UC Santa Barbara

Graphical models reveal the conditional dependence structure between random variables. By estimating the joint density or conditional density, we can detect edges and recover the structure of a graphical model. We propose new nonparametric methods to learn edges for graphical models under a consolidated framework of smoothing spline ANOVA (SS ANOVA) decomposition. We first develop an automatic nonparametric edge detection method by estimating the joint density function with an L1 penalty to interactions in the SS ANOVA decomposition. In the second project, we work directly on the conditional dependence structure and develop a fully nonparametric neighborhood selection method. We detect edges by applying an L1 regularization to interactions in the SS ANOVA decomposition of conditional density functions. These two methods are flexible and contain many existing models as special cases. They also provide a unified framework without any restrictions on the type of each random variable. The joint density approach requires a large computer memory and is thus computationally feasible only when the dimension is small. The neighborhood selection approach overcomes this disadvantage and is more computationally efficient.

We propose iterative procedures to compute the estimates and establish the convergence rates for both the joint and conditional density as well as interactions. Simulations indicate that both joint and neighborhood selection methods perform well under Gaussian and non-Gaussian settings. We illustrate the proposed methods using real data examples.