Tree density estimation

We study the problem of estimating the density $f(\boldsymbol x)$ of a random vector ${\boldsymbol X}$ 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 $fn$ such that, without any regularity conditions on the density $f$, one has $\lim{n\to \infty} \int |fn(\boldsymbol x)-f{T^{*}(\boldsymbol} x)|d\boldsymbol x=0$ a.s. For Lipschitz $f$ with bounded support, $\mathbb E \left{ \int |fn(\boldsymbol x)-f{T^{*}(\boldsymbol} x)|d\boldsymbol x\right}=O\big(n^{{-1/4}\big)$,} a dimension-free rate.