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\begin{abstract}
  The problem of computing the marginal probability of a tuple in the result of a query over set-probabilistic databases (PDBs) can be reduced to calculating the probability of the \emph{lineage formula} of the result, a Boolean formula over random variables representing the existence of tuples in the database's possible worlds.
  The analog for bag semantics is a natural number-valued polynomial over random variables that evaluates to the multiplicity of the tuple in each world.
  In this work, we study the problem of calculating the expectation of such polynomials (a tuple's expected  multiplicity) exactly and approximately.
  We are specifically interested in the fine-grained complexity of this problem relative to the complexity of deterministic query evaluation --- if these complexities are comparable, it opens the door to practical deployment of probabilistic databases.
  Unfortunately, we show the reverse; our results imply that computing probabilities for Bag-PDB based on the results produced by such algorithms introduces super-linear overhead.
  % Such factorized representations are necessary to realize the performance of modern join algorithms (e.g., worst-case optimal joins), and so our results imply that a Bag-PDB doing exact computations (via these factorized representations) can never be as fast as a classical (deterministic) database.
  The problem stays hard even if all input tuples have a fixed probability $\prob$ (s.t. $\prob \in (0,1)$).
  We proceed to study polynomials of result tuples of positive relational algebra queries ($\raPlus$) over TIDBs and for a non-trivial subclass of block-independent databases (BIDBs).
  We develop a sampling algorithm that computes a $1 \pm \epsilon$-approximation of the expected multiplicity of an output tuple in linear time in the runtime of a comparable deterministic query.
  % By removing Bag-PDB's reliance on the sum-of-products representation of polynomials, this result paves the way for future work on PDBs that are competitive with deterministic databases.
\end{abstract}

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