Tweedie's Formula and Score-Driven Updating

Score-driven models update time-varying parameters using conditional likelihood scores. This paper develops a Bayesian interpretation of such updates through Tweedie's formula, which connects posterior mean corrections with marginal scores. In Gaussian signal extraction, this gives an exact posterior-correction identity. For natural exponential families, related identities characterize posterior means in natural- and expectation-parameter spaces. Building on these identities, we show that conjugate Bayesian filtering in expectation space coincides exactly with an inverse-Fisher-scaled conditional score update under local precision discounting. For general conditional densities, the exact Bayesian correction involves a generally unavailable predictive-marginal score. A local Gaussian approximation shows that the conditional likelihood score provides the leading approximation to this posterior correction; under local precision discounting, the predictive covariance becomes proportional to inverse Fisher information, yielding the familiar inverse-Fisher-scaled score recursion. The results clarify when score-driven updates are exact Bayesian filters and when they should instead be viewed as tractable local approximations.