This research-oriented course will explore the theoretical underpinnings of graphical modeling and causality. A graphical model is a mathematical structure that describes complex dependencies between random variables. More precisely, given a (directed or undirected) graph, we envision the random variables as living at each of its vertices, and the conditional independence statements that hold among them can be read off from the graph structure. We will study both undirected and directed graphical models. Given samples from a graphical model, we will discuss model selection: the problem of finding the graph that the data arose from, and inference: the problem of estimating the distribution assuming we know the graph. We will explore different types of algorithms used to solve these questions as well as the mathematical theory involved.

Building on the theory of graphical models, we will study causal inference. Here, we are interested in finding a directed graph that depicts the causal relationships among the observed random variables (e.g., X --> Y if X causes Y). We will discuss how to solve this problem in both the observational and interventional (e.g. randomized control trials) settings. We will conclude with theory and algorithms for the case of hidden variables as well as directed cycles in the graph.