The geometry of causal probability trees that are algebraically constrained

In this chapter we show how algebraic geometry can be used to define and then analyse the properties of certain important classes of discrete probability models described through probability trees. Our aim is to show how much wider classes of discrete statistical models than have been considered previously have a useful algebraic formulation and unlike its competitors this class is closed under discovery of arbitrary marginal distributions. We proceed to illustrate how the identifiability of certain causal functions can be articulated and analysed within this framework which generalises the causal Bayesian network formulation. We note that as with causal Bayesian networks the most convenient polynomial parametrisation is one that is based on conditional rather than marginal probabilities. In Section 1 we introduce the probability tree representation of a discrete model and show that discrete Bayesian networks and some of their recent generalisations are special subclasses of these models. We then introduce an algebraic representation of important classes of probability tree models called algebraic constraint models (ACTs). In Section 3 we proceed to examine how ACTs are closed under the discovery of the marginal distribution of a random variable measurable with respect to the path sigma algebra of its underlying probability tree. Probability tree representations are especially useful to specify and study the implications of certain causal hypotheses. In Sections 4 and 5 we relate these causal models to ACTs and give a formal discussion of the conditional probability graphs of discrete models that can also be expressed as Bayesian networks. In Section 6 we illustrate these ideas from the perspective of a simple modelling context.