We consider importance sampling to estimate the probability μ of a union
of J rare events Hj​ defined by a random variable x. The
sampler we study has been used in spatial statistics, genomics and
combinatorics going back at least to Karp and Luby (1983). It works by sampling
one event at random, then sampling x conditionally on that event
happening and it constructs an unbiased estimate of μ by multiplying an
inverse moment of the number of occuring events by the union bound. We prove
some variance bounds for this sampler. For a sample size of n, it has a
variance no larger than μ(μˉ​−μ)/n where μˉ​ is the union
bound. It also has a coefficient of variation no larger than
(J+J−1−2)/(4n)​ regardless of the overlap pattern among the J
events. Our motivating problem comes from power system reliability, where the
phase differences between connected nodes have a joint Gaussian distribution
and the J rare events arise from unacceptably large phase differences. In the
grid reliability problems even some events defined by 5772 constraints in
326 dimensions, with probability below 10−22, are estimated with a
coefficient of variation of about 0.0024 with only n=10,000 sample
values