 
Summary: An adaptive version for the Metropolis adjusted Langevin
algorithm with a truncated drift
Yves F. Atchadé1
(First draft March 2005; revised November 2005)
Abstract
This paper extends some adaptive schemes that have been developed for the Random Walk
Metropolis algorithm to more general versions of the MetropolisHastings (MH) algorithm,
particularly to the Metropolis Adjusted Langevin algorithm of Roberts and Tweedie (1996).
Our simulations show that the adaptation drastically improves the performance of such MH
algorithms. We study the convergence of the algorithm. Our proves are based on a new
approach to the analysis of stochastic approximation algorithms based on mixingales theory.
Key words: Adaptive Markov Chain Monte Carlo, Langevin algorithms, MetropolisHastings
algorithms, Stochastic approximation algorithms.
MSC Numbers: 65C05, 65C40, 60J27, 60J35
1 Introduction
Markov Chain Monte Carlo (MCMC) is a wellestablished probabilistic tool to sample from proba
bility measures. A MCMC algorithm is designed by specifying a transition kernel with a predened
invariant probability measure (the target distribution). Such transition kernel typically depends on
various parameters to be provided by the user. Finding the optimal values of the parameters for a
given target distribution is a dicult analytical problem. As a consequence, many netunings of
