Polynomial Bell inequalities

It is a recent realization that many of the concepts and tools of causal discovery in machine learning are highly relevant to problems in quantum information, in particular quantum nonlocality. The crucial ingredient in the connection between both fields is the tool of Bayesian networks, a graphical model used to reason about probabilistic causation. Indeed, Bell's theorem concerns a particular kind of a Bayesian network and Bell inequalities are a special case of linear constraints following from such models. It is thus natural to look for generalized Bell scenarios involving more complex Bayesian networks. The problem, however, relies on the fact that such generalized scenarios are characterized by polynomial Bell inequalities and no current method is available to derive them beyond very simple cases. In this work, we make a significant step in that direction, providing a general and practical method for the derivation of polynomial Bell inequalities in a wide class of scenarios, applying it to a few cases of interest. We also show how our construction naturally gives rise to a notion of non-signalling in generalized networks.Comment: 9 pages (including appendix

Efficient computational strategies to learn the structure of probabilistic graphical models of cumulative phenomena

Structural learning of Bayesian Networks (BNs) is a NP-hard problem, which is further complicated by many theoretical issues, such as the I-equivalence among different structures. In this work, we focus on a specific subclass of BNs, named Suppes-Bayes Causal Networks (SBCNs), which include specific structural constraints based on Suppes' probabilistic causation to efficiently model cumulative phenomena. Here we compare the performance, via extensive simulations, of various state-of-the-art search strategies, such as local search techniques and Genetic Algorithms, as well as of distinct regularization methods. The assessment is performed on a large number of simulated datasets from topologies with distinct levels of complexity, various sample size and different rates of errors in the data. Among the main results, we show that the introduction of Suppes' constraints dramatically improve the inference accuracy, by reducing the solution space and providing a temporal ordering on the variables. We also report on trade-offs among different search techniques that can be efficiently employed in distinct experimental settings. This manuscript is an extended version of the paper "Structural Learning of Probabilistic Graphical Models of Cumulative Phenomena" presented at the 2018 International Conference on Computational Science

Equivalence Classes of Staged Trees

In this paper we give a complete characterization of the statistical equivalence classes of CEGs and of staged trees. We are able to show that all graphical representations of the same model share a common polynomial description. Then, simple transformations on that polynomial enable us to traverse the corresponding class of graphs. We illustrate our results with a real analysis of the implicit dependence relationships within a previously studied dataset.Comment: 18 pages, 4 figure