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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.
  For tuple-independent databases (TIDBs), the expected multiplicity of a query result tuple can trivially be computed in linear time in the size of the tuple's lineage, if this polynomial is encoded as a sum of products (the standard operating procedure for Set-PDBs).
  However, using a reduction from the problem of counting $k$-matchings, we demonstrate that calculating the expectation is \sharpwonehard when the polynomial is compressed, for example through factorization.
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 for polynomials generated by conjunctive queries (CQs) if all input tuples have a fixed probability $\prob$ (s.t. $\prob \in (0,1)$).
  We proceed to study polynomials of result tuples of union of conjunctive queries (UCQs) 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 expectation of polynomial circuits in linear time in the size of the polynomial.
  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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