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Nudging state-space models for Bayesian filtering under misspecified dynamics

Fabian Gonzalez, O. Deniz Akyildiz, Dan Crisan, Joaquin Miguez

TL;DR

This paper tackles misspecification in Bayesian filtering for state-space models by introducing Nudging, a data-driven modification of transition kernels that yields a nudged model $\mathcal{M}^\alpha$. It proves, under mild regularity conditions, that one can choose nudging parameters so that the marginal likelihood $p_T(y_{1:T}\vert \mathcal{M}^\alpha)$ exceeds the base $p_T(y_{1:T}\vert \mathcal{M})$, and it specializes to gradient-based nudging with explicit formulas in linear-Gaussian settings, including a modified Kalman filter. The authors demonstrate the approach numerically on a four-dimensional linear-Gaussian SSM and a stochastic Lorenz 63 model, showing improved Bayesian evidence and robustness to parameter mismatch, even when the original model is misspecified. The work suggests that nudging can serve as a flexible, data-informed tool for robust filtering in the presence of dynamical misspecification, with clear practical procedures for implementation and potential extensions to constrained or non-differentiable likelihoods. The findings have implications for improving filtering performance in high-dimensional, complex systems where dynamics are uncertain or drift away from modeled behavior.

Abstract

Nudging is a popular algorithmic strategy in numerical filtering to deal with the problem of inference in high-dimensional dynamical systems. We demonstrate in this paper that general nudging techniques can also tackle another crucial statistical problem in filtering, namely the misspecification of the transition kernel. Specifically, we rely on the formulation of nudging as a general operation increasing the likelihood and prove analytically that, when applied carefully, nudging techniques implicitly define state-space models that have higher marginal likelihoods for a given (fixed) sequence of observations. This provides a theoretical justification of nudging techniques as data-informed algorithmic modifications of state-space models to obtain robust models under misspecified dynamics. To demonstrate the use of nudging, we provide numerical experiments on linear Gaussian state-space models and a stochastic Lorenz 63 model with misspecified dynamics and show that nudging offers a robust filtering strategy for these cases.

Nudging state-space models for Bayesian filtering under misspecified dynamics

TL;DR

This paper tackles misspecification in Bayesian filtering for state-space models by introducing Nudging, a data-driven modification of transition kernels that yields a nudged model . It proves, under mild regularity conditions, that one can choose nudging parameters so that the marginal likelihood exceeds the base , and it specializes to gradient-based nudging with explicit formulas in linear-Gaussian settings, including a modified Kalman filter. The authors demonstrate the approach numerically on a four-dimensional linear-Gaussian SSM and a stochastic Lorenz 63 model, showing improved Bayesian evidence and robustness to parameter mismatch, even when the original model is misspecified. The work suggests that nudging can serve as a flexible, data-informed tool for robust filtering in the presence of dynamical misspecification, with clear practical procedures for implementation and potential extensions to constrained or non-differentiable likelihoods. The findings have implications for improving filtering performance in high-dimensional, complex systems where dynamics are uncertain or drift away from modeled behavior.

Abstract

Nudging is a popular algorithmic strategy in numerical filtering to deal with the problem of inference in high-dimensional dynamical systems. We demonstrate in this paper that general nudging techniques can also tackle another crucial statistical problem in filtering, namely the misspecification of the transition kernel. Specifically, we rely on the formulation of nudging as a general operation increasing the likelihood and prove analytically that, when applied carefully, nudging techniques implicitly define state-space models that have higher marginal likelihoods for a given (fixed) sequence of observations. This provides a theoretical justification of nudging techniques as data-informed algorithmic modifications of state-space models to obtain robust models under misspecified dynamics. To demonstrate the use of nudging, we provide numerical experiments on linear Gaussian state-space models and a stochastic Lorenz 63 model with misspecified dynamics and show that nudging offers a robust filtering strategy for these cases.

Paper Structure

This paper contains 32 sections, 19 theorems, 142 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction
  2. State space models
  3. Model misspecification
  4. Contributions
  5. Outline of the paper
  6. Summary of notation
  7. Background and problem statement
  8. State space models
  9. Bayesian filter
  10. Model assessment: the Bayesian evidence
  11. Problem statement
  12. Nudging schemes
  13. Nudging
  14. Parametric nudging scheme
  15. Parametric models
  16. ...and 17 more sections

Key Result

Theorem 3.1

Let $\{\alpha_t\}_{t \in \mathbb{N}}$ be a family of nudging parametric transformations as in Definition def:NudParT. If Assumption A2 holds, then there exists a sequence of positive parameters $\gamma_{1:T},$ such that i.e., model $\mathcal{M}^{\alpha}$ has a higher Bayesian evidence than model $\mathcal{M}$.

Figures (16)

  • Figure 1: Comparison of marginal likelihoods for the step-size interval $\gamma \in [5 \times 10^{-3}, 1.5 \times 10^{-1}]$ where $\gamma_t := \gamma$ for all $t = 1, \ldots, T$. The figure shows that the nudged Kalman filter attains a higher likelihood than the original (correctly specified) Kalman filter for a range of step-size values and attains much higher likelihood than the misspecified Kalman filter across all step-sizes.
  • Figure 2: Comparison of the NMSEs for the step-size interval $\gamma \in [5 \times 10^{-3}, 1.5 \times 10^{-1}]$ where $\gamma_t := \gamma$ for all $t = 1, \ldots, T$. The figure shows, similarly, the nudged Kalman filter attains a lower NMSE than the misspecified Kalman filter.
  • Figure 3: Coordinate $x_1$ of the state and its PF estimates with models $\mathcal{M}_\theta$ and $\mathcal{M}_\theta^\alpha$.
  • Figure 4: Coordinate $x_2$ of the state and its PF estimates with models $\mathcal{M}_\theta$ and $\mathcal{M}_\theta^\alpha$.
  • Figure 5: Coordinate $x_3$ of the state and its PF estimates with models $\mathcal{M}_\theta$ and $\mathcal{M}_\theta^\alpha$.
  • ...and 11 more figures

Theorems & Definitions (47)

  • Remark 1
  • Definition 1
  • Remark 2
  • Theorem 3.1
  • Remark 3
  • Corollary 3.2
  • proof
  • Remark 4
  • Proposition 3.3
  • Corollary 3.4
  • ...and 37 more