Unbiased Estimation of the Gradient of the Log-Likelihood for a Class of Continuous-Time State-Space Models

TL;DR

This paper introduces an unbiased gradient estimation method for continuous-time state-space models using a novel coupled conditional particle filter, enabling more accurate gradient-based parameter estimation in stochastic models.

Contribution

The paper presents a new coupled conditional particle filter that provides unbiased gradient estimates, improving parameter estimation in continuous-time state-space models.

Findings

Unbiased gradient estimates facilitate more accurate stochastic gradient methods.

The proposed estimator outperforms the Rhee & Glynn estimator in numerical experiments.

Application to stochastic gradient descent demonstrates practical effectiveness.

Abstract

In this paper, we consider static parameter estimation for a class of continuous-time state-space models. Our goal is to obtain an unbiased estimate of the gradient of the log-likelihood (score function), which is an estimate that is unbiased even if the stochastic processes involved in the model must be discretized in time. To achieve this goal, we apply a doubly randomized scheme, that involves a novel coupled conditional particle filter (CCPF) on the second level of randomization. Our novel estimate helps facilitate the application of gradient-based estimation algorithms, such as stochastic-gradient Langevin descent. We illustrate our methodology in the context of stochastic gradient descent (SGD) in several numerical examples and compare with the Rhee & Glynn estimator.