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Tweedie’s Formula and Score-Driven Updating
\maketitle
\begin{abstract}
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.
\end{abstract}
{\noindent}{\small\textit{Keywords:}}{\small{}
Tweedie's formula, score-driven models, empirical Bayes, observation-driven models, exponential dispersion models, filtering, variance functions.}{\small\par}
\setstretch{1.5}
\newpage
\section{Introduction}
Score-driven models provide a parsimonious way to introduce time variation in the parameters of conditional distributions. In the terminology of \citet{Cox:1981}, they are observation-driven models: the current observation directly updates the parameter governing the next conditional density. Their defining feature is that the update is proportional to the score of the conditional log likelihood with respect to the time-varying parameter. This framework was developed systematically by \citet{CrealKoopmanLucas:2013} and \citet{Harvey:2013}; see also \citet{ArtemovaBlasquesVanBrummelenKoopman:2022} for a recent overview. It also has roots in earlier score-based and robust filtering ideas, including the approximate non-Gaussian filter of \citet{Masreliez:1975}.
The appeal of score-driven models is well established. They generate fully observation-driven recursions, so the likelihood is available by prediction-error decomposition. They adapt automatically to the conditional density, since the score reflects the local shape of the likelihood. They also provide robust updating in heavy-tailed models, where unusual observations may have bounded or attenuated influence; see, for example, \citet{HarveyLuati:2014}. In addition, score-driven updates have information-theoretic foundations based on Kullback-Leibler criteria. \citet{BlasquesKoopmanLucas:2015} provide an influential local optimality argument, and \citet{GorgiLauriaLuati:2024} further develop this perspective. More recently, \citet{dePunderDimitriadisLange:2026} refine this interpretation by showing that the appropriate guarantee is expected Kullback-Leibler improvement, characterized by alignment between the expected update direction and the expected score.
This paper offers a complementary Bayesian interpretation. The starting point is Tweedie's formula.\footnote{To the best of our knowledge, Tweedie's formula has not previously been used to interpret GAS, DCS, or score-driven updating. As a check, the curated GAS bibliography at gasmodel.com listed 453 papers and articles as of May 2026, and a search of this list did not identify any prior use of the phrase ``Tweedie's formula'' in this context.} In the Gaussian location model, Tweedie's formula shows that the posterior mean correction from a noisy observation to the latent signal is exactly proportional to the score of the marginal predictive density.
The identity was reported by \citet{Robbins:1956}, who credited personal correspondence with M.~C.~K.~Tweedie, and was later emphasized by \citet{Efron:2011} in the context of empirical Bayes and selection bias. Thus, in this setting, the score is not merely a local likelihood direction; it is the exact Bayesian correction that moves the observation toward the latent quantity of interest.
The same idea extends beyond the Gaussian location case. For natural exponential families, one Tweedie identity expresses the posterior mean of the natural parameter as the score of the marginal density after correcting for the base measure. A companion identity characterizes the posterior mean of the expectation parameter \(\mu=\psi'(\theta)\). Under conjugate priors, this expectation-parameter identity has a simple closed form and becomes the algebraic source of the exact recursive filtering benchmark developed below. A distinct but related contribution of Tweedie is the variance-function index, which classifies exponential dispersion models by their variance functions \citep{Tweedie:1984,Jorgensen:1997book}. In the present context, this index clarifies how conditional scores normalize forecast errors across observation densities.
The connection to score-driven models requires distinguishing two different scores. The identities considered here involve marginal predictive scores, either with respect to the observation or with respect to a predictive parameter. Score-driven models typically use a different score: the score of the conditional density with respect to the time-varying parameter. In a dynamic model, the exact Bayesian update generally requires integration over the latent state and is unavailable in closed form. Score-driven models can therefore be viewed as replacing an unavailable marginal-score correction by a tractable scaled conditional-score correction evaluated at the current predictive parameter.
We make this connection precise by separating exact expectation-space filtering from local score-driven approximations. First, in the exact conjugate case, natural exponential-family observation densities and conjugate priors imply a closed-form posterior mean update for the expectation parameter \(\mu_t=\psi'(\theta_t)\). Under local precision discounting, this update is exactly an inverse-Fisher-scaled conditional-score correction before transition dynamics are imposed.
