Normalizing flows transform a latent distribution through an invertible neural network for a flexible and pleasingly simple approach to generative modelling, while preserving an exact likelihood. We propose FlowGMM, an end-to-end approach to generative semi supervised learning with normalizing flows, using a latent Gaussian mixture model. FlowGMM is distinct in its simplicity, unified treatment of labelled and unlabelled data with an exact likelihood, interpretability, and broad applicability beyond image data. We show promising results on a wide range of applications, including AG-News and Yahoo Answers text data, tabular data, and semi-supervised image classification. We also show that FlowGMM can discover interpretable structure, provide real-time optimization-free feature visualizations, and specify well calibrated predictive distributions.

Data-based discovery of effective, coarse-grained (CG) models of high-dimensional dynamical systems presents a unique challenge in computational physics and particularly in the context of multiscale problems. The present paper offers a data-based, probablistic perspective that enables the quantification of predictive uncertainties. One of the outstanding problems has been the introduction of physical constraints in the probabilistic machine learning objectives. The primary utility of such constraints stems from the undisputed physical laws such as conservation of mass, energy etc that they represent. Furthermore and apart from leading to physically realistic predictions, they can significantly reduce the requisite amount of training data which for high-dimensional, multiscale systems are expensive to obtain (Small Data regime). We formulate the coarse-graining process by employing a probabilistic state-space model and account for the aforementioned equality constraints as virtual observables in the associated densities. We demonstrate how probabilistic inference tools can be employed to identify the coarse-grained variables in combination with deep neural nets and their evolution model without ever needing to define a fine-to-coarse (restriction) projection and without needing time-derivatives of state variables. The formulation adopted enables the quantification of a crucial, and often neglected, component in the CG process, i.e. the predictive uncertainty due to information loss. Furthermore, it is capable of reconstructing the evolution of the full, fine-scale system and therefore the observables of interest need not be selected a priori. We demonstrate the efficacy of the proposed framework by applying it to systems of interacting particles and an image series of a nonlinear pendulum. In both cases we identify the underlying coarse dynamics and can generate extrap-olative predicitions including the forming and propagation of a shock for the particle systems and a stable trajectory in the phase space for the pendulum. Keywords: Bayesian machine learning, virtual observables, multiscale modeling, reduced order modeling, coarse graining1. Introduction High-dimensional, nonlinear dynamical systems are ubiquitous in applied physics and engineering. The computational resources needed for their solution can grow exponentially with the dimension of the state-space as well as with the smallest timescale that needs to be resolved as this determines the discretization time-step.

Describing or calculating the conditional probabilities of multiple events is exponentially expensive. In this work, a natural generalization of Bayesian belief network is proposed by incorporating with tensor network, which is dubbed as Bayesian tensor network (BTN), to efficiently describe the conditional probabilities among multiple sets of events. The complexity of BTN that gives the conditional probabilities of $M$ sets of events scales only polynomially with $M$. To testify its validity, BTN is implemented to capture the conditional probabilities between images and their classifications, where each feature is mapped to a probability distribution of a set of mutually exclusive events. A rotation optimization method is suggested to update BTN, which avoids gradient vanishing problem and exhibits high efficiency. With a simple tree network structures, BTN exhibits competitive performances on fashion-MNIST dataset. Analogous to the tensor network simulations of quantum systems, the validity of BTN implies an "area law" of fluctuations in image recognition problems.

Deep neural networks (DNN) are versatile parametric models utilised successfully in a diverse number of tasks and domains. However, they have limitations---particularly from their lack of robustness and over-sensitivity to out of distribution samples. Bayesian Neural Networks, due to their formulation under the Bayesian framework, provide a principled approach to building neural networks that address these limitations. This paper describes a study that empirically evaluates and compares Bayesian Neural Networks to their equivalent point estimate Deep Neural Networks to quantify the predictive uncertainty induced by their parameters, as well as their performance in view of this uncertainty. In this study, we evaluated and compared three point estimate deep neural networks against comparable Bayesian neural network alternatives using two well-known benchmark image classification datasets (CIFAR-10 and SVHN).

