  Multiple-try Metropolis (MTM) is a popular Markov chain Monte Carlo method
with the appealing feature of being amenable to parallel computing. At each
iteration, it samples several candidates for the next state of the Markov chain
and randomly selects one of them based on a weight function. The canonical
weight function is proportional to the target density. We show both
theoretically and empirically that this weight function induces pathological
behaviours in high dimensions, especially during the convergence phase. We
propose to instead use weight functions akin to the locally-balanced proposal
distributions of Zanella (2020), thus yielding MTM algorithms that do not
exhibit those pathological behaviours. To theoretically analyse these
algorithms, we study the high-dimensional performance of ideal schemes that can
be thought of as MTM algorithms which sample an infinite number of candidates
at each iteration, as well as the discrepancy between such schemes and the MTM
algorithms which sample a finite number of candidates. Our analysis unveils a
strong distinction between the convergence and stationary phases: in the
former, local balancing is crucial and effective to achieve fast convergence,
while in the latter, the canonical and novel weight functions yield similar
performance. Numerical experiments include an application in precision medicine
involving a computationally-expensive forward model, which makes the use of
parallel computing within MTM iterations beneficial.
