Langevin diffusions and the Metropolis-adjusted Langevin algorithm

By Tatiana Xifara, Chris Sherlock, Samuel Livingstone, Simon Byrne, Mark A. Girolami. 2013

143 citations recorded by the source corpus.

We describe a Langevin diffusion with a target stationary density with respect to Lebesgue measure, as opposed to the volume measure of a previously-proposed diffusion. The two are sometimes equivalent but in general distinct and lead to different Metropolis-adjusted Langevin algorithms, which we compare.

Subfields: Mathematical Physics, Molecular Biology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/0ea79b613808e7ca00f8ec9838f5583eab3cb36e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mathematical Physics

1. Sharmila Karumuri
2. Yian Ma
3. Bin Yu
4. Christian P. Robert
5. Peter J. Green
6. Marcelo Pereyra
7. Martin J. Wainwright
8. Jitesh H. Panchal

In Molecular Biology

1. Bin Yu
2. Christian P. Robert
3. Peter J. Green
4. Marcelo Pereyra
5. Martin J. Wainwright
6. Tianqi Chen
7. Arnak S. Dalalyan
8. K. Latuszy'nski

In Statistics and Probability

1. Martin J. Wainwright
2. Bin Yu
3. Peter J. Green
4. Arnak S. Dalalyan
5. Francis R. Bach
6. Michael I. Jordan
7. K. Latuszy'nski
8. Marcelo Pereyra

Citations in the corpus

• Simulator-free solution of high-dimensional stochastic elliptic partial differential equations using deep neural networks (2019, methodology): “We, therefore resort to the Metropolis-adjusted Langevin algorithm (MALA) algorithm [77, 78], a celebrated Markov-Chain Monte Carlo (MCMC) technique which leverages the gradient of the unnormalized log posterior to simulate a random walk whose invariant measure is the target posterior.”
• A Complete Recipe for Stochastic Gradient MCMC (2015, methodology): “this correction term corresponds to the distribution function with respect to a non-Lebesgue measure [18]; for the Lebesgue measure, the revised Γi(θ) was as determined by our framework [18].”
• Bayesian computation: a perspective on the current state, and sampling backwards and forwards (2015, methodology): “One particular direction of active research is considering a more general version of equation (1) with state dependent drift and diffusion coefficient which also has π as invariant distribution (Xifara et al. (2014), c.f. Kent (1978)).”
• Theoretical guarantees for approximate sampling from smooth and log‐concave densities (2014, methodology): “…essentially on the Metropolis-adjusted version of the LMC algorithm, known as the Metropolis-adjusted Langevin algorithm, and its modifications (Roberts and Rosenthal, 1998; Stramer and Tweedie, 1999a,b; Jarner and Hansen, 2000; Roberts and Stramer, 2002; Pillai et al., 2012; Xifara et al., 2014).”
• Log-concave sampling: Metropolis-Hastings algorithms are fast! (2018, methodology): “Algorithms related to the Langevin diffusion include the Metropolis adjusted Langevin Algorithm (MALA) [41, 40, 2], the unadjusted Langevin algorithm (ULA) [33, 17, 41, 11], underdamped Langevin MCMC [8], Riemannian MALA [45], Proximal-MALA [34, 13], Metropolis adjusted Langevin truncated algorithm ”