  In recent years, various interacting particle samplers have been developed to
sample from complex target distributions, such as those found in Bayesian
inverse problems. These samplers are motivated by the mean-field limit
perspective and implemented as ensembles of particles that move in the product
state space according to coupled stochastic differential equations. The
ensemble approximation and numerical time stepping used to simulate these
systems can introduce bias and affect the invariance of the particle system
with respect to the target distribution. To correct for this, we investigate
the use of a Metropolization step, similar to the Metropolis-adjusted Langevin
algorithm. We examine Metropolization of either the whole ensemble or smaller
subsets of the ensemble, and prove basic convergence of the resulting ensemble
Markov chain to the target distribution. Our numerical results demonstrate the
benefits of this correction in numerical examples for popular interacting
particle samplers such as ALDI, CBS, and stochastic SVGD.
