On the use of local optimizations within Metropolis–Hastings updates

Summary.  We propose new Metropolis–Hastings algorithms for sampling from multimodal dis- tributions on ℜ ⁿ . Tjelmeland and Hegstad have obtained direct mode jumping proposals by optimization within Metropolis–Hastings updates and different proposals for 'forward' and 'backward' steps. We generalize their scheme by allowing the probability distribution for forward and backward kernels to depend on the current state. We use the new setting to combine mode jumping proposals and proposals from a prior approximation. We obtain that the frequency of proposals from the different proposal kernels is automatically adjusted to their quality. Mode jumping proposals include local optimizations. When combining this with a prior approximation it is tempting to use local optimization results not only for mode jumping proposals but also to improve the prior approximation. We show how this idea can be implemented. The resulting algorithm is adaptive but has a Markov structure. We evaluate the effectiveness of the proposed algorithms in two simulation examples.