A characterization of the log-density smoothing spline ANOVA model

In this article, we introduce a characterization of the log-density smoothing spline ANOVA model. We show that in a log-density ANOVA model of order r (consisting of the main effects and all the interactions of order up to r), the joint density function is uniquely determined by the collection of all r-dimensional marginal densities. Furthermore, the order r model is the largest log-density ANOVA model under which the joint density function is uniquely determined by the r-dimensional marginals. Our results are valid for log-density ANOVA model with other general structures.