In this work, we develop Gaussian process regression (GPR) models of hyperelastic material behavior. First, we consider the direct approach of modeling the components of the Cauchy stress tensor as a function of the components of the Finger stretch tensor in a Gaussian process. We then consider an improvement on this approach that embeds rotational invariance of the stress-stretch constitutive relation in the GPR representation. This approach requires fewer training examples and achieves higher accuracy while maintaining invariance to rotations exactly. Finally, we consider an approach that recovers the strain-energy density function and derives the stress tensor from this potential. Although the error of this model for predicting the stress tensor is higher, the strain-energy density is recovered with high accuracy from limited training data. The approaches presented here are examples of physics-informed machine learning. They go beyond purely data-driven approaches by embedding the physical system constraints directly into the Gaussian process representation of materials models.

We introduce a procedure for predictive conditional density estimation under logarithmic loss, which we call SMP (Sample Minmax Predictor). This predictor minimizes a new general excess risk bound, which critically remains valid under model misspecification. On standard examples, this bound scales as $d/n$ where $d$ is the dimension of the model and $n$ the sample size, regardless of the true distribution. The SMP, which is an improper (out-of-model) procedure, improves over proper (within-model) estimators (such as the maximum likelihood estimator), whose excess risk can degrade arbitrarily in the misspecified case. For density estimation, our bounds improve over approaches based on online-to-batch conversion, by removing suboptimal $\log n$ factors, addressing an open problem from Gr{\"u}nwald and Kot{\l}owski (2011) for the considered models. For the Gaussian linear model, the SMP admits an explicit expression, and its expected excess risk in the general misspecified case is at most twice the minimax excess risk in the \emph{well-specified case}, but without any condition on the noise variance or approximation error of the linear model. For logistic regression, a penalized SMP can be computed efficiently by training two logistic regressions, and achieves a non-asymptotic excess risk of $O((d + B^2R^2)/n)$, where $R$ is a bound on the norm of the features and $B$ the norm of the comparison linear predictor. This improves the rates of proper (within-model) estimators, since such procedures can achieve no better rate than $\min(BR/\sqrt{n},de^{BR}/n)$ in general. This also provides a computationally more efficient alternative to approaches based on online-to-batch conversion of Bayesian mixture procedures, which require approximate posterior sampling, thereby partly answering a question by Foster et al. (2018).

Evaluation of Bayesian deep learning (BDL) methods is challenging. We often seek to evaluate the methods' robustness and scalability, assessing whether new tools give `better' uncertainty estimates than old ones. These evaluations are paramount for practitioners when choosing BDL tools on-top of which they build their applications. Current popular evaluations of BDL methods, such as the UCI experiments, are lacking: Methods that excel with these experiments often fail when used in application such as medical or automotive, suggesting a pertinent need for new benchmarks in the field. We propose a new BDL benchmark with a diverse set of tasks, inspired by a real-world medical imaging application on \emph{diabetic retinopathy diagnosis}. Visual inputs (512x512 RGB images of retinas) are considered, where model uncertainty is used for medical pre-screening---i.e. to refer patients to an expert when model diagnosis is uncertain. Methods are then ranked according to metrics derived from expert-domain to reflect real-world use of model uncertainty in automated diagnosis. We develop multiple tasks that fall under this application, including out-of-distribution detection and robustness to distribution shift. We then perform a systematic comparison of well-tuned BDL techniques on the various tasks. From our comparison we conclude that some current techniques which solve benchmarks such as UCI `overfit' their uncertainty to the dataset---when evaluated on our benchmark these underperform in comparison to simpler baselines. The code for the benchmark, its baselines, and a simple API for evaluating new BDL tools are made available at https://github.com/oatml/bdl-benchmarks.

Consider a Bayesian inference problem where a variable of interest does not take values in a Euclidean space. These "non-standard" data structures are in reality fairly common. They are frequently used in problems involving latent discrete factor models, networks, and domain specific problems such as sequence alignments and reconstructions, pedigrees, and phylogenies. In principle, Bayesian inference should be particularly well-suited in such scenarios, as the Bayesian paradigm provides a principled way to obtain confidence assessment for random variables of any type. However, much of the recent work on making Bayesian analysis more accessible and computationally efficient has focused on inference in Euclidean spaces. In this paper, we introduce Blang, a domain specific language (DSL) and library aimed at bridging this gap. Blang allows users to perform Bayesian analysis on arbitrary data types while using a declarative syntax similar to BUGS. Blang is augmented with intuitive language additions to invent data types of the user's choosing. To perform inference at scale on such arbitrary state spaces, Blang leverages recent advances in parallelizable, non-reversible Markov chain Monte Carlo methods.

