  The need to calibrate increasingly complex statistical models requires a
persistent effort for further advances on available, computationally intensive
Monte Carlo methods. We study here an advanced version of familiar Markov Chain
Monte Carlo (MCMC) algorithms that sample from target distributions defined as
change of measures from Gaussian laws on general Hilbert spaces. Such a model
structure arises in several contexts: we focus here at the important class of
statistical models driven by diffusion paths whence the Wiener process
constitutes the reference Gaussian law. Particular emphasis is given on
advanced Hybrid Monte-Carlo (HMC) which makes large, derivative-driven steps in
the state space (in contrast with local-move Random-walk-type algorithms) with
analytical and experimental results. We illustrate it's computational
advantages in various diffusion processes and observation regimes; examples
include stochastic volatility and latent survival models. In contrast with
their standard MCMC counterparts, the advanced versions have mesh-free mixing
times, as these will not deteriorate upon refinement of the approximation of
the inherently infinite-dimensional diffusion paths by finite-dimensional ones
used in practice when applying the algorithms on a computer.
