  We explore whether splitting and killing methods can improve the accuracy of
Markov chain Monte Carlo (MCMC) estimates of rare event probabilities, and we
make three contributions. First, we prove that "weighted ensemble" is the only
splitting and killing method that provides asymptotically consistent estimates
when combined with MCMC. Second, we prove a lower bound on the asymptotic
variance of weighted ensemble's estimates. Third, we give a constructive proof
and numerical examples to show that weighted ensemble can approach this optimal
variance bound, in many cases reducing the variance of MCMC estimates by
multiple orders of magnitude.
