Convergence Rate of Multiple-try Metropolis Independent sampler

The Multiple-try Metropolis (MTM) method is an interesting extension of the classical Metropolis-Hastings algorithm. However, theoretical understandings of its convergence behavior as well as whether and how it may help are still unknown. This paper derives the exact convergence rate for Multiple-try Metropolis Independent sampler (MTM-IS) via an explicit eigen analysis. As a by-product, we prove that MTM-IS is less efficient than the simpler approach of repeated independent Metropolis-Hastings method at the same computational cost. We further explore more variations and find it possible to design more efficient MTM algorithms by creating correlated multiple trials.