Methodology And Computing In Applied Probability

, Volume 1, Issue 3, pp 283-306

First online:

Langevin-Type Models I: Diffusions with Given Stationary Distributions and their Discretizations*

  • O. StramerAffiliated withDepartment of Statistics and Actuarial Science, University of Iowa
  • , R. L. TweedieAffiliated withDivision of Biostatistics, University of Minnesota

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We describe algorithms for estimating a given measure π known up to a constant of proportionality, based on a large class of diffusions (extending the Langevin model) for which π is invariant. We show that under weak conditions one can choose from this class in such a way that the diffusions converge at exponential rate to π, and one can even ensure that convergence is independent of the starting point of the algorithm. When convergence is less than exponential we show that it is often polynomial at verifiable rates. We then consider methods of discretizing the diffusion in time, and find methods which inherit the convergence rates of the continuous time process. These contrast with the behavior of the naive or Euler discretization, which can behave badly even in simple cases. Our results are described in detail in one dimension only, although extensions to higher dimensions are also briefly described.

Markov chain Monte Carlo diffusions Langevin models posterior distributions irreducible Markov processes exponential ergodicity uniform ergodicity Euler schemes