Parameter Estimation for Partially Observed Hypoelliptic Diffusions

TL;DR

This paper develops a Gibbs sampling method for estimating parameters in partially observed hypoelliptic diffusion models, addressing challenges of ill-conditioning and partial data, with demonstrated consistency and application to molecular dynamics.

Contribution

It introduces a novel Gibbs sampling approach tailored for hypoelliptic diffusions with partial observations, improving parameter estimation accuracy.

Findings

Method shows asymptotic consistency on simulated data

Effective handling of ill-conditioning in parameter estimation

Successful application to molecular dynamics data

Abstract

Hypoelliptic diffusion processes can be used to model a variety of phenomena in applications ranging from molecular dynamics to audio signal analysis. We study parameter estimation for such processes in situations where we observe some components of the solution at discrete times. Since exact likelihoods for the transition densities are typically not known, approximations are used that are expected to work well in the limit of small inter-sample times $\Delta t$ and large total observation times $N\Delta t$. Hypoellipticity together with partial observation leads to ill-conditioning requiring a judicious combination of approximate likelihoods for the various parameters to be estimated. We combine these in a deterministic scan Gibbs sampler alternating between missing data in the unobserved solution components, and parameters. Numerical experiments illustrate asymptotic consistency of the method when applied to simulated data. The paper concludes with application of the Gibbs sampler to molecular dynamics data.