Tree density estimation

• URL: http://arxiv.org/abs/2111.11971v1
• Date: Tue, 23 Nov 2021 16:05:59 GMT
• Title: Tree density estimation
• Authors: L\'aszl\'o Gy\"orfi and Aryeh Kontorovich and Roi Weiss
• Abstract summary: density estimation for a random vector $boldsymbol X$ in $mathbb Rd$ with probability density $f(boldsymbol x)$. For Lipschitz continuous $f$ with bounded support, $mathbb E int |f_n(boldsymbol x)-fT*(boldsymbol x)|dboldsymbol x=0$ a.s. For Lipschitz continuous $f$ with bounded support, $mathbb E int |f_n(boldsymbol x)-f
• Score: 12.831051269764115
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study the problem of density estimation for a random vector ${\boldsymbol X}$ in $\mathbb R^d$ with probability density $f(\boldsymbol x)$. 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. The optimal spanning tree $T^*$ is the spanning tree $T$, for which the Kullback-Leibler divergence of $f$ and $f_{T}$ is the smallest. From i.i.d. data we identify the optimal tree $T^*$ and computationally efficiently construct a tree density estimate $f_n$ such that, without any regularity conditions on the density $f$, one has that $\lim_{n\to \infty} \int |f_n(\boldsymbol x)-f_{T^*}(\boldsymbol x)|d\boldsymbol x=0$ a.s. For Lipschitz continuous $f$ with bounded support, $\mathbb E\{ \int |f_n(\boldsymbol x)-f_{T^*}(\boldsymbol x)|d\boldsymbol x\}=O(n^{-1/4})$.

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