  We propose a weighting scheme for the proposals within Markov chain Monte
Carlo algorithms and show how this can improve statistical efficiency at no
extra computational cost. These methods are most powerful when combined with
multi-proposal MCMC algorithms such as multiple-try Metropolis, which can
efficiently exploit modern computer architectures with large numbers of cores.
The locally weighted Markov chain Monte Carlo method also improves upon a
partial parallelization of the Metropolis-Hastings algorithm via
Rao-Blackwellization. We derive the effective sample size of the output of our
algorithm and show how to estimate this in practice. Illustrations and examples
of the method are given and the algorithm is compared in theory and
applications with existing methods.
