Mode jumping proposals in MCMC

by Håkon Tjelmeland (Norwegian University of Science and Technology, Norway) and Bjørn Kåre Hegstad (Statoil, Norway), Scandinavian Journal of Statistics, 2001, 28, 205-223.

Markov chain Monte Carlo algorithms generate samples from a target distribution by simulating a Markov chain. Large flexibility exists in specification of transition matrix of the chain. In practice, however, most algorithms used only allow small changes in the state vector in each iteration. This choice typically causes problems for multi-modal distributions as moves between modes become rare and, in turn, results in slow convergence to the target distribution.

In this paper we consider continuous distributions on $\Re^n$ and specify how optimisation for local maxima of the target distribution can be incorporated in the specification of the Markov chain. Thereby, we obtain a chain with frequent jumps between modes. We demonstrate the effectiveness of the approach in three examples. The first considers a simple mixture of bivariate normal distributions, whereas the two last examples consider sampling from posterior distributions based on previously analysed data sets.