Delayed Rejection schemes for efficient Markov chain Monte Carlo sampling of multimodal distributions

A number of problems in a variety of fields are characterised by target distributions with a multimodal structure in which the presence of several isolated local maxima dramatically reduces the efficiency of Markov chain Monte Carlo sampling algorithms. Several solutions, such as simulated tempering or the use of parallel chains, have been proposed to facilitate the exploration of the relevant parameter space. They provide effective strategies in the cases in which the dimension of the parameter space is small and/or the computational costs are not a limiting factor. These approaches fail however in the case of high-dimensional spaces where the multimodal structure is induced by degeneracies between regions of the parameter space. In this paper we present a fully Markovian way to efficiently sample this kind of distribution based on the general Delayed Rejection scheme with an arbitrary number of steps, and provide details for an efficient numerical implementation of the algorithm.