  Although the Metropolis algorithm is simple to implement, it often has
difficulties exploring multimodal distributions. We propose the
repelling-attracting Metropolis (RAM) algorithm that maintains the
simple-to-implement nature of the Metropolis algorithm, but is more likely to
jump between modes. The RAM algorithm is a Metropolis-Hastings algorithm with a
proposal that consists of a downhill move in density that aims to make local
modes repelling, followed by an uphill move in density that aims to make local
modes attracting. The downhill move is achieved via a reciprocal Metropolis
ratio so that the algorithm prefers downward movement. The uphill move does the
opposite using the standard Metropolis ratio which prefers upward movement.
This down-up movement in density increases the probability of a proposed move
to a different mode. Because the acceptance probability of the proposal
involves a ratio of intractable integrals, we introduce an auxiliary variable
which creates a term in the acceptance probability that cancels with the
intractable ratio. Using several examples, we demonstrate the potential for the
RAM algorithm to explore a multimodal distribution more efficiently than a
Metropolis algorithm and with less tuning than is commonly required by
tempering-based methods.
