Weighting function for non-Gaussian exponential-family observation models in WoLF

Investigate and specify the observation-dependent weighting function W(y_t, \hat{y}_t) when the measurement model p(y_t | \theta_t) belongs to a non-Gaussian exponential family within the Weighted Observation Likelihood Filter (WoLF) and exponential-family EKF framework; derive the corresponding update rules and criteria for selecting W(y_t, \hat{y}_t) in this setting.

Background

The paper introduces the Weighted Observation Likelihood Filter (WoLF), which replaces the standard log-likelihood with a weighted log-likelihood using a data-dependent weight W(y_t, \hat{y}_t). Robustness results and closed-form updates are developed primarily for Gaussian observation models (and their EKF/EnKF variants), with specific weight functions proposed and analyzed in that context.

An extension to exponential-family observation models is outlined by approximating the likelihood with a moment-matched Gaussian and inserting the weighting term into the log-likelihood. However, the authors explicitly state that determining the appropriate form of W(y_t, \hat{y}_t) for non-Gaussian exponential families is left for future work, indicating the need to define principled choices of W and associated update rules in this broader class.