Second, we study what remains outside such exact conjugate cases. We derive a parameter-space Tweedie identity for dynamic filtering when the predictive distribution of the state belongs to a natural exponential family. This identity expresses the posterior correction as the score of the predictive marginal density with respect to the predictive mean. For general conditional densities, the predictive marginal score is typically unavailable. A local expansion shows that the conditional likelihood score gives the leading covariance-scaled approximation to the exact posterior correction. Combined with a local precision-discounting covariance specification, this yields the inverse-Fisher-scaled score correction used in score-driven models. Thus score-driven updating has a local Bayesian interpretation: the conditional
score provides the leading correction direction, while local precision
discounting yields the inverse-Fisher scaling used in score-driven recursions.
The examples illustrate the scope of the argument. In the Gaussian location model, Tweedie's formula reproduces the Kalman update. In Gaussian variance models, the relevant sufficient statistic is the squared observation, and inverse-Fisher scaling converts the variance score into the innovation in squared observations; embedded in an observation-driven transition equation, this yields GARCH-type recursions. In Poisson and Gamma models, the variance function determines how the conditional score normalizes raw forecast errors, while conjugate-prior updating in expectation space delivers the exact posterior-mean recursion under local precision discounting.
The contribution is therefore both interpretive and formal. The filtering interpretation of \citet{Masreliez:1975}, the score-driven model classes of \citet{CrealKoopmanLucas:2013} and \citet{Harvey:2013}, and the Kullback-Leibler optimality results of \citet{BlasquesKoopmanLucas:2015}, \citet{GorgiLauriaLuati:2024}, and \citet{dePunderDimitriadisLange:2026} remain central. Tweedie's formula adds a Bayesian signal-extraction perspective: in canonical empirical-Bayes problems, posterior corrections are marginal-score objects. The exact conjugate benchmark identifies a setting in which Bayesian updating can be written directly as an inverse-Fisher-scaled conditional-score correction in expectation space. The local expansion then explains why conditional scores arise as approximate Bayesian filtering corrections outside exact conjugate cases, while local precision discounting gives the inverse-Fisher scaling. Together, these results link empirical Bayes denoising, conjugate Bayesian updating, approximate filtering, and observation-driven time-series modeling.
\section{Tweedie's Formula for Gaussian and Natural Exponential Families}
\subsection{The Gaussian location case}
We begin with Tweedie's formula, which links the posterior expectation of a latent signal to the score of the marginal density of the observation. Consider the Gaussian location model
$$
Y=\mu+\varepsilon,\qquad \varepsilon\sim N(0,\sigma^2),
$$
where $\sigma^2$ is known and $\mu$ has prior density $\pi$.
Let
$$
f(y)=\int \phi_\sigma(y-\mu)\pi(\mu)d\mu
$$
denote the marginal density of $Y$, where $\phi_\sigma$ is the $N(0,\sigma^2)$ density.
\begin{lemma}[Tweedie's formula]
\label{lem:tweedie_gaussian}
Suppose $f(y)>0$. Suppose also that there is an open neighborhood $U$ of $y$ and an integrable function $H$ such that, for all $u\in U$,
$
\left|\partial_u\phi_\sigma(u-\mu)\pi(\mu)\right|\leq H(\mu)
$
for almost all $\mu$. Then
$$
\mathbb{E}[\mu|Y=y]
=
y+\sigma^2\partial_y\log f(y).
$$
Equivalently,
$\mathbb{E}[\mu|Y=y]-y=\sigma^2\partial_y\log f(y)$ and $\mathbb{E}[\varepsilon|Y=y]=-\sigma^2\partial_y\log f(y)$.
\end{lemma}
\begin{proof}
By the stated domination condition, differentiation may be passed under the integral sign. Hence
$
f^\prime(y)
=
\int -\frac{y-\mu}{\sigma^2}\phi_\sigma(y-\mu)\pi(\mu)d\mu
=
\frac{1}{\sigma^2}
\left\{
\int \mu\phi_\sigma(y-\mu)\pi(\mu)d\mu-yf(y)
\right\}$.
Dividing by $f(y)$ yields
$\sigma^2 f^\prime(y)/f(y)=\mathbb{E}[\mu|Y=y]-y$, which proves the result.
\end{proof}
Thus, in the Gaussian location model, the Bayesian correction from the noisy observation $y$ to the latent signal $\mu$ is exactly the score of the marginal density, scaled by the noise variance. The prior enters only through the marginal density $f$. Consequently, if the marginal density, or its score, can be specified or estimated, the posterior mean can be recovered directly. This is the empirical-Bayes interpretation emphasized by \citet{Efron:2011}.
