Bias of Particle Approximations to Optimal Filter Derivative

TL;DR

This paper analyzes the bias of particle approximations to the optimal filter derivative in state-space models, providing bounds that are uniform over time and inversely proportional to the number of particles, applicable to many practical models.

Contribution

It derives tight bias bounds for particle approximations of the optimal filter derivative, extending understanding of their accuracy in practical, nonlinear, and non-tractable models.

Findings

Bias bounds are inversely proportional to the number of particles.

Bounds are uniform in time under strong mixing conditions.

Results apply broadly to practical state-space models.

Abstract

In many applications, a state-space model depends on a parameter which needs to be inferred from a data set. Quite often, it is necessary to perform the parameter inference online. In the maximum likelihood approach, this can be done using stochastic gradient search and the optimal filter derivative. However, the optimal filter and its derivative are not analytically tractable for a non-linear state-space model and need to be approximated numerically. In [Poyiadjis, Doucet and Singh, Biometrika 2011], a particle approximation to the optimal filter derivative has been proposed, while the corresponding $L_{p}$ error bonds and the central limit theorem have been provided in [Del Moral, Doucet and Singh, SIAM Journal on Control and Optimization 2015]. Here, the bias of this particle approximation is analyzed. We derive (relatively) tight bonds on the bias in terms of the number of particles. Under (strong) mixing conditions, the bounds are uniform in time and inversely proportional to the number of particles. The obtained results apply to a (relatively) broad class of state-space models met in practice.