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:
new states are proposed using (overdamped) Langevin dynamics, which use evaluations of the gradient of the target probability density function; these proposals are accepted or rejected using the Metropolis–Hastings algorithm, which uses evaluations of the target probability density (but not its gradient). 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, (although the method Smart Monte Carlo was already introduced in 1978 ) and its properties were examined in detail by Gareth Roberts together with Richard Tweedie and Jeff Rosenthal. Many variations and refinements have been introduced since then, e.g. the manifold variant of Girolami and Calderhead (2011). The method is equivalent to using the Hamiltonian Monte Carlo (hybrid Monte Carlo) algorithm with only a single discrete time step.
Further details Let π {\displaystyle \pi } denote a probability density function on R d {\displaystyle \mathbb {R} ^{d}} , one from which it is desired to draw an ensemble of independent and identically distributed samples. We consider the overdamped Langevin Itô diffusion
X ˙ = ∇ log π ( X ) + 2 W ˙ {\displaystyle {\dot {X}}=\nabla \log \pi (X)+{\sqrt {2}}{\dot {W}}}
driven by the time derivative of a standard Brownian motion W {\displaystyle W} . (Note that another commonly used normalization for this diffusion is
X ˙ = 1 2 ∇ log π ( X ) + W ˙ , {\displaystyle {\dot {X}}={\frac {1}{2}}\nabla \log \pi (X)+{\dot {W}},}
which generates the same dynamics.) In the limit as t → ∞ {\displaystyle t\to \infty } , this probability distribution ρ ( t ) {\displaystyle \rho (t)} of X ( t ) {\displaystyle X(t)} approaches a stationary distribution, which is also invariant under the diffusion, which we denote ρ ∞ {\displaystyle \rho _{\infty }} . It turns out that, in fact, ρ ∞ = π {\displaystyle \rho _{\infty }=\pi } . 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 τ > 0 {\displaystyle \tau >0} . We set X 0 := x 0 {\displaystyle X_{0}:=x_{0}} and then recursively define an approximation X k {\displaystyle X_{k}} to the true solution X ( k τ ) {\displaystyle X(k\tau )} by
X k + 1 := X k + τ ∇ log π ( X k ) + 2 τ ξ k , {\displaystyle X_{k+1}:=X_{k}+\tau \nabla \log \pi (X_{k})+{\sqrt {2\tau }}\xi _{k},}
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