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\documentclass[twoside,11pt]{article} | ||
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% Any additional packages needed should be included after jmlr2e. | ||
% Note that jmlr2e.sty includes epsfig, amssymb, natbib and graphicx, | ||
% and defines many common macros, such as 'proof' and 'example'. | ||
% | ||
% It also sets the bibliographystyle to plainnat; for more information on | ||
% natbib citation styles, see the natbib documentation, a copy of which | ||
% is archived at https://www.jmlr.org/format/natbib.pdf | ||
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\usepackage{jmlr2e} | ||
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% Definitions of handy macros can go here | ||
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\newcommand{\dataset}{{\cal D}} | ||
\newcommand{\fracpartial}[2]{\frac{\partial #1}{\partial #2}} | ||
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% Heading arguments are {volume}{year}{pages}{date submitted}{date published}{paper id}{author-full-names} | ||
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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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% Short headings should be running head and authors last names | ||
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\ShortHeadings{Learning with Mixtures of Trees}{Meil\u{a} and Jordan} | ||
\firstpageno{1} | ||
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\begin{document} | ||
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\title{Learning with Mixtures of Trees} | ||
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\author{\name Marina Meil\u{a} \email [email protected] \\ | ||
\addr Department of Statistics\\ | ||
University of Washington\\ | ||
Seattle, WA 98195-4322, USA | ||
\AND | ||
\name Michael I.\ Jordan \email [email protected] \\ | ||
\addr Division of Computer Science and Department of Statistics\\ | ||
University of California\\ | ||
Berkeley, CA 94720-1776, USA} | ||
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\editor{Kevin Murphy and Bernhard Sch{\"o}lkopf} | ||
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\maketitle | ||
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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} | ||
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\begin{keywords} | ||
Bayesian Networks, Mixture Models, Chow-Liu Trees | ||
\end{keywords} | ||
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\section{Introduction} | ||
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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...\\ | ||
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{\noindent \em Remainder omitted in this sample. See https://www.jmlr.org/papers/ for full paper.} | ||
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% Acknowledgements should go at the end, before appendices and references | ||
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\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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% Manual newpage inserted to improve layout of sample file - not | ||
% needed in general before appendices/bibliography. | ||
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\newpage | ||
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\appendix | ||
\section*{Appendix A.} | ||
\label{app:theorem} | ||
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% Note: in this sample, the section number is hard-coded in. Following | ||
% proper LaTeX conventions, it should properly be coded as a reference: | ||
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%In this appendix we prove the following theorem from | ||
%Section~\ref{sec:textree-generalization}: | ||
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In this appendix we prove the following theorem from | ||
Section~6.2: | ||
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\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 | ||
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\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 $\fracpartial{I_{uv}}{P_{v0}} < 0$....\\ | ||
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{\noindent \em Remainder omitted in this sample. See https://www.jmlr.org/papers/ for full paper.} | ||
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\vskip 0.2in | ||
\bibliography{sample} | ||
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\end{document} |