This Final Exam will be conducted via Zoom. Here is the information below:

Meeting ID: 880-921-183


Graphical models specify conditional independence relations between variables. These include undirected graphical models and directed graphical models, the latter of which also capture causal relationships. In this talk, we discuss challenges but also opportunities brought about by non-Gaussianity in undirected and directed graphical models.

A common challenge in estimating parameters of probability density functions is the intractability of the normalizing constant, which hinders the use of maximum likelihood estimation, especially for non-Gaussian graphical models or when the density is supported on a subspace of $\mathbb{R}^m$. In the first part of the talk, we present a generalized form of score matching for distributions supported on general domains. The proposed approach generalizes the original forms proposed in Hyvärinen (2005) for $\mathbb{R}^m$ and Hyvärinen (2007) for $\mathbb{R}_+^m$, while avoiding direct calculation of the normalizing constant, and yielding closed-form estimates for exponential families of continuous distributions. We generalize the regularized score matching method of Lin et al (2016), and apply it to a general class of pairwise interaction graphical models, establishing strong theoretical guarantees.

In the second part of the talk, motivated by modern RNA sequencing technologies that provide gene expression measurements from single cells, we address the challenge of causal discovery from zero-inflated expression patterns in single cell data. In particular, we propose directed graphical models based on Hurdle conditional distributions to explore cause-effect relationships among the genes. We show that, under a natural and weak assumption, the exact directed acyclic graph for our model can be identified. We propose methods for graph recovery and show simulated experiments that validate the identifiability and graph estimation methods in practice.