This paper considers the problem of estimating a low-rank matrix from the observation of all, or a subset, of its entries in the presence of Poisson noise. When we observe all the entries, this is a problem of matrix denoising; when we observe only a subset of the entries, this is a problem of matrix completion. In both cases, we exploit an assumption that the underlying matrix is low-rank. Specifically, we analyze several estimators, including a constrained nuclear-norm minimization program, nuclear-norm regularized least squares, and a nonconvex constrained low-rank optimization problem. We show that for all three estimators, with high probability, we have an upper error bound (in the Frobenius norm error metric) that depends on the matrix rank, the fraction of the elements observed, and maximal row and column sums of the true matrix. We furthermore show that the above results are minimax optimal (within a universal constant) in classes of matrices with low rank and bounded row and column sums. We also extend these results to handle the case of matrix multinomial denoising and completion.

We compute exact literal-weighted model counts of CNF formulas. Our algorithm employs dynamic programming, with Algebraic Decision Diagrams as the primary data structure. This technique is implemented in ADDMC, a new model counter. We empirically evaluate various heuristics that can be used with ADDMC. We also compare ADDMC to state-of-the-art exact model counters (Cachet, c2d, d4, miniC2D, and sharpSAT) on the two largest CNF model counting benchmark families (BayesNet and Planning). ADDMC solves the most benchmarks in total within the given timeout.

We propose a new framework of XGBoost that predicts the entire conditional distribution of a univariate response variable. In particular, XGBoostLSS models all moments of a parametric distribution, i.e., mean, location, scale and shape (LSS), instead of the conditional mean only. Choosing from a wide range of continuous, discrete and mixed discrete-continuous distribution, modelling and predicting the entire conditional distribution greatly enhances the flexibility of XGBoost, as it allows to gain additional insight into the data generating process, as well as to create probabilistic forecasts from which prediction intervals and quantiles of interest can be derived. We present both a simulation study and real world examples that demonstrate the benefits of our approach.

Many methods for machine learning rely on approximate inference from intractable probability distributions. Variational inference approximates such distributions by tractable models that can be subsequently used for approximate inference. Learning sufficiently accurate approximations requires a rich model family and careful exploration of the relevant modes of the target distribution. We propose a method for learning accurate GMM approximations of intractable probability distributions based on insights from policy search by establishing information-geometric trust regions for principled exploration. For efficient improvement of the GMM approximation, we derive a lower bound on the corresponding optimization objective enabling us to update the components independently. The use of the lower bound ensures convergence to a local optimum of the original objective. The number of components is adapted online by adding new components in promising regions and by deleting components with negligible weight. We demonstrate on several domains that we can learn approximations of complex, multi-modal distributions with a quality that is unmet by previous variational inference methods, and that the GMM approximation can be used for drawing samples that are on par with samples created by state-of-the-art MCMC samplers while requiring up to three orders of magnitude less computational resources.

We provide here a general mathematical framework to model attitudes towards ambiguity which uses the formalism of quantum theory as a "purely mathematical formalism, detached from any physical interpretation". We show that the quantum-theoretic framework enables modelling of the "Ellsberg paradox", but it also successfully applies to more concrete human decision-making (DM) tests involving financial, managerial and medical decisions. In particular, we provide a faithful mathematical representation of various empirical studies which reveal that attitudes of managers towards uncertainty shift from "ambiguity seeking" to "ambiguity aversion", and viceversa, thus exhibiting "hope effects" and "fear effects" in management decisions. The present framework provides a new bold and promising direction towards the development of a unified theory of decisions in the presence of uncertainty.

We derive the functional form of mutual information (MI) from a set of design criteria and a principle of maximal sufficiency. The (MI) between two sets of propositions is a global quantifier of correlations and is implemented as a tool for ranking joint probability distributions with respect to said correlations. The derivation parallels the derivations of relative entropy with an emphasis on the behavior of independent variables. By constraining the functional $I$ according to special cases, we arrive at its general functional form and hence establish a clear meaning behind its definition. We also discuss the notion of sufficiency and offer a new definition which broadens its applicability.