This paper introduces a new approach to learning from i.i.d. data under model miss-specification. This approach casts the problem of learning as minimizing the expected code-length of a Bayesian mixture code. To solve this problem, we build on PAC-Bayes bounds, information theory and a new family of second-order Jensen bounds. The key insight of this paper is that the use of the standard (first-order) Jensen bounds in learning is suboptimal when our model class is miss-specified (i.e. it does not contain the data generating distribution). As a consequence of this insight, this work provides strong theoretical arguments explaining why the Bayesian posterior is not optimal for making predictions that generalize under model miss-specification because the Bayesian posterior is directly related to the use of first-order Jensen bounds. We then argue for the use of second-order Jensen bounds, which leads to new families of learning algorithms. In this work, we introduce novel variational and ensemble learning methods based on the minimization of a novel family of second-order PAC-Bayes bounds over the expected code-length of a Bayesian mixture code. Using this new framework, we also provide novel hypotheses of why parameters in a flat minimum generalize better than parameters in a sharp minimum.

The family of methods collectively known as classifier chains has become a popular approach to multi-label learning problems. This approach involves linking together off-the-shelf binary classifiers in a chain structure, such that class label predictions become features for other classifiers. Such methods have proved flexible and effective and have obtained state-of-the-art empirical performance across many datasets and multi-label evaluation metrics. This performance led to further studies of how exactly it works, and how it could be improved, and in the recent decade numerous studies have explored classifier chains mechanisms on a theoretical level, and many improvements have been made to the training and inference procedures, such that this method remains among the state-of-the-art options for multi-label learning. Given this past and ongoing interest, which covers a broad range of applications and research themes, the goal of this work is to provide a review of classifier chains, a survey of the techniques and extensions provided in the literature, as well as perspectives for this approach in the domain of multi-label classification in the future. We conclude positively, with a number of recommendations for researchers and practitioners, as well as outlining a number of areas for future research.

We propose a general method for constructing hypothesis tests and confidence sets that have finite sample guarantees without regularity conditions. We refer to such procedures as ``universal.'' The method is very simple and is based on a modified version of the usual likelihood ratio statistic, that we call ``the split likelihood ratio test'' (split LRT). The method is especially appealing for irregular statistical models. Canonical examples include mixture models and models that arise in shape-constrained inference. %mixture models and shape-constrained models are just two examples. Constructing tests and confidence sets for such models is notoriously difficult. Typical inference methods, like the likelihood ratio test, are not useful in these cases because they have intractable limiting distributions. In contrast, the method we suggest works for any parametric model and also for some nonparametric models. The split LRT can also be used with profile likelihoods to deal with nuisance parameters, and it can also be run sequentially to yield anytime-valid $p$-values and confidence sequences.

Crowdsourcing is a relatively economic and efficient solution to collect annotations from the crowd through online platforms. Answers collected from workers with different expertise may be noisy and unreliable, and the quality of annotated data needs to be further maintained. Various solutions have been attempted to obtain high-quality annotations. However, they all assume that workers' label quality is stable over time (always at the same level whenever they conduct the tasks). In practice, workers' attention level changes over time, and the ignorance of which can affect the reliability of the annotations. In this paper, we focus on a novel and realistic crowdsourcing scenario involving attention-aware annotations. We propose a new probabilistic model that takes into account workers' attention to estimate the label quality. Expectation propagation is adopted for efficient Bayesian inference of our model, and a generalized Expectation Maximization algorithm is derived to estimate both the ground truth of all tasks and the label-quality of each individual crowd worker with attention. In addition, the number of tasks best suited for a worker is estimated according to changes in attention. Experiments against related methods on three real-world and one semi-simulated datasets demonstrate that our method quantifies the relationship between workers' attention and label-quality on the given tasks, and improves the aggregated labels.

We start with the distinction of outcome- and belief-based Bayesian models of the sequential update of agents' beliefs and subjective reliability of sources (trust). We then focus on discussing the influential Bayesian model of belief-based trust update by Eric Olsson, which models dichotomic events and explicitly represents anti-reliability. After sketching some disastrous recent results for this perhaps most promising model of belief update, we show new simulation results for the temporal dynamics of learning belief with and without trust update and with and without communication. The results seem to shed at least a somewhat more positive light on the communicating-and-trust-updating agents. This may be a light at the end of the tunnel of belief-based models of trust updating, but the interpretation of the clear findings is much less clear.

These challenges can be addressed by enforcing sparsity in the series representation through retaining only the most important basis terms. In this work, we present a novel sparse Bayesian learning technique for obtaining sparse Polynomial Chaos expansions which is based on a Relevance Vector Machine model and is trained using Variational Inference. The methodology shows great potential in high-dimensional data-driven settings using relatively few data points and achieves user-controlled sparse levels that are comparable to other methods such as compressive sensing. The proposed approach is illustrated on two numerical examples, a synthetic response function that is explored for validation purposes and a low-carbon steel plate with random Young's modulus and random loading, which is modelled by stochastic finite element with 38 input random variables.