Adaptive Metropolis Adjusted Langevin Algorithm with a truncated drift

Implementation of the Truncated-MALA HM sampler¹ (T-MALA) for the course of Computational Statistics given by Stéphanie Allassonière for Master MVA.

Presentation

This sampler belongs to the family of Hastings-Metropolis algorithms, and more precisely to the class of Metropolis-adjusted Langevin algorithm (MALA) samplers which are Markov chain Monte Carlo (MCMC) methods for obtaining a sequence of random samples from a probability distribution from which direct sampling is difficult.

Samplers

This repository notably contains implementation of the following samplers (hasting_metropolis.py):

• Symmetric Random Walk (with normal proposal distribution whose scale parameter is adaptive)
• Fully adaptive Symmetric Random Walk (with normal proposal distribution whose scale parameter and covariance matrix are adaptive)
• Fully adaptive T-MALA (implementation of [1])
• T-MALA (same as above except only the scale parameter is adaptive)

Target distributions

The following are suggested as target distributions (sampler_utils.py):

• Multivariate gaussian
• "Banana shape" distribution, whose density is given by: $$\pi(x) \propto \text{exp}(x_1^2 / 200 - 0.5 (x_2 + B x_1^2 - 100 B)^2 - 0.5 (x_3^2 + \dots + x_d^2))$$

Sampling examples

• Gaussian target distribution:

• "Banana shape" target distribution:

1. An adaptive version for the Metropolis adjusted Langevin algorithm with a truncated drift, Yves F. Atchadé, Methodology and Computing in Applied Probability, 2006.