My intuition is that you'll get a better reduction in false positives by using the additional space that the anti-filter would have occupied to just expand the positive filter.
As for resources, the papers referenced for March 8 from my course syllabus would be useful.
A Bloom filter is a space-efficient probabilistic data structure, conceived by Burton Howard Bloom in 1970, that is used to test whether an element is a member of a set. False positive matches are possible, but false negatives are not – in other words, a query returns either "possibly in set" or "definitely not in…
Wikipedia