We consider problems of approximate inference in which the query of interest is given by a complex formula (such as a formula in disjunctive formal form (DNF)) over a joint distribution given by a graphical model. We give a general reduction showing that (approximate) marginal inference for a class of distributions yields approximate inference for DNF queries, and extend our techniques to accommodate even more complex queries, and dense graphical models with variational inference, under certain

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... conditions. Our results unify and generalize classical inference techniques (which are generally restricted to simple marginal queries) and approximate counting methods such as those introduced by Karp, Luby and Madras (which are generally restricted to product distributions). 1 This result follows from the fact that standard DNFs are a subclass of threshold DNFs (see Section 2).