[Paper Review] A Probabilistic Perspective on Gaussian Filtering and Smoothing

Marc Peter Deisenroth, Henrik Ohlsson|arXiv (Cornell University)|Jun 10, 2010

Target Tracking and Data Fusion in Sensor Networks21 references3 citations

TL;DR

This paper presents a unified probabilistic framework for Gaussian filtering and smoothing, showing that all such methods differ only in how they compute moments of key joint distributions. It introduces the cubature Kalman smoother and a novel Gibbs sampling-based filter/smoothers, demonstrating improved robustness and moment preservation in nonlinear systems.

ABSTRACT

We present a general probabilistic perspective on Gaussian filtering and smoothing. This allows us to show that common approaches to Gaussian filtering/smoothing can be distinguished solely by their methods of computing/approximating the means and covariances of joint probabilities. This implies that novel filters and smoothers can be derived straightforwardly by providing methods for computing these moments. Based on this insight, we derive the cubature Kalman smoother and propose a novel robust filtering and smoothing algorithm based on Gibbs sampling.

Motivation & Objective

• To provide a general probabilistic perspective on Gaussian filtering and smoothing, unifying existing approaches under a common framework.
• To identify that differences among Gaussian filters/smoothers stem solely from their methods of computing moments of joint distributions.
• To enable the straightforward derivation of new filtering and smoothing algorithms by specifying moment computation methods.
• To propose a novel Gibbs sampling-based filtering and Rauch-Tung-Striebel (RTS) smoothing algorithm that does not require known noise covariances.
• To evaluate the performance of the proposed methods in terms of accuracy, robustness, and moment preservation in nonlinear systems.

Proposed method

• Formalizes Gaussian filtering and smoothing as approximations to the joint posterior distributions $p({oldsymbol{oldsymbol{x}}}_{t-1},{oldsymbol{oldsymbol{x}}}_{t}|{oldsymbol{oldsymbol{z}}}_{1:t-1})$ and $p({oldsymbol{oldsymbol{x}}}_{t},{oldsymbol{oldsymbol{z}}}_{t}|{oldsymbol{oldsymbol{z}}}_{1:t-1})$.
• Derives the cubature Kalman smoother (CKS) by applying cubature rules to compute moments of these joint distributions.
• Proposes a Gibbs sampling-based filter and RTS smoother that infer the means and covariances of the joint distributions directly from data.
• Uses Gibbs sampling to iteratively sample from full conditional distributions, enabling moment estimation without requiring differentiability of system functions.
• Employs a joint sampling approach for $p({oldsymbol{oldsymbol{x}}}_{t-1},{oldsymbol{oldsymbol{x}}}_{t},{oldsymbol{oldsymbol{z}}}_{t}|{oldsymbol{oldsymbol{z}}}_{1:t-1})$ to improve consistency between marginals.
• Relies only on function evaluations of the system and measurement models, avoiding assumptions of differentiability or known noise matrices.

Experimental results

Research questions

• RQ1 How can Gaussian filtering and smoothing be unified under a single probabilistic framework?
• RQ2 What distinguishes different Gaussian filters and smoothers at the level of joint distribution moment computation?
• RQ3 Can a novel, robust filtering and smoothing algorithm be derived by specifying a new method for moment estimation?
• RQ4 How does a Gibbs sampling-based approach compare to standard filters in terms of moment preservation and accuracy in nonlinear systems?
• RQ5 What are the implications of using joint distributions involving both states and measurements for improving smoothing consistency?

Key findings

• The cubature Kalman smoother (CKS) is derived as a direct application of the proposed framework, providing a new, accurate smoothing algorithm.
• The Gibbs-filter and Gibbs-RTSS achieve moment preservation, avoiding incoherent filtering and smoothing distributions that affect non-moment-preserving methods.
• The Gibbs-based methods do not require knowledge of the noise covariance matrices $\mathbf{Q}$ and $\mathbf{R}$, which can instead be inferred from the data.
• Numerical evaluations show that the Gibbs-filter/RTSS performs comparably to state-of-the-art Gaussian filters and smoothers in terms of accuracy and robustness.
• Relative differences in marginal distributions from inconsistent joint approximations were smaller than $10^{-5}$, indicating high consistency when using the extended joint $p({\boldsymbol{x}}_{t-1},{\boldsymbol{x}}_{t},{\boldsymbol{z}}_{t}|{\boldsymbol{z}}_{1:t-1})$.
• The proposed sampling-based approach avoids degeneracy issues common in particle filters and does not require differentiability of the system functions.