Metropolis-adjusted Langevin algorithm

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In computational statistics, the Metropolis-adjusted Langevin algorithm (MALA) or Langevin Monte Carlo (LMC) is a Markov chain Monte Carlo (MCMC) method for obtaining random samples – sequences of random observations – from a probability distribution for which direct sampling is difficult. As the name suggests, MALA uses a combination of two mechanisms to generate the states of a random walk that has the target probability distribution as an invariant measure:

Informally, the Langevin dynamics drive the random walk towards regions of high probability in the manner of a gradient flow, while the Metropolis–Hastings accept/reject mechanism improves the mixing and convergence properties of this random walk. MALA was originally proposed by Julian Besag in 1994,[1] (although the method Smart Monte Carlo was already introduced in 1978 [2]) and its properties were examined in detail by Gareth Roberts together with Richard Tweedie[3] and Jeff Rosenthal.[4] Many variations and refinements have been introduced since then, e.g. the manifold variant of Girolami and Calderhead (2011).[5] The method is equivalent to using the Hamiltonian Monte Carlo (hybrid Monte Carlo) algorithm with only a single discrete time step.[5]

Further details

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Let   denote a probability density function on  , one from which it is desired to draw an ensemble of independent and identically distributed samples. We consider the overdamped Langevin Itô diffusion

 

driven by the time derivative of a standard Brownian motion  . (Note that another commonly-used normalization for this diffusion is

 

which generates the same dynamics.) In the limit as  , this probability distribution   of   approaches a stationary distribution, which is also invariant under the diffusion, which we denote  . It turns out that, in fact,  .

Approximate sample paths of the Langevin diffusion can be generated by many discrete-time methods. One of the simplest is the Euler–Maruyama method with a fixed time step  . We set   and then recursively define an approximation   to the true solution   by

 

where each   is an independent draw from a multivariate normal distribution on   with mean 0 and covariance matrix equal to the   identity matrix. Note that   is normally distributed with mean   and covariance equal to   times the   identity matrix.

In contrast to the Euler–Maruyama method for simulating the Langevin diffusion, which always updates   according to the update rule

 

MALA incorporates an additional step. We consider the above update rule as defining a proposal   for a new state,

 

This proposal is accepted or rejected according to the Metropolis-Hastings algorithm: set

 

where

 

is the transition probability density from   to   (note that, in general  ). Let   be drawn from the continuous uniform distribution on the interval  . If  , then the proposal is accepted, and we set  ; otherwise, the proposal is rejected, and we set  .

The combined dynamics of the Langevin diffusion and the Metropolis–Hastings algorithm satisfy the detailed balance conditions necessary for the existence of a unique, invariant, stationary distribution  . Compared to naive Metropolis–Hastings, MALA has the advantage that it usually proposes moves into regions of higher   probability, which are then more likely to be accepted. On the other hand, when   is strongly anisotropic (i.e. it varies much more quickly in some directions than others), it is necessary to take   in order to properly capture the Langevin dynamics; the use of a positive-definite preconditioning matrix   can help to alleviate this problem, by generating proposals according to

 

so that   has mean   and covariance  .

For limited classes of target distributions, the optimal acceptance rate for this algorithm can be shown to be  ; if it is discovered to be substantially different in practice,   should be modified accordingly.[4]

References

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  1. ^ J. Besag (1994). "Comments on "Representations of knowledge in complex systems" by U. Grenander and MI Miller". Journal of the Royal Statistical Society, Series B. 56: 591–592.
  2. ^ Rossky, P.J.; Doll, J.D.; Friedman, H.L. (1978). "Brownian Dynamics as smart Monte Carlo simulation". Journal of Chemical Physics. 69 (10): 4628. Bibcode:1978JChPh..69.4628R. doi:10.1063/1.436415.
  3. ^ G. O. Roberts and R. L. Tweedie (1996). "Exponential convergence of Langevin distributions and their discrete approximations". Bernoulli. 2 (4): 341–363. doi:10.2307/3318418. JSTOR 3318418.
  4. ^ a b G. O. Roberts and J. S. Rosenthal (1998). "Optimal scaling of discrete approximations to Langevin diffusions". Journal of the Royal Statistical Society, Series B. 60 (1): 255–268. doi:10.1111/1467-9868.00123. S2CID 5831882.
  5. ^ a b M. Girolami and B. Calderhead (2011). "Riemann manifold Langevin and Hamiltonian Monte Carlo methods". Journal of the Royal Statistical Society, Series B. 73 (2): 123–214. CiteSeerX 10.1.1.190.580. doi:10.1111/j.1467-9868.2010.00765.x.