Tree Density Estimation

摘要

We study the problem of estimating the density $f({mathbf {x}})$ of a random vector ${ {mathbf {X}}}$ in ${mathbb R}^{d}$ . For a spanning tree $T$ defined on the vertex set ${1, {dots },d}$ , 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 $lim _{nto infty } int f_{n}({mathbf {x}})-f_{T^{*}}({mathbf {x}})d {mathbf {x}}=0$ a.s. For Lipschitz $f$ with bounded support, ${mathbb E}left {{ int f_{n}({mathbf {x}})-f_{T^{*}}({mathbf {x}})d {mathbf {x}}}right }=Obig (n^{-1/4}big)$ , a dimension-free rate.