Approximate Methods for State-Space Models

TL;DR

This paper introduces the Laplace-Gaussian filter, a fast and stable approximate nonlinear filtering method for state-space models that outperforms traditional methods like sequential Monte Carlo in speed and accuracy.

Contribution

The paper presents the Laplace-Gaussian filter, a novel deterministic approximation technique for nonlinear/non-Gaussian state-space models, improving speed and stability over existing methods.

Findings

LGF provides fast, recursive state estimates.

LGF achieves higher accuracy than sequential Monte Carlo in experiments.

LGF is stable over time and effective in neural decoding applications.

Abstract

State-space models provide an important body of techniques for analyzing time-series, but their use requires estimating unobserved states. The optimal estimate of the state is its conditional expectation given the observation histories, and computing this expectation is hard when there are nonlinearities. Existing filtering methods, including sequential Monte Carlo, tend to be either inaccurate or slow. In this paper, we study a nonlinear filter for nonlinear/non-Gaussian state-space models, which uses Laplace's method, an asymptotic series expansion, to approximate the state's conditional mean and variance, together with a Gaussian conditional distribution. This {\em Laplace-Gaussian filter} (LGF) gives fast, recursive, deterministic state estimates, with an error which is set by the stochastic characteristics of the model and is, we show, stable over time. We illustrate the estimation ability of the LGF by applying it to the problem of neural decoding and compare it to sequential Monte Carlo both in simulations and with real data. We find that the LGF can deliver superior results in a small fraction of the computing time.