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%!TEX root=./main.tex
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\section{Introduction (Rewrite - 070921)}\label{sec:intro-rewrite-070921}
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\section{Introduction}\label{sec:intro}
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\input{two-step-model}
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\input{two-step-model}
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A probabilistic database (PDB) $\pdb$ is a tuple $\inparen{\idb, \pd}$, where $\idb$ is a set of deterministic database instances called possible worlds and $\pd$ is a probability distribution over $\idb$.
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A probabilistic database (PDB) $\pdb$ is a tuple $\inparen{\idb, \pd}$, where $\idb$ is a set of deterministic database instances called possible worlds and $\pd$ is a probability distribution over $\idb$.
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A commonly studied problem in probabilistic databases is, given a query $\query$, PDB $\pdb$, and possible query result tuple $\tup$, to compute the tuple's \textit{marginal probability} of being in the query's result, i.e., computing the expectation of a Boolean random variable over $\pd$ that is $1$ for every $\db \in \idb$ for which $\tup \in \query(\db)$ and $0$ otherwise. In this work, we are interested in bag semantics where each tuple $\tup$ is associated with a multiplicity $\db(\tup)$ from $\semN$ in each possible world\footnote{We find it convenient to use the notation from~\cite{DBLP:conf/pods/GreenKT07} which models bag relations as functions that map tuples to their multiplicity.}.
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A commonly studied problem in probabilistic databases is, given a query $\query$, PDB $\pdb$, and possible query result tuple $\tup$, to compute the tuple's \textit{marginal probability} of being in the query's result, i.e., computing the expectation of a Boolean random variable over $\pd$ that is $1$ for every $\db \in \idb$ for which $\tup \in \query(\db)$ and $0$ otherwise. In this work, we are interested in bag semantics where each tuple $\tup$ is associated with a multiplicity $\db(\tup)$ from $\semN$ in each possible world\footnote{We find it convenient to use the notation from~\cite{DBLP:conf/pods/GreenKT07} which models bag relations as functions that map tuples to their multiplicity.}.
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