\relax \bibstyle{plainnat} \citation{pearl88} \citation{howard84} \citation{henrion91} \@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{295}} \citation{ransohoff78} \citation{knottnerus87} \citation{pearl88} \citation{howard84} \@writefile{toc}{\contentsline {section}{\numberline {2}Conceptual Framework}{297}} \newlabel{sec:bns}{{2}{297}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Fundamentals of Bayesian Networks}{297}} \newlabel{sec:def-bn}{{2.1}{297}} \newlabel{factorization}{{1}{297}} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Example Bayesian network consisting of four variables: $D$, $S$, $T$, and $H$.}}{298}} \newlabel{fig:BN_example}{{1}{298}} \citation{berkson46} \citation{cooper92} \citation{pearl91} \citation{spirtes93} \citation{berkson46} \@writefile{toc}{\contentsline {subsection}{\numberline {2.2}Subpopulations and Selection Variables}{299}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.3}Building Bayesian Networks}{299}} \citation{diez97} \citation{druzdzel00b} \citation{onisko01} \citation{greenland99} \citation{pearl00} \citation{hernan02} \citation{knottnerus87} \@writefile{toc}{\contentsline {section}{\numberline {3}Problems of Naive Use of Data}{301}} \newlabel{sec:examples}{{3}{301}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.1}The Risk of Combining Knowledge from Different Sources }{301}} \newlabel{sec:first-example}{{3.1}{301}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Example model under construction.}}{301}} \newlabel{fig:model}{{2}{301}} \@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Distribution of the disease and test result in the hospital population.}}{301}} \newlabel{tab:hospital}{{1}{301}} \newlabel{sensitivity-error}{{2}{302}} \newlabel{specificity-error}{{3}{302}} \newlabel{pd-s-error}{{4}{302}} \@writefile{toc}{\contentsline {paragraph}{Analysis of the problem}{302}} \newlabel{pd-sh=pd-s}{{5}{302}} \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces A causal model for the hospital data set. Note that since each entry in the data is collected for a hospital patient, the model is effectively conditioned on $H=+h$, presence in the hospital, which is the selection variable of the hospital data set. 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