Statistical algorithms for low-frequency diffusion data: A PDE approach

TL;DR

This paper introduces a PDE-based computational approach for nonparametric inference in multi-dimensional diffusions from low-frequency data, enabling efficient likelihood evaluation and Bayesian inference.

Contribution

It develops a novel PDE-inspired method to compute likelihood gradients and perform statistical inference, reducing complex problems to elliptic eigenvalue solutions.

Findings

Efficient likelihood and gradient computation using elliptic eigenvalue problems.

Successful application of the method in nonparametric Bayesian models with Gaussian priors.

Demonstrated effectiveness through extensive simulation studies.

Abstract

We consider the problem of making nonparametric inference in a class of multi-dimensional diffusions in divergence form, from low-frequency data. Statistical analysis in this setting is notoriously challenging due to the intractability of the likelihood and its gradient, and computational methods have thus far largely resorted to expensive simulation-based techniques. In this article, we propose a new computational approach which is motivated by PDE theory and is built around the characterisation of the transition densities as solutions of the associated heat (Fokker-Planck) equation. Employing optimal regularity results from the theory of parabolic PDEs, we prove a novel characterisation for the gradient of the likelihood. Using these developments, for the nonlinear inverse problem of recovering the diffusivity, we then show that the numerical evaluation of the likelihood and its gradient can be reduced to standard elliptic eigenvalue problems, solvable by powerful finite element methods. This enables the efficient implementation of a large class of popular statistical algorithms, including (i) preconditioned Crank-Nicolson and Langevin-type methods for posterior sampling, and (ii) gradient-based descent optimisation schemes to compute maximum likelihood and maximum-a-posteriori estimates. We showcase the effectiveness of these methods via extensive simulation studies in a nonparametric Bayesian model with Gaussian process priors, in which both the proposed optimisation and sampling schemes provide good numerical recovery. The reproducible code is available online at https://github.com/MattGiord/LF-Diffusion.