We report on the potential for using algorithms for non-negative matrix factorization (NMF) to improve parameter estimation in topic models. While several papers have studied connections between NMF and topic models, none have suggested leveraging these connections to develop new algorithms for fitting topic models. Importantly, NMF avoids the "sum-to-one" constraints on the topic model parameters, resulting in an optimization problem with simpler structure and more efficient computations. Building on recent advances in optimization algorithms for NMF, we show that first solving the NMF problem then recovering the topic model fit can produce remarkably better fits, and in less time, than standard algorithms for topic models. While we focus primarily on maximum likelihood estimation, we show that this approach also has the potential to improve variational inference for topic models. Our methods are implemented in the R package fastTopics.

In a recent work we have shown how to construct an information algebra of coherent sets of gambles defined on general possibility spaces. Here we analyze the connection of such an algebra with the set algebra of subsets of the possibility space on which gambles are defined and the set algebra of sets of its atoms. Set algebras are particularly important information algebras since they are their prototypical structures. Furthermore, they are the algebraic counterparts of classical propositional logic. As a consequence, this paper also details how propositional logic is naturally embedded into the theory of imprecise probabilities.

We present a Bayesian treatment for deep regression using an Errors-in-Variables model which accounts for the uncertainty associated with the input to the employed neural network. It is shown how the treatment can be combined with already existing approaches for uncertainty quantification that are based on variational inference. Our approach yields a decomposition of the predictive uncertainty into an aleatoric and epistemic part that is more complete and, in many cases, more consistent from a statistical perspective. We illustrate and discuss the approach along various toy and real world examples.

The viral load of patients infected with SARS-CoV-2 varies on logarithmic scales and possibly with age. Controversial claims have been made in the literature regarding whether the viral load distribution actually depends on the age of the patients. Such a dependence would have implications for the COVID-19 spreading mechanism, the age-dependent immune system reaction, and thus for policymaking. We hereby develop a method to analyze viral-load distribution data as a function of the patients' age within a flexible, non-parametric, hierarchical, Bayesian, and causal model. This method can be applied to other contexts as well, and for this purpose, it is made freely available. The developed reconstruction method also allows testing for bias in the data. This could be due to, e.g., bias in patient-testing and data collection or systematic errors in the measurement of the viral load. We perform these tests by calculating the Bayesian evidence for each implied possible causal direction. When applying these tests to publicly available age and SARS-CoV-2 viral load data, we find a statistically significant increase in the viral load with age, but only for one of the two analyzed datasets. If we consider this dataset, and based on the current understanding of viral load's impact on patients' infectivity, we expect a non-negligible difference in the infectivity of different age groups. This difference is nonetheless too small to justify considering any age group as noninfectious.

Ordinary differential equations (ODEs), commonly used to characterize the dynamic systems, are difficult to propose in closed-form for many complicated scientific applications, even with the help of domain expert. We propose a fast and accurate data-driven method, MAGI-X, to learn the unknown dynamic from the observation data in a non-parametric fashion, without the need of any domain knowledge. Unlike the existing methods that mainly rely on the costly numerical integration, MAGI-X utilizes the powerful functional approximator of neural network to learn the unknown nonlinear dynamic within the MAnifold-constrained Gaussian process Inference (MAGI) framework that completely circumvents the numerical integration. Comparing against the state-of-the-art methods on three realistic examples, MAGI-X achieves competitive accuracy in both fitting and forecasting while only taking a fraction of computational time. Moreover, MAGI-X provides practical solution for the inference of partial observed systems, which no previous method is able to handle.

Deep ensembles can be seen as the current state-of-the-art for uncertainty quantification in deep learning. While the approach was originally proposed as an non-Bayesian technique, arguments towards its Bayesian footing have been put forward as well. We show that deep ensembles can be viewed as an approximate Bayesian method by specifying the corresponding assumptions. Our finding leads to an improved approximation which results in an increased epistemic part of the uncertainty. Numerical examples suggest that the improved approximation can lead to more reliable uncertainties. Analytical derivations ensure easy calculation of results.

"The key distinguishing property of a Bayesian approach is marginalization instead of optimization, where we represent solutions given by all settings of parameters weighted by their posterior probabilities, rather than bet everything on a single setting of parameters." The time is ripe to dig into marginalization vs optimization, and broaden our general understanding of the Bayesian approach.

Graphical models provide a natural framework for probabilistic modeling and for inference with a large number of random variables. The framework has numerous applications including in computer vision, artificial intelligence, error correcting codes, and statistical physics. Exact inference in graphical models is NP-hard so it is essential to develop approximate inference algorithms that are computationally tractable. Belief propagation is a widely used message passing algorithm that can be used to perform either exact or approximate inference in a graphical model depending on the topology of the graph. This algorithm was first introduced by Judea Pearl in the early 1980s as a method to perform exact inference on a tree-structured graph in polynomial time [15].

This research paper proposes a novel Neighbourhood Rough Set based approach for supervised Multi-document Text Summarization (MDTS) with analysis and impact on the summarization results for MDTS. Here, Rough Set based LERS algorithm is improved using Neighborhood Rough Set which is itself a novel combination called Neighborhood-LERS to be experimented for evaluations of efficacy and efficiency. In this paper, we shall apply and evaluate the proposed Neighborhood-LERS for Multi-document Summarization which here is proved experimentally to be superior to the base LERS technique for MDTS.

In this work, we have proposed augmented KRnets including both discrete and continuous models. One difficulty in flow-based generative modeling is to maintain the invertibility of the transport map, which is often a trade-off between effectiveness and robustness. The exact invertibility has been achieved in the real NVP using a specific pattern to exchange information between two separated groups of dimensions. KRnet has been developed to enhance the information exchange among data dimensions by incorporating the Knothe-Rosenblatt rearrangement into the structure of the transport map. Due to the maintenance of the exact invertibility, a full nonlinear update of all data dimensions needs three iterations in KRnet. To alleviate this issue, we will add augmented dimensions that act as a channel for the data dimensions to exchange information. In the augmented KRnet, a fully nonlinear update is achieved in two iterations. We also show that the augmented KRnet can be reformulated as the discretization of a neural ODE, where the exact invertibility is kept such that the adjoint method can be formulated with respect to the discretized ODE to obtain the exact gradient. Numerical experiments have been implemented to demonstrate the effectiveness of our models.