Tree density estimation

Abstract

We study the problem of estimating the density f( x) of a random vector X in R^d. For a spanning tree T defined on the vertex set \1, ,d\, the tree density f_T is a product of bivariate conditional densities. An optimal spanning tree minimizes the Kullback-Leibler divergence between f and f_T. From i.i.d. data we identify an optimal tree T^* and efficiently construct a tree density estimate f_n such that, without any regularity conditions on the density f, one has _n |f_n( x)-f_T^*( x)|d x=0 a.s. For Lipschitz f with bounded support, E \ |f_n( x)-f_T^*( x)|d x\=O(n^-1/4), a dimension-free rate.

Tasks

Reproductions