nuisance observations must be filtered out. However, identifying the source of observations when large amounts of observations are made is difficult. To address this, we present a hierarchical Bayesian model that incorporates both the expert's and the confounding elements' observations thereby explicitly modeling the diverse observations a robot may receive. We extend an existing IRL algorithm originally designed to work under partial occlusion of the expert to consider the diverse observations. In a simulated robotic sorting domain containing both occlusion and confounding elements, we demonstrate the model's effectiveness. In particular, our technique outperforms several other comparative methods, second only to having perfect knowledge of the subject's trajectory.

This paper studies the estimation of high-dimensional, discrete, possibly sparse, mixture models in topic models. The data consists of observed multinomial counts of $p$ words across $n$ independent documents. In topic models, the $p\times n$ expected word frequency matrix is assumed to be factorized as a $p\times K$ word-topic matrix $A$ and a $K\times n$ topic-document matrix $T$. Since columns of both matrices represent conditional probabilities belonging to probability simplices, columns of $A$ are viewed as $p$-dimensional mixture components that are common to all documents while columns of $T$ are viewed as the $K$-dimensional mixture weights that are document specific and are allowed to be sparse. The main interest is to provide sharp, finite sample, $\ell_1$-norm convergence rates for estimators of the mixture weights $T$ when $A$ is either known or unknown. For known $A$, we suggest MLE estimation of $T$. Our non-standard analysis of the MLE not only establishes its $\ell_1$ convergence rate, but reveals a remarkable property: the MLE, with no extra regularization, can be exactly sparse and contain the true zero pattern of $T$. We further show that the MLE is both minimax optimal and adaptive to the unknown sparsity in a large class of sparse topic distributions. When $A$ is unknown, we estimate $T$ by optimizing the likelihood function corresponding to a plug in, generic, estimator $\hat{A}$ of $A$. For any estimator $\hat{A}$ that satisfies carefully detailed conditions for proximity to $A$, the resulting estimator of $T$ is shown to retain the properties established for the MLE. The ambient dimensions $K$ and $p$ are allowed to grow with the sample sizes. Our application is to the estimation of 1-Wasserstein distances between document generating distributions. We propose, estimate and analyze new 1-Wasserstein distances between two probabilistic document representations.

Bayesian models based on the Dirichlet process and other stick-breaking priors have been proposed as core ingredients for clustering, topic modeling, and other unsupervised learning tasks. Prior specification is, however, relatively difficult for such models, given that their flexibility implies that the consequences of prior choices are often relatively opaque. Moreover, these choices can have a substantial effect on posterior inferences. Thus, considerations of robustness need to go hand in hand with nonparametric modeling. In the current paper, we tackle this challenge by exploiting the fact that variational Bayesian methods, in addition to having computational advantages in fitting complex nonparametric models, also yield sensitivities with respect to parametric and nonparametric aspects of Bayesian models. In particular, we demonstrate how to assess the sensitivity of conclusions to the choice of concentration parameter and stick-breaking distribution for inferences under Dirichlet process mixtures and related mixture models. We provide both theoretical and empirical support for our variational approach to Bayesian sensitivity analysis.

State-of-the-art artificial neural networks (ANNs) require labelled data or feedback between layers, are often biologically implausible, and are vulnerable to adversarial attacks that humans are not susceptible to. On the other hand, Hebbian learning in winner-take-all (WTA) networks, is unsupervised, feed-forward, and biologically plausible. However, an objective optimization theory for WTA networks has been missing, except under very limiting assumptions. Here we derive formally such a theory, based on biologically plausible but generic ANN elements. Through Hebbian learning, network parameters maintain a Bayesian generative model of the data. There is no supervisory loss function, but the network does minimize cross-entropy between its activations and the input distribution. The key is a "soft" WTA where there is no absolute "hard" winner neuron, and a specific type of Hebbian-like plasticity of weights and biases. We confirm our theory in practice, where, in handwritten digit (MNIST) recognition, our Hebbian algorithm, SoftHebb, minimizes cross-entropy without having access to it, and outperforms the more frequently used, hard-WTA-based method. Strikingly, it even outperforms supervised end-to-end backpropagation, under certain conditions. Specifically, in a two-layered network, SoftHebb outperforms backpropagation when the training dataset is only presented once, when the testing data is noisy, and under gradient-based adversarial attacks. Adversarial attacks that confuse SoftHebb are also confusing to the human eye. Finally, the model can generate interpolations of objects from its input distribution.