Tweedie's Formula, Variance Functions, and Score-Driven Updating

Abstract: Score-driven models update time-varying parameters using conditional likelihood scores. This paper gives a Bayesian interpretation based on Tweedie's formula. In Gaussian signal extraction, Tweedie's formula expresses the posterior correction as a scaled score of the marginal predictive density; in natural exponential families, the corresponding identity includes a base-measure adjustment. For general conditional densities, we show that inverse-Fisher-scaled conditional scores arise as local Gaussian posterior corrections based on Fisher scoring and precision discounting. For conjugate natural exponential families, the classical discounted Bayesian recursion has an exact score-driven representation: with steady-state precision discounting and expectation-space inverse-Fisher scaling, the score-driven correction equals the Bayesian posterior mean before transition dynamics are imposed. Tweedie's variance-function index further clarifies how conditional scores normalize forecast errors. The results link empirical Bayes, approximate filtering, dynamic generalized linear models, and score-driven models while distinguishing exact Bayesian updating from local score-based approximation.