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Conjugate continuous-discrete projection filter via sparse-Grid quadrature

Muhammad F. Emzir, Zaid A. Sawlan, Sami El Ferik

TL;DR

This work develops a continuous-discrete projection filter on exponential-family manifolds with conjugate likelihoods, leveraging sparse-grid quadrature and an adaptive bijection to enable multivariate filtering. It proves that exact Bayesian updates are possible under Gaussian additive-noise conjugacy, and provides a practical numerical framework to propagate square-root densities via projected Fokker–Planck dynamics. The approach is demonstrated on a nonlinear stochastic van der Pol filtering problem, where it outperforms EnKF and Gaussian-sum-based methods in density accuracy and computational efficiency, while using far fewer degrees of freedom. Overall, the method offers a scalable, information-geometry–driven alternative for high-dimensional, non-Gaussian continuous-discrete filtering with strong practical implications for real-time inference. Future work points to robustness in higher-dimensional regimes and further refinement of the conjugate-extended statistic design.

Abstract

In this article, we study the continuous-discrete projection filter for the exponential-family manifolds with conjugate likehoods. We first derive the local projection error of the prediction step of the continuous-discrete projection filter. We then derive the exact Bayesian update algorithm for a class of discrete measurement processes with additive Gaussian noise. Lastly, we present a numerical simulation of the stochastic van der Pol filtering problem with a nonlinear measurement process. The proposed projection filter shows superior performance compared to several state-of-the-art parametric continuous-discrete filtering methods.

Conjugate continuous-discrete projection filter via sparse-Grid quadrature

TL;DR

This work develops a continuous-discrete projection filter on exponential-family manifolds with conjugate likelihoods, leveraging sparse-grid quadrature and an adaptive bijection to enable multivariate filtering. It proves that exact Bayesian updates are possible under Gaussian additive-noise conjugacy, and provides a practical numerical framework to propagate square-root densities via projected Fokker–Planck dynamics. The approach is demonstrated on a nonlinear stochastic van der Pol filtering problem, where it outperforms EnKF and Gaussian-sum-based methods in density accuracy and computational efficiency, while using far fewer degrees of freedom. Overall, the method offers a scalable, information-geometry–driven alternative for high-dimensional, non-Gaussian continuous-discrete filtering with strong practical implications for real-time inference. Future work points to robustness in higher-dimensional regimes and further refinement of the conjugate-extended statistic design.

Abstract

In this article, we study the continuous-discrete projection filter for the exponential-family manifolds with conjugate likehoods. We first derive the local projection error of the prediction step of the continuous-discrete projection filter. We then derive the exact Bayesian update algorithm for a class of discrete measurement processes with additive Gaussian noise. Lastly, we present a numerical simulation of the stochastic van der Pol filtering problem with a nonlinear measurement process. The proposed projection filter shows superior performance compared to several state-of-the-art parametric continuous-discrete filtering methods.

Paper Structure

This paper contains 12 sections, 4 theorems, 30 equations, 7 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Exponential-Family Manifold $\text{EM}(c)$ and Projection onto $\text{EM}(c)^{1/2}$
  4. Local projection error of projection filter
  5. On Augmenting Natural Statistics
  6. Exact Bayesian Update Algorithm
  7. Numerical Implementation
  8. Numerical Simulations
  9. Metrics
  10. Benchmark Methods
  11. Simulation Results
  12. Conclusions

Key Result

Lemma 1

Consider an $m$-dimensional exponential-family manifold $\text{EM}(c) = \{ p_\theta = \exp(c^\top \theta - \psi(\theta)) : \theta \in \Theta \}$ with $c^\top = [c_1^\top, c_2^\top]$, where $c_i : \mathbb{R}^d \to \mathbb{R}^{m_i}$ for $i = 1, 2$, $m = m_1 + m_2$, and $c$ is second-order continuously

Figures (7)

  • Figure 1: The initial densities $p_0$, the initial projection densities $p_{\theta_0}$, and the initial Gaussian-mixture densities for GSF, SP-GSF, PGM (K-means), and PGM (EM). In these plots, the $\times$ sign indicates the location of the real state value. The dash-dotted lines represent the equipotential lines of the densities.
  • Figure 2: Comparison of the posterior empirical densities from particle filter samples and the approximated posterior densities at $k=1$.
  • Figure 3: Similar to Figure \ref{['fig:density_comparison_med_time_index_1']}, but at $k=4$.
  • Figure 4: Comparison of $\text{nMSE}$ for different methods. The values are normalized against the $\text{nMSE}$ of the particle filter.
  • Figure 5: Quartile plots of the ratio $E_1(\sqrt{p_\theta})^2/\mathbb{E}_{\theta}\left[ \left(\frac{\mathcal{L}^\ast (p_\theta)}{p_\theta}\right)^2 \right]$.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Definition 1
  • Lemma 2
  • Proposition 2
  • proof