A splitting method to reduce MCMC variance

We explore the possibility of combining Markov chain Monte Carlo (MCMC) with the rare event sampling approach known as ``splitting and killing." First, we prove there is a unique splitting method providing asymptotically consistent estimates when combined with MCMC. This splitting method, called weighted ensemble, has been known since 1997, yet its advantages over MCMC have not yet been precisely quantified. Our second contribution, therefore, is to clarify weighted ensemble's properties by providing a lower bound on the asymptotic variance of the method's estimates. We give a constructive proof and numerical examples to show weighted ensemble can approach this optimal variance bound, in many cases reducing the variance of MCMC estimates by multiple orders of magnitude.