Tweedie's Formula and Score-Driven Updating
Abstract
Tweedie's formula expresses a conditional mean in terms of a score function, and this connects score-driven models with Bayesian updating. We show that the score-driven update is identical to a Bayesian update in some special cases. More generally, we show that score-driven updating is a local approximation to Bayesian updating, with an error of second order in the predictive variance. The second-order approximation error has two components, one from the curvature of the log likelihood and one from the asymmetry of the posterior distribution. Implicit score-driven (ISD) updating, which evaluates the score at the updated parameter, corrects for the former but not the latter. A simple correction for the second component leads to an improved third-order approximation, which we denote ISD*. Numerical experiments show substantial gains from ISD and ISD*.
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BibTeX
@misc{hansen2026tweedies,
title = {{Tweedie's Formula and Score-Driven Updating}},
author = {Peter Reinhard Hansen and Chen Tong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.15902},
eprint = {2605.15902},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.15902},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Tweedie's Formula and Score-Driven Updating
AU - Peter Reinhard Hansen
AU - Chen Tong
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2605.15902
UR - https://arxiv.org/abs/2605.15902
AB - Tweedie's formula expresses a conditional mean in terms of a score function, and this connects score-driven models with Bayesian updating. We show that the score-driven update is identical to a Bayesian update in some special cases. More generally, we show that score-driven updating is a local approximation to Bayesian updating, with an error of second order in the predictive variance. The second-order approximation error has two components, one from the curvature of the log likelihood and one from the asymmetry of the posterior distribution. Implicit score-driven (ISD) updating, which evaluates the score at the updated parameter, corrects for the former but not the latter. A simple correction for the second component leads to an improved third-order approximation, which we denote ISD*. Numerical experiments show substantial gains from ISD and ISD*.
ER -