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

TL;DR

This paper provides a Bayesian interpretation of score-driven models using Tweedie's formula, linking them to empirical Bayes, filtering, and generalized linear models, and clarifying their theoretical foundations.

Contribution

It introduces a Bayesian perspective on score-driven models through Tweedie's formula, connecting various statistical frameworks and deriving exact and approximate updating formulas.

Findings

Tweedie's formula expresses posterior correction as a scaled score of the predictive density.

Inverse-Fisher-scaled scores arise as local Gaussian posterior corrections.

Classical Bayesian recursion has an exact score-driven representation in conjugate exponential families.

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.