The Metropolis-Hastings algorithm for estimating a distribution Ï€ is based on choosing a candidate Markov chain and then accepting or rejecting moves of the candidate to produce a chain known to have Ï€ as the invariant measure. The traditional methods use candidates essentially unconnected to Ï€. Based on diffusions for which Ï€ is invariant (such as Langevin diffusions), we develop for one-dimensional distributions a class of candidate distributions that "self-target" towards the high density areas of Ï€. These produce Metropolis-Hastings algorithms with convergence rates that appear to be considerably better than those known for the traditional candidate choices, such as random walk.
In particular, for wide classes of Ï€ these choices may effectively help reduce the "burn-in" problem. We illustrate this behaviour for examples with exponential and polynomial tails, and for a logistic regression model using a Gibbs sampling algorithm.