5/19/2022 CMSA Interdisciplinary Science Seminar

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

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