Nested particle filters for online parameter estimation in discrete-time state-space Markov models

TL;DR

This paper introduces a recursive nested particle filter method for online estimation of static parameters in state-space models, offering constant computational complexity and proven asymptotic accuracy.

Contribution

The paper proposes a novel recursive nested particle filter approach that simplifies computation compared to existing methods like SMC$^2$, with theoretical convergence guarantees.

Findings

Asymptotic error vanishes at rate 1/√N + 1/√M

Method operates with constant recursive computational complexity

Illustrated with a simple example and simulations

Abstract

We address the problem of approximating the posterior probability distribution of the fixed parameters of a state-space dynamical system using a sequential Monte Carlo method. The proposed approach relies on a nested structure that employs two layers of particle filters to approximate the posterior probability measure of the static parameters and the dynamic state variables of the system of interest, in a vein similar to the recent "sequential Monte Carlo square" (SMC$^2$) algorithm. However, unlike the SMC$^2$ scheme, the proposed technique operates in a purely recursive manner. In particular, the computational complexity of the recursive steps of the method introduced herein is constant over time. We analyse the approximation of integrals of real bounded functions with respect to the posterior distribution of the system parameters computed via the proposed scheme. As a result, we prove, under regularity assumptions, that the approximation errors vanish asymptotically in $L_p$ ($p \ge 1$) with convergence rate proportional to $\frac{1}{\sqrt{N}} + \frac{1}{\sqrt{M}}$, where $N$ is the number of Monte Carlo samples in the parameter space and $N\times M$ is the number of samples in the state space. This result also holds for the approximation of the joint posterior distribution of the parameters and the state variables. We discuss the relationship between the SMC$^2$ algorithm and the new recursive method and present a simple example in order to illustrate some of the theoretical findings with computer simulations.