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Multivariate density estimation and graphical models play important roles instatistical learning. The estimated density can be used to construct agraphical model that reveals conditional relationships whereas a graphicalstructure can be used to build models for density estimation. Our goal is toconstruct a consolidated framework that can perform both density and graphestimation. Denote $\bm{Z}$ as the random vector of interest with densityfunction $f(\bz)$. Splitting $\bm{Z}$ into two parts,$\bm{Z}=(\bm{X}^T,\bm{Y}^T)^T$ and writing $f(\bz)=f(\bx)f(\by|\bx)$ where$f(\bx)$ is the density function of $\bm{X}$ and $f(\by|\bx)$ is theconditional density of $\bm{Y}|\bm{X}=\bx$. We propose a semiparametricframework that models $f(\bx)$ nonparametrically using a smoothing spline ANOVA(SS ANOVA) model and $f(\by|\bx)$ parametrically using a conditional Gaussiangraphical model (cGGM). Combining flexibility of the SS ANOVA model withsuccinctness of the cGGM, this framework allows us to deal withhigh-dimensional data without assuming a joint Gaussian distribution. Wepropose a backfitting estimation procedure for the cGGM with a computationallyefficient approach for selection of tuning parameters. We also develop ageometric inference approach for edge selection. We establish asymptoticconvergence properties for both the parameter and density estimation. Theperformance of the proposed method is evaluated through extensive simulationstudies and two real data applications.

Combining Smoothing Spline with Conditional Gaussian Graphical Model for Density and Graph Estimation