Pith Number

pith:L74BNFNC

pith:2026:L74BNFNC3SVGHZP6OUUZGZFECZ

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

Chen Tong, Peter Reinhard Hansen

Score-driven updates match Bayesian posterior corrections exactly for conjugate natural exponential families under steady-state precision discounting and inverse-Fisher scaling.

arxiv:2605.15902 v1 · 2026-05-15 · econ.EM · stat.ME

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Claims

C1strongest claim

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.

C2weakest assumption

The derivations rely on the conditional densities belonging to natural exponential families for the exact equivalence and on Fisher scoring yielding a valid local Gaussian posterior correction for general densities; if these modeling choices do not hold, the claimed representation between score-driven and Bayesian updates fails.

C3one line summary

Score-driven models equal Bayesian posterior mean updates in conjugate natural exponential families under steady-state precision discounting and inverse-Fisher scaling.

References

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[1] Artemova, M., Blasques, F., van Brummelen, J., and Koopman, S. J. (2022). Score-driven models: Methods and applications. InOxford Research Encyclopedia of Economics and Finance. Oxford University Pres 2022

[2] Barndorff-Nielsen, O. E. (1978).Information and Exponential Families in Statistical Theory. John Wiley and Sons, New York. Blasques,F.,Koopman,S.J.,andLucas, A.(2015). Information-theoreticoptimalityo 1978

[3] Bollerslev, T. (1986). Generalized autoregressive heteroskedasticity.Journal of Econometrics, 31:307–327. Buccheri,G.,Bormetti,G.,Corsi,F.,andLillo,F.(2019). Filteringandsmoothingwithscore-driven mode 1986

[4] Cox, D. R. (1981). Statistical analysis of time series: Some recent developments [with discussion and reply].Scandinavian Journal of Statistics, 8:93–115 1981

[5] J., and Lucas, A 2013

Receipt and verification

First computed	2026-05-20T00:01:24.624708Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

5ff81695a2dcaa63e5fe75299364a416489b9251288938ccbd7e18430a66359e

Aliases

arxiv: 2605.15902 · arxiv_version: 2605.15902v1 · doi: 10.48550/arxiv.2605.15902 · pith_short_12: L74BNFNC3SVG · pith_short_16: L74BNFNC3SVGHZP6 · pith_short_8: L74BNFNC

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/L74BNFNC3SVGHZP6OUUZGZFECZ \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 5ff81695a2dcaa63e5fe75299364a416489b9251288938ccbd7e18430a66359e

Canonical record JSON

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    "cross_cats_sorted": [
      "stat.ME"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "econ.EM",
    "submitted_at": "2026-05-15T12:41:05Z",
    "title_canon_sha256": "838a800fc06054c5659f8d297f398df643142fd8733279232936e91e40e5b7a3"
  },
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  "source": {
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    "kind": "arxiv",
    "version": 1
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}