Unbiased estimation of the Hessian for partially observed diffusions

Abstract

In this article, we consider the development of unbiased estimators of the Hessian, of the log-likelihood function with respect to parameters, for partially observed diffusion processes. These processes arise in numerous applications, where such diffusions require derivative information, either through the Jacobian or Hessian matrix. As time-discretizations of diffusions induce a bias, we provide an unbiased estimator of the Hessian. This is based on using Girsanov’s Theorem and randomization schemes developed through Mcleish (2011 Monte Carlo Methods Appl. 17 , 301–315 ( doi:10.1515/mcma.2011.013 )) and Rhee & Glynn (2016 Op. Res. 63 , 1026–1043). We demonstrate our developed estimator of the Hessian is unbiased, and one of finite variance. We numerically test and verify this by comparing the methodology here to that of a newly proposed particle filtering methodology. We test this on a range of diffusion models, which include different Ornstein–Uhlenbeck processes and the Fitzhugh–Nagumo model, arising in neuroscience.

Ordinal Outcome State-Space Models for Intensive Longitudinal Data

“…However, this coverage was not at the nominal level of 95%, indicating that the slice-likelihood standard errors are overly liberal. There are a number of approaches for obtaining accurate SE estimates from models that are estimated using a particle filtering approach such as profile likelihoods (Ionides et al, 2015) or particle filter approximations of the Hessian (Spall, 2005;Chada et al, 2022) . However, these approaches are orders of magnitude more computationally expensive than slice-likelihood, requiring in the case of the profile likelihoods re-estimation of thousands of models with slight variations in fixed parameter values.…”