Conjugate continuous-discrete projection filter via sparse-Grid quadrature

In this article, we study the continuous-discrete projection filter for exponential-family manifolds with conjugate likelihoods. 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. To control the stiffness of the natural parameters’ ordinary differential equations, we introduce a regularization method via projection to the Fisher information metric’s eigenspace. Lastly, we apply the proposed method to approximate the filtering density of a modified Van der Pol oscillator problem and a coupled stochastic FitzHugh–Nagumo system. The proposed projection filter shows superior performance compared to several state-of-the-art parametric continuous-discrete filtering methods.

💡 Research Summary

This paper presents a novel continuous‑discrete projection filter that exploits the structure of exponential‑family manifolds together with conjugate measurement models. The authors first derive a local projection error bound for the prediction step, showing that the error incurred by projecting the square‑root density onto the tangent space of the square‑root exponential‑family manifold is essentially the same as the projection error of the underlying Fokker‑Planck equation. By splitting the natural statistics vector c into two parts—c₁, which captures moments needed for approximating the Fokker‑Planck dynamics, and c₂, which is chosen so that the measurement likelihood is conjugate—they guarantee that the Bayesian update can be performed exactly in closed form, without any numerical integration or optimization.

The prediction dynamics are expressed as an ODE for the natural parameters θ:
dθ/dt = g⁻¹(θ) E_θ