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

Discovery of statistical equivalence classes using computer algebra

Discrete statistical models supported on labelled event trees can be specified using so-called interpolating polynomials which are generalizations of generating functions. These admit a nested representation. A new algorithm exploits the primary decomposition of monomial ideals associated with an interpolating polynomial to quickly compute all nested representations of that polynomial. It hereby determines an important subclass of all trees representing the same statistical model. To illustrate this method we analyze the full polynomial equivalence class of a staged tree representing the best fitting model inferred from a real-world dataset.Comment: 26 pages, 9 figure

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

Parameter identifiability of discrete Bayesian networks with hidden variables

Identifiability of parameters is an essential property for a statistical model to be useful in most settings. However, establishing parameter identifiability for Bayesian networks with hidden variables remains challenging. In the context of finite state spaces, we give algebraic arguments establishing identifiability of some special models on small DAGs. We also establish that, for fixed state spaces, generic identifiability of parameters depends only on the Markov equivalence class of the DAG. To illustrate the use of these results, we investigate identifiability for all binary Bayesian networks with up to five variables, one of which is hidden and parental to all observable ones. Surprisingly, some of these models have parameterizations that are generically 4-to-one, and not 2-to-one as label swapping of the hidden states would suggest. This leads to interesting difficulties in interpreting causal effects.Comment: 23 page

Efficient algorithms for conditional independence inference

The topic of the paper is computer testing of (probabilistic) conditional independence (CI) implications by an algebraic method of structural imsets. The basic idea is to transform (sets of) CI statements into certain integral vectors and to verify by a computer the corresponding algebraic relation between the vectors, called the independence implication. We interpret the previous methods for computer testing of this implication from the point of view of polyhedral geometry. However, the main contribution of the paper is a new method, based on linear programming (LP). The new method overcomes the limitation of former methods to the number of involved variables. We recall/describe the theoretical basis for all four methods involved in our computational experiments, whose aim was to compare the efficiency of the algorithms. The experiments show that the LP method is clearly the fastest one. As an example of possible application of such algorithms we show that testing inclusion of Bayesian network structures or whether a CI statement is encoded in an acyclic directed graph can be done by the algebraic method

Equations defining probability tree models

Coloured probability tree models are statistical models coding conditional independence between events depicted in a tree graph. They are more general than the very important class of context-specific Bayesian networks. In this paper, we study the algebraic properties of their ideal of model invariants. The generators of this ideal can be easily read from the tree graph and have a straightforward interpretation in terms of the underlying model: they are differences of odds ratios coming from conditional probabilities. One of the key findings in this analysis is that the tree is a convenient tool for understanding the exact algebraic way in which the sum-to-1 conditions on the parameter space translate into the sum-to-one conditions on the joint probabilities of the statistical model. This enables us to identify necessary and sufficient graphical conditions for a staged tree model to be a toric variety intersected with a probability simplex.Comment: 22 pages, 4 figure