Advances in Learning Bayesian Networks of Bounded Treewidth

This work presents novel algorithms for learning Bayesian network structures with bounded treewidth. Both exact and approximate methods are developed. The exact method combines mixed-integer linear programming formulations for structure learning and treewidth computation. The approximate method consists in uniformly sampling $k$-trees (maximal graphs of treewidth $k$), and subsequently selecting, exactly or approximately, the best structure whose moral graph is a subgraph of that $k$-tree. Some properties of these methods are discussed and proven. The approaches are empirically compared to each other and to a state-of-the-art method for learning bounded treewidth structures on a collection of public data sets with up to 100 variables. The experiments show that our exact algorithm outperforms the state of the art, and that the approximate approach is fairly accurate.Comment: 23 pages, 2 figures, 3 table

Efficient Predictive Uncertainty Estimators for Deep Probabilistic Models

Deep Probabilistic Models (DPM) based on arithmetic circuits representation, such as Sum-Product Networks (SPN) and Probabilistic Sentential Decision Diagrams (PSDD), have shown competitive performance in several machine learning tasks with interesting properties (Poon and Domingos 2011; Kisa et al. 2014). Due to the high number of parameters and scarce data, DPMs can produce unreliable and overconfident inference. This research aims at increasing the robustness of predictive inference with DPMs by obtaining new estimators of the predictive uncertainty. This problem is not new and the literature on deep models contains many solutions. However the probabilistic nature of DPMs offer new possibilities to achieve accurate estimates at low computational costs, but also new challenges, as the range of different types of predictions is much larger than with traditional deep models. To cope with such issues, we plan on investigating two different approaches. The first approach is to perform a global sensitivity analysis on the parameters, measuring the variability of the output to perturbations of the model weights. The second approach is to capture the variability of the prediction with respect to changes in the model architecture. Our approaches shall be evaluated on challenging tasks such as image completion, multilabel classification

Credal Sum-Product Networks

Sum-product networks are a relatively new and increasingly popular class of (precise) probabilistic graphical models that allow for marginal inference with polynomial effort. As with other probabilistic models, sum-product networks are often learned from data and used to perform classification. Hence, their results are prone to be unreliable and overconfident. In this work, we develop credal sum-product networks, an imprecise extension of sum-product networks. We present algorithms and complexity results for common inference tasks. We apply our algorithms on realistic classification task using images of digits and show that credal sum-product networks obtained by a perturbation of the parameters of learned sum-product networks are able to distinguish between reliable and unreliable classifications with high accuracy