In this work, we develop a new approximation method to solve the analytically intractable Bayesian inference for Gaussian process models with factorizable Gaussian likelihoods and single-output latent functions. Our method -- dubbed QP -- is similar to the expectation propagation (EP), however it minimizes the $L^2$ Wasserstein distance instead of the Kullback-Leibler (KL) divergence. We consider the specific case in which the non-Gaussian likelihood is approximated by the Gaussian likelihood. We show that QP has the following properties: (1) QP matches quantile functions rather than moments in EP; (2) QP and EP have the same local update for the mean of the approximate Gaussian likelihood; (3) the local variance estimate for the approximate likelihood is smaller for QP than for EP's, addressing EP's over-estimation of the variance; (4) the optimal approximate Gaussian likelihood enjoys a univariate parameterization, reducing memory consumption and computation time. Furthermore, we provide a unified interpretations of EP and QP -- both are coordinate descent algorithms of a KL and an $L^2$ Wasserstein global objective function respectively, under the same assumptions. In the performed experiments, we employ eight real world datasets and we show that QP outperforms EP for the task of Gaussian process binary classification.

We make the following striking observation: fully convolutional VAE models trained on 32x32 ImageNet can generalize well, not just to 64x64 but also to far larger photographs, with no changes to the model. We use this property, applying fully convolutional models to lossless compression, demonstrating a method to scale the VAE-based 'Bits-Back with ANS' algorithm for lossless compression to large color photographs, and achieving state of the art for compression of full size ImageNet images. We release Craystack, an open source library for convenient prototyping of lossless compression using probabilistic models, along with full implementations of all of our compression results.

Time perception is the phenomenological experience of time by an individual. In this paper, we study how to replicate neural mechanisms involved in time perception, allowing robots to take a step towards temporal cognition. Our framework follows a twofold biologically inspired approach. The first step consists of estimating the passage of time from sensor measurements, since environmental stimuli influence the perception of time. Sensor data is modeled as Gaussian processes that represent the second-order statistics of the natural environment. The estimated elapsed time between two events is computed from the maximum likelihood estimate of the joint distribution of the data collected between them. Moreover, exactly how time is encoded in the brain remains unknown, but there is strong evidence of the involvement of dopaminergic neurons in timing mechanisms. Since their phasic activity has a similar behavior to the reward prediction error of temporal-difference learning models, the latter are used to replicate this behavior. The second step of this approach consists therefore of applying the agent's estimate of the elapsed time in a reinforcement learning problem, where a feature representation called Microstimuli is used. We validate our framework by applying it to an experiment that was originally conducted with mice, and conclude that a robot using this framework is able to reproduce the timing mechanisms of the animal's brain.

There exists a dichotomy between classical probabilistic graphical models, such as Bayesian networks (BNs), and modern tractable models, such as sum-product networks (SPNs). The former have generally intractable inference, but allow a high level of interpretability, while the latter admits a wide range of tractable inference routines, but are typically harder to interpret. Due to this dichotomy, tools to convert between BNs and SPNs are desirable. While one direction -- compiling BNs into SPNs -- is well discussed in Darwiche's seminal work on arithmetic circuit compilation, the converse direction -- decompiling SPNs into BNs -- has received surprisingly little attention. In this paper, we fill this gap by proposing SPN2BN, an algorithm that decompiles an SPN into a BN. SPN2BN has several salient features when compared to the only other two works decompiling SPNs. Most significantly, the BNs returned by SPN2BN are minimal independence-maps. Secondly, SPN2BN is more parsimonious with respect to the introduction of latent variables. Thirdly, the output BN produced by SPN2BN can be precisely characterized with respect to the compiled BN. More specifically, a certain set of directed edges will be added to the input BN, giving what we will call the moral-closure. It immediately follows that there is a set of BNs related to the input BN that will also return the same moral closure. Lastly, it is established that our compilation-decompilation process is idempotent. We confirm our results with systematic experiments on a number of synthetic BNs.

Continuous-time event sequences represent discrete events occurring in continuous time. Such sequences arise frequently in real-life and cover a wide variety of natural events, such as earthquakes, or events corresponding to human actions, such as medical administrations. Usually we expect the event sequences to follow some regular pattern over time. However, sometimes these regular patterns may be interrupted by unexpected absence or unexpected occurrences of events. Identification of these unexpected cases can be very important as they may point to abnormal situations that need human attention. In this work, we study and develop methods for detecting outliers in continuous-time event sequences, including unexpected absence and unexpected occurrences of events. Since the patterns that event sequences tend to follow may change in different contexts, we develop outlier detection methods based on point processes that take into account different contexts. Our outlier scoring methods are based on Bayesian decision theory and hypothesis testing with theoretical guarantees. To test the performance of the methods, we conduct experiments on both synthetic data and real-world clinical data and show the effectiveness of the proposed methods.

In this work, we consider cases with missing certain kinds of anomalies in the training dataset, while significant statistics for the normal class is available. For such scenarios, conventional supervised methods might suffer from the class imbalance, while unsupervised methods tend to ignore difficult anomalous examples. We extend the idea of the supervised classification approach for class-imbalanced datasets by exploiting normalizing flows for proper Bayesian inference of the posterior probabilities. Index Terms --Machine Learning, Neural Nets, Anomaly Detection, Imbalanced Data Set, Generate Potential Outliers, Normalizing Flow null 1 I NTRODUCTION The anomaly detection problem is one of the important tasks in the analysis of real-world data. Possible applications range from the data-quality certification [1] to finding the rare specific cases of the diseases in medicine [2].