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\jmlrheading{1}{2000}{1-48}{4/00}{10/00}{meila00a}{Marina Meil\u{a} and Michael I. Jordan}
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\ShortHeadings{Learning with Mixtures of Trees}{Meil\u{a} and Jordan}
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\begin{document}
\title{Learning with Mixtures of Trees}
\author{\name Marina Meil\u{a} \email mmp@stat.washington.edu \\
\addr Department of Statistics\\
University of Washington\\
Seattle, WA 98195-4322, USA
\AND
\name Michael I.\ Jordan \email jordan@cs.berkeley.edu \\
\addr Division of Computer Science and Department of Statistics\\
University of California\\
Berkeley, CA 94720-1776, USA}
\editor{Kevin Murphy and Bernhard Sch{\"o}lkopf}
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\begin{abstract}% <- trailing '%' for backward compatibility of .sty file
This paper describes the mixtures-of-trees model, a probabilistic
model for discrete multidimensional domains. Mixtures-of-trees
generalize the probabilistic trees of \citet{chow:68}
in a different and complementary direction to that of Bayesian networks.
We present efficient algorithms for learning mixtures-of-trees
models in maximum likelihood and Bayesian frameworks.
We also discuss additional efficiencies that can be
obtained when data are ``sparse,'' and we present data
structures and algorithms that exploit such sparseness.
Experimental results demonstrate the performance of the
model for both density estimation and classification.
We also discuss the sense in which tree-based classifiers
perform an implicit form of feature selection, and demonstrate
a resulting insensitivity to irrelevant attributes.
\end{abstract}
\begin{keywords}
Bayesian Networks, Mixture Models, Chow-Liu Trees
\end{keywords}
\section{Introduction}
Probabilistic inference has become a core technology in AI,
largely due to developments in graph-theoretic methods for the
representation and manipulation of complex probability
distributions~\citep{pearl:88}. Whether in their guise as
directed graphs (Bayesian networks) or as undirected graphs (Markov
random fields), \emph{probabilistic graphical models} have a number
of virtues as representations of uncertainty and as inference engines.
Graphical models allow a separation between qualitative, structural
aspects of uncertain knowledge and the quantitative, parametric aspects
of uncertainty...\\
{\noindent \em Remainder omitted in this sample. See http://www.jmlr.org/papers/ for full paper.}
% Acknowledgements should go at the end, before appendices and references
\acks{We would like to acknowledge support for this project
from the National Science Foundation (NSF grant IIS-9988642)
and the Multidisciplinary Research Program of the Department
of Defense (MURI N00014-00-1-0637). }
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\appendix
\section*{Appendix A.}
\label{app:theorem}
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%In this appendix we prove the following theorem from
%Section~\ref{sec:textree-generalization}:
In this appendix we prove the following theorem from
Section~6.2:
\noindent
{\bf Theorem} {\it Let $u,v,w$ be discrete variables such that $v, w$ do
not co-occur with $u$ (i.e., $u\neq0\;\Rightarrow \;v=w=0$ in a given
dataset $\dataset$). Let $N_{v0},N_{w0}$ be the number of data points for
which $v=0, w=0$ respectively, and let $I_{uv},I_{uw}$ be the
respective empirical mutual information values based on the sample
$\dataset$. Then
\[
N_{v0} \;>\; N_{w0}\;\;\Rightarrow\;\;I_{uv} \;\leq\;I_{uw}
\]
with equality only if $u$ is identically 0.} \hfill\BlackBox
\noindent
{\bf Proof}. We use the notation:
\[
P_v(i) \;=\;\frac{N_v^i}{N},\;\;\;i \neq 0;\;\;\;
P_{v0}\;\equiv\;P_v(0)\; = \;1 - \sum_{i\neq 0}P_v(i).
\]
These values represent the (empirical) probabilities of $v$
taking value $i\neq 0$ and 0 respectively. Entropies will be denoted
by $H$. We aim to show that $\fracpartial{I_{uv}}{P_{v0}} < 0$....\\
{\noindent \em Remainder omitted in this sample. See http://www.jmlr.org/papers/ for full paper.}
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