One of the main problems related to Gaussian graphical models is learning, i.e. the estimation of model parameters and structure. In this talk, we apply algebraic geometry to analyze two widely used methods for learning, namely the maximum likelihood approach for parameter estimation and the PC-algorithm for structure estimation.

First, we give an algebraic criterion to find exact lower bounds on the number of observations needed for the existence of the maximum likelihood estimator in undirected Gaussian graphical models. This is particularly important for applications such as gene association networks, where the number of random variables is much larger than the number of observations. We also find a first instance of a graph for which only treewidth-many observations are needed; an encouraging result. We then turn to structure estimation in directed Gaussian graphical models and give a rather discouraging result. We analyze the so-called "strong-faithfulness condition", one of the main assumptions of the PC-algorithm, and show that this assumption is in fact extremely restrictive, implying fundamental limitations for the PC-algorithm and other algorithms based on partial correlations.