This problem has a practical application in probabilistic databases over multisets, where it corresponds to calculating the expected multiplicity of a query result tuple.
Interesting directions for future work include development of a dichotomy for bag \abbrPDB\xplural. While we handle higher moments in \Cref{app:sec-cicuits}, more general approximations are an interesting area for exploration, including those for more general data models. % beyond what we consider in this paper.
% Furthermore, it would be interesting to see whether our approximation algorithm can be extended to support queries with negations, perhaps using circuits with monus as a representation system.
% \BG{I am not sure what interesting future work is here. Some wild guesses, if anybody agrees I'll try to flesh them out:
% \textbullet{More queries: what happens with negation can circuits with monus be used?}
% \textbullet{More databases: can we push beyond BIDBs? E.g., C-tables / aggregate semimodules or just TIDBs where each input tuple is a random variable over $\mathbb{N}$?}
% \textbullet{Other results: can we extend the work to approximate $P(R(t) = n)$}