The time series cluster kernel (TCK) provides a powerful tool for analysing multivariate time series subject to missing data. TCK is designed using an ensemble learning approach in which Bayesian mixture models form the base models. Because of the Bayesian approach, TCK can naturally deal with missing values without resorting to imputation and the ensemble strategy ensures robustness to hyperparameters, making it particularly well suited for unsupervised learning. However, TCK assumes missing at random and that the underlying missingness mechanism is ignorable, i.e. uninformative, an assumption that does not hold in many real-world applications, such as e.g. medicine. To overcome this limitation, we present a kernel capable of exploiting the potentially rich information in the missing values and patterns, as well as the information from the observed data. In our approach, we create a representation of the missing pattern, which is incorporated into mixed mode mixture models in such a way that the information provided by the missing patterns is effectively exploited. Moreover, we also propose a semi-supervised kernel, capable of taking advantage of incomplete label information to learn more accurate similarities. Experiments on benchmark data, as well as a real-world case study of patients described by longitudinal electronic health record data who potentially suffer from hospital-acquired infections, demonstrate the effectiveness of the proposed methods.

The framework of variational autoencoders allows us to efficiently learn deep latent-variable models, such that the model's marginal distribution over observed variables fits the data. Often, we're interested in going a step further, and want to approximate the true joint distribution over observed and latent variables, including the true prior and posterior distributions over latent variables. This is known to be generally impossible due to unidentifiability of the model. We address this issue by showing that for a broad family of deep latent-variable models, identification of the true joint distribution over observed and latent variables is actually possible up to a simple transformation, thus achieving a principled and powerful form of disentanglement. Our result requires a factorized prior distribution over the latent variables that is conditioned on an additionally observed variable, such as a class label or almost any other observation. We build on recent developments in nonlinear ICA, which we extend to the case with noisy, undercomplete or discrete observations, integrated in a maximum likelihood framework. The result also trivially contains identifiable flow-based generative models as a special case.

We consider probabilistic programming for birth-death models of evolution and introduce a new widely-applicable inference method that combines an extension of the alive particle filter (APF) with automatic Rao-Blackwellization via delayed sampling. Birth-death models of evolution are an important family of phylogenetic models of the diversification processes that lead to evolutionary trees. Probabilistic programming languages (PPLs) give phylogeneticists a new and exciting tool: their models can be implemented as probabilistic programs with just a basic knowledge of programming. The general inference methods in PPLs reduce the need for external experts, allow quick prototyping and testing, and accelerate the development and deployment of new models. We show how these birth-death models can be implemented as simple programs in existing PPLs, and demonstrate the usefulness of the proposed inference method for such models. For the popular BiSSE model the method yields an increase of the effective sample size and the conditional acceptance rate by a factor of 30 in comparison with a standard bootstrap particle filter. Although concentrating on phylogenetics, the extended APF is a general inference method that shows its strength in situations where particles are often assigned zero weight. In the case when the weights are always positive, the extra cost of using the APF rather than the bootstrap particle filter is negligible, making our method a suitable drop-in replacement for the bootstrap particle filter in probabilistic programming inference.

Recently, there has been much research into modelling of uncertainty in human perception through Fuzzy Sets (FSs). Most of this research has focused on allowing respondents to express their (intra) uncertainty using intervals. Here, depending on the technique used and types of uncertainties being modelled different types of FSs can be obtained (e.g., Type-1, Interval Type-2, General Type-2). Arguably, one of the most flexible techniques is the Interval Agreement Approach (IAA) as it allows to model the perception of all respondents without making assumptions such as outlier removal or predefined membership function types (e.g. Gaussian). A key aspect in the analysis of interval-valued data and indeed, IAA based agreement models of said data, is to determine the position and strengths of agreement across all the sources/participants. While previously, the Agreement Ratio was proposed to measure the strength of agreement in fuzzy set based models of interval data, said measure has only been applicable to type-1 fuzzy sets. In this paper, we extend the Agreement Ratio to capture the degree of inter-group agreement modelled by a General Type-2 Fuzzy Set when using the IAA. This measure relies on using a similarity measure to quantitatively express the relation between the different levels of agreement in a given FS. Synthetic examples are provided in order to demonstrate both behaviour and calculation of the measure. Finally, an application to real-world data is provided in order to show the potential of this measure to assess the divergence of opinions for ambiguous concepts when heterogeneous groups of participants are involved.