On directional Metropolis-Hastings algorithms

Metropolis–Hastings algorithms are used to simulate Markov chains with limiting distribution equal to a specified target distribution. The current paper studies target densities on R. In directional Metropolis–Hastings algorithms each iteration consists of three steps i) generate a line by sampling an auxiliary variable, ii) propose a new state along the line, and iii) accept/reject according to the Metropolis–Hastings acceptance probability. We consider two classes of algorithms. The first uses a point in R as auxiliary variable, the second uses an auxiliary direction vector. The directional Metropolis–Hastings algorithms considered here generalize previously proposed directional algorithms in that we allow the distribution of the auxiliary variable to depend on properties of the target at the current state. By letting the proposal distribution along the line depend on the density of the auxiliary variable, we then identify proposal mechanisms that give unit acceptance rate. Especially when we use direction vector as auxiliary variable, we get the advantageous effect of large moves in the Markov chain and the autocorrelation length of the chain is small. We illustrate the algorithms for a Gaussian example and in a Bayesian spatial model for seismic data.