Refining a Bayesian network using a chain event graph

The search for a useful explanatory model based on a Bayesian Network (BN) now has a long and successful history. However, when the dependence structure between the variables of the problem is asymmetric then this cannot be captured by the BN. The Chain Event Graph (CEG) provides a richer class of models which incorporates these types of dependence structures as well as retaining the property that conclusions can be easily read back to the client. We demonstrate on a real health study how the CEG leads us to promising higher scoring models and further enables us to make more refined conclusions than can be made from the BN. Further we show how these graphs can express causal hypotheses about possible interventions that could be enforced

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

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

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