  Latent Gaussian processes are widely applied in many fields like, statistics,
inverse problems and machine learning. A popular method for inference is
through the posterior distribution, which is typically carried out by Markov
Chain Monte Carlo (MCMC) algorithms. Most Gaussian processes can be represented
as a Gaussian measure in a infinite dimensional space. This is an issue for
standard algorithms as they break down in an infinite dimensional setting, thus
the need for appropriate infinite dimensional samplers for implementing
probabilistic inference in such framework. In this paper, we introduce several
adaptive versions of the preconditioned Crank-Nicolson Langevin (pCNL)
algorithm, which can be viewed as an infinite dimensional version of the well
known Metropolis adjusted Langevin algorithm (MALA) algorithm for Gaussian
processes. The basic premise for all our proposals lies in the idea of
implementing change of measure formulation to adapt the algorithms to greatly
improve their efficiency. A gradient-free version of pCNL is introduced, which
is a hybrid of an adaptive independence sampler and an adaptive random walk
sampler, and is shown to outperform the standard preconditioned Crank-Nicolson
(pCN) scheme. Finally, we demonstrate the efficiency of our proposed algorithm
for three different statistical models.
