Minimising MCMC variance via diffusion limits, with an application to simulated tempering

We derive new results comparing the asymptotic variance of diffusions by writing them as appropriate limits of discrete-time birth.death chains which themselves satisfy Peskun orderings. We then apply our results to simulated tempering algorithms to establish which choice of inverse temperatures minimises the asymptotic variance of all functionals and thus leads to the most efficient MCMC algorithm.