Influence diagrams (IDs) are well-known formalisms extending Bayesian networks to model decision situations under uncertainty. Although they are convenient as a decision theoretic tool, their knowledge representation ability is limited in capturing other crucial notions such as logical consistency. We complement IDs with the light-weight description logic (DL) EL to overcome such limitations. We consider a setup where DL axioms hold in some contexts, yet the actual context is uncertain. The framework benefits from the convenience of using DL as a domain knowledge representation language and the modelling strength of IDs to deal with decisions over contexts in the presence of contextual uncertainty. We define related reasoning problems and study their computational complexity.

We can use driving data collected over a long period of time to extract rich information about how vehicles behave in different areas of the roads. In this paper, we introduce the concept of directional primitives, which is a representation of prior information of road networks. Specifically, we represent the uncertainty of directions using a mixture of von Mises distributions and associated speeds using gamma distributions. These location-dependent primitives can be combined with motion information of surrounding vehicles to predict their future behavior in the form of probability distributions. Experiments conducted on highways, intersections, and roundabouts in the Carla simulator, as well as real-world urban driving datasets, indicate that primitives lead to better uncertainty-aware motion estimation.

Estimating the score, i.e., the gradient of log density function, from a set of samples generated by an unknown distribution is a fundamental task in inference and learning of probabilistic models that involve flexible yet intractable densities. Kernel estimators based on Stein's methods or score matching have shown promise, however their theoretical properties and relationships have not been fully-understood. We provide a unifying view of these estimators under the framework of regularized nonparametric regression. It allows us to analyse existing estimators and construct new ones with desirable properties by choosing different hypothesis spaces and regularizers. A unified convergence analysis is provided for such estimators. Finally, we propose score estimators based on iterative regularization that enjoy computational benefits from curl-free kernels and fast convergence.

Virtual screening aims to find desirable compounds from chemical library by using computational methods. For this purpose with machine learning, model outputs that can be interpreted as predictive probability will be beneficial, in that a high prediction score corresponds to high probability of correctness. In this work, we present a study on the prediction performance and reliability of graph neural networks trained with the recently proposed Bayesian learning algorithms. Our work shows that Bayesian learning algorithms allow well-calibrated predictions for various GNN architectures and classification tasks. Also, we show the implications of reliable predictions on virtual screening, where Bayesian learning may lead to higher success in finding hit compounds.

We study Sinkhorn EM (sEM), a variant of the expectation maximization (EM) algorithm for mixtures based on entropic optimal transport. sEM differs from the classic EM algorithm in the way responsibilities are computed during the expectation step: rather than assign data points to clusters independently, sEM uses optimal transport to compute responsibilities by incorporating prior information about mixing weights. Like EM, sEM has a natural interpretation as a coordinate ascent procedure, which iteratively constructs and optimizes a lower bound on the log-likelihood. However, we show theoretically and empirically that sEM has better behavior than EM: it possesses better global convergence guarantees and is less prone to getting stuck in bad local optima. We complement these findings with experiments on simulated data as well as in an inference task involving C. elegans neurons and show that sEM learns cell labels significantly better than other approaches.

Learning the joint probability of random variables (RVs) lies at the heart of statistical signal processing and machine learning. However, direct nonparametric estimation for high-dimensional joint probability is in general impossible, due to the curse of dimensionality. Recent work has proposed to recover the joint probability mass function (PMF) of an arbitrary number of RVs from three-dimensional marginals, leveraging the algebraic properties of low-rank tensor decomposition and the (unknown) dependence among the RVs. Nonetheless, accurately estimating three-dimensional marginals can still be costly in terms of sample complexity, affecting the performance of this line of work in practice in the sample-starved regime. Using three-dimensional marginals also involves challenging tensor decomposition problems whose tractability is unclear. This work puts forth a new framework for learning the joint PMF using only pairwise marginals, which naturally enjoys a lower sample complexity relative to the third-order ones. A coupled nonnegative matrix factorization (CNMF) framework is developed, and its joint PMF recovery guarantees under various conditions are analyzed. Our method also features a Gram-Schmidt (GS)-like algorithm that exhibits competitive runtime performance. The algorithm is shown to provably recover the joint PMF up to bounded error in finite iterations, under reasonable conditions. It is also shown that a recently proposed economical expectation maximization (EM) algorithm guarantees to improve upon the GS-like algorithm's output, thereby further lifting up the accuracy and efficiency. Real-data experiments are employed to showcase the effectiveness.

