  Markov Chain Monte Carlo (MCMC) is one of the most powerful methods to sample
from a given probability distribution, of which the Metropolis Adjusted
Langevin Algorithm (MALA) is a variant wherein the gradient of the distribution
is used towards faster convergence. However, being set up in the Euclidean
framework, MALA might perform poorly in higher dimensional problems or in those
involving anisotropic densities as the underlying non-Euclidean aspects of the
geometry of the sample space remain unaccounted for. We make use of concepts
from differential geometry and stochastic calculus on Riemannian manifolds to
geometrically adapt a stochastic differential equation with a non-trivial drift
term. This adaptation is also referred to as a stochastic development. We apply
this method specifically to the Langevin diffusion equation and arrive at a
geometrically adapted Langevin dynamics. This new approach far outperforms
MALA, certain manifold variants of MALA, and other approaches such as
Hamiltonian Monte Carlo (HMC), its adaptive variant the no-U-turn sampler
(NUTS) implemented in Stan, especially as the dimension of the problem
increases where often GALA is actually the only successful method. This is
evidenced through several numerical examples that include parameter estimation
of a broad class of probability distributions and a logistic regression
problem.
