Richard Nickl (University of Cambridge)

Nonparametric Bayesian inference for discretely sampled diffusions

We consider the nonlinear statistical inverse problem of
making inference on the unknown parameters of a diffusion process
describing the solution of a stochastic differential equation. The
observation regime is such that the process is sampled at discrete
time points that are a fixed distance apart, and we investigate the
asymptotic regime when more samples accrue in the time horizon (thus
avoiding unrealistic `high frequency’ assumptions). We shall briefly
review frequentist estimation techniques and then turn to Bayesian
nonparametric approaches to the problem, which have recently been
shown to be computationally tractable (work of A. Stuart, G. Roberts
and co-authors). A theory of frequentist contraction rates for the
posterior distribution has been elusive for several years, and we
present first rigorous, minimax optimal contraction rates for
posterior distributions arising from natural prior distributions on
infinite-dimensional parameter spaces on the drift and diffusion
coefficient. We will discuss what can be learnt from these results for
the choice of prior in such problems, as well as for the Bayesian
analysis of general nonlinear inverse problems. In the proofs we
obtain some new (functional) concentration inequalities for additive
functionals of Markov chains arising from diffusions that are of
independent interest.

This is joint work with Jakob Soehl (Cambridge),
see http://arxiv.org/abs/1510.05526