Over the past years, the task of learning in the presence of outliers has become an increasingly important objective in both statistics and machine learning. Indeed, in many situations, training data can be contaminated by undesired samples, which may badly affect the resulting learning task, especially in adversarial settings. Building robust estimators and algorithms that are resilient to outliers is therefore becoming crucial in many learning procedures. In particular, the inference of a probability density function from a contaminated random sample is of major concerns. Density estimation methods are mostly divided into parametric and nonparametric techniques. Among the nonparametric family, the Kernel Density Estimator (KDE) is probably the most known and used for both univariate and multivariate densities [Parzen, 1962; Silverman, 1986; Scott, 2015], but it also known to be sensitive to dataset contaminated by outliers [Kim and Scott, 2011, 2012; Vandermeulen and Scott, 2014].

The study of network formation is pervasive in economics, sociology, and many other fields. In this paper, we model network formation as a ``choice'' that is made by nodes in a network to connect to other nodes. We study these ``choices'' using discrete-choice models, in which an agent chooses between two or more discrete alternatives. One framework for studying network formation is the multinomial logit (MNL) model. We highlight limitations of the MNL model on networks that are constructed from empirical data. We employ the ``repeated choice'' (RC) model to study network formation \cite{TrainRevelt97mixedlogit}. We argue that the RC model overcomes important limitations of the MNL model and is well-suited to study network formation. We also illustrate how to use the RC model to accurately study network formation using both synthetic and real-world networks. Using synthetic networks, we also compare the performance of the MNL model and the RC model; we find that the RC model estimates the data-generation process of our synthetic networks more accurately than the MNL model. We provide examples of qualitatively interesting questions -- the presence of homophily in a teen friendship network and the fact that new patents are more likely to cite older, more cited, and similar patents -- for which the RC model allows us to achieve insights.

The Variational Auto-Encoder (VAE) belongs to a class of models, which we will refer to as deep maximum likelihood models, that uses a deep neural network to learn a maximum likelihood model for some input data. They are perhaps the most simple and efficient deep maximum likelihood model available, and have thus gained popularity in representation learning and generative image modeling. Unfortunately, in my opinion, in some circles the term "VAE" has become somewhat synonymous with "an auto-encoder with stochastic regularization that generates useful or beautiful samples", which has led to various misconceptions about VAEs. In this tutorial, we will return to the probabilistic and information theoretic roots of VAEs, clarify common misconceptions about VAEs, and look at a toy example on 2D data that will illustrate the capabilities and limitations of VAEs. In Section 2, we will give an overview of what is a maximum likelihood model and what a VAE looks like.

We propose a unified framework for adaptive connection sampling in graph neural networks (GNNs) that generalizes existing stochastic regularization methods for training GNNs. The proposed framework not only alleviates over-smoothing and over-fitting tendencies of deep GNNs, but also enables learning with uncertainty in graph analytic tasks with GNNs. Instead of using fixed sampling rates or hand-tuning them as model hyperparameters in existing stochastic regularization methods, our adaptive connection sampling can be trained jointly with GNN model parameters in both global and local fashions. GNN training with adaptive connection sampling is shown to be mathematically equivalent to an efficient approximation of training Bayesian GNNs. Experimental results with ablation studies on benchmark datasets validate that adaptively learning the sampling rate given graph training data is the key to boost the performance of GNNs in semi-supervised node classification, less prone to over-smoothing and over-fitting with more robust prediction.