An effective method for high-dimensionallog-density anova estimation, with application to nonparametric graphical model building

The log-density functional ANOVA model provides a powerful framework for the estimation and interpretation of high-dimensional densities. Existing methods for fitting such a model require repeated numerical integration of high-dimensional functions, and are infeasible in problems of dimension larger than four. We propose a new method for fitting the log-density ANOVA model based on a penalized M-estimation formulation with a novel loss function. Solving the penalized M-estimation problem does not require high-dimensional integration: only one-dimensional integrals are required and they can be computed quickly by using the cumulative distribution function of familiar one-dimensional densities. Simulations indicate that the proposed method is statistically very efficient and computationally practical in high-dimensional problems. We apply the new method to the construction and estimation of (undirected) nonparametric graphical models. The graphical models use graphs to display the conditional dependence among random variables and have become very popular, but have mostly been studied parametrically. Our method provides a practical way to construct and estimate nonparametric graphical models.