Speaker: Fatemeh Mohammadi, Ghent University

**Title:** The geometry of conditional independence models with hidden variables

**Abstract:** Conditional independence (CI) is an important tool instatistical modeling, as, for example, it gives a statistical

interpretation to graphical models. In general, given a list of

dependencies among random variables, it is difficult to say which

constraints are implied by them. Moreover, it is important to know

what constraints on the random variables are caused by hidden

variables. On the other hand, such constraints are corresponding to

some determinantal conditions on the tensor of joint probabilities of

the observed random variables. Hence, the inference question in

statistics relates to understanding the algebraic and geometric

properties of determinantal varieties such as their irreducible

decompositions or determining their defining equations. I will explain

some recent progress that arises by uncovering the link to point

configurations in matroid theory and incidence geometry. This

connection, in particular, leads to effective computational approaches

for (1) giving a decomposition for each CI variety; (2) identifying

each component in the decomposition as a matroid variety; (3)

determining whether the variety has a real point or equivalently there

is a statistical model satisfying a given collection of dependencies.

The talk is based on joint works with Oliver Clarke, Kevin Grace, and

Harshit Motwani.

The papers are available on arxiv: https://arxiv.org/pdf/2011.02450

and https://arxiv.org/pdf/2103.16550.pdf**–**