The Gaussian-convolution identity is especially transparent because the latent signal may have an arbitrary prior distribution.\footnote{For example, if $Y=X+Z$, where $Z\sim N(0,\sigma^2)$ and $X$ is Cauchy, then $Y$ has a Voigt density. Lemma \ref{lem:tweedie_gaussian} gives $\mathbb{E}[X|Y=y]=y+\sigma^2\partial_y\log f(y)$, and hence $\mathbb{E}[Z|Y=y]=-\sigma^2\partial_y\log f(y)$. In \citet{HansenTong:2026Voigt}, this identity yields the conditional Gaussian component in the Gauss-Cauchy convolution and produces a redescending update.} At the same time, the Gaussian location identity is recovered as a special case of the natural-exponential-family identity below after writing the Gaussian density in natural-parameter form.
For the purposes of this paper, the key implication is that the score appears as the posterior-mean correction that denoises $y$. In the Gaussian location model, moving in the direction of the marginal score is not merely a local likelihood adjustment or a robustness device; it is the exact posterior-mean correction. This observation provides the first step toward the connection with score-driven models. In dynamic settings, the exact marginal score is typically unavailable because it requires integrating over the latent state. Score-driven models replace this generally intractable Bayesian correction by a tractable recursive update based on the conditional likelihood score evaluated at the current predictive parameter. The distinction between these two scores is central to the argument developed below.
\subsection{Natural exponential families}
The Gaussian identity in Lemma~\ref{lem:tweedie_gaussian} is a special case of a natural-exponential-family identity. This version of Tweedie's formula is attributed by \citet{Efron:2011} to \citet{Robbins:1956} and is closely related to standard exponential-family identities; see also \citet{Barndorff-Nielsen:1978book}. In this setting, the posterior expectation of the natural parameter is determined by a marginal score after correcting for the base measure.
Consider the case in which the conditional observation distribution of $Y$ given $\theta$ belongs to a natural exponential family,
$$
p(y|\theta)=h(y)\exp\{\theta y-\psi(\theta)\},
$$
where $\theta$ is the natural parameter and $\psi$ is the cumulant generating function.
Let
$$
f(y)=\int p(y|\theta)\pi(\theta)d\theta
$$
denote the marginal density of $Y$ induced by a prior distribution $\pi$ for $\theta$.
\begin{lemma}[Natural-exponential-family Tweedie identity]
\label{lem:tweedie_nef}
Suppose $f(y)>0$ and $h(y)>0$. Suppose also that there is an open neighborhood $U$ of $y$ and an integrable function $H$ such that, for all $u\in U$,
$\left|\theta\exp\{\theta u-\psi(\theta)\}\pi(\theta)\right|\leq H(\theta)$ for almost all $\theta$. Then
$$
\mathbb E[\theta|Y=y]
=
\partial_y\log\frac{f(y)}{h(y)}
=
\partial_y\log f(y)-\partial_y\log h(y).
$$
\end{lemma}
\begin{proof}
Since we know that
$f(y)/h(y)=\int \exp\{\theta y-\psi(\theta)\}\pi(\theta)d\theta$,
differentiating its logarithm gives
$
\partial_y\log\frac{f(y)}{h(y)}
=
\frac{\int \theta\exp\{\theta y-\psi(\theta)\}\pi(\theta)d\theta}
{\int \exp\{\theta y-\psi(\theta)\}\pi(\theta)d\theta}
=
\mathbb{E}[\theta|Y=y]$.
This proves the identity.
\end{proof}
Lemma \ref{lem:tweedie_nef} shows that posterior learning about the natural parameter is governed by the score of the marginal density, adjusted by the score of the base measure. The adjustment is essential: outside special cases, the posterior mean of the natural parameter is not obtained from the marginal score $\partial_y\log f(y)$ alone.
The Gaussian location model illustrates the connection with Lemma \ref{lem:tweedie_gaussian}. If
$Y|\mu\sim N(\mu,\sigma^2)$, then the natural parameter is $\theta=\mu/\sigma^2$ and
$h(y)\propto\exp\left\{-y^2/(2\sigma^2)\right\}$.
Hence $-\partial_y\log h(y)=y/\sigma^2$, and Lemma \ref{lem:tweedie_nef} gives
$$
\mathbb{E}\left[\frac{\mu}{\sigma^2}|Y=y\right]
=
\partial_y\log f(y)+\frac{y}{\sigma^2}.
$$
Multiplying by $\sigma^2$ yields the Gaussian form of Tweedie's formula in Lemma \ref{lem:tweedie_gaussian}.
For the recursive filtering result below, the expectation parameter
\(\mu=\psi'(\theta)\) is also important. The following identity targets
\(\mathbb E(\mu\mid Y=y)\). Unlike Lemma \ref{lem:tweedie_nef}, it is obtained
by integration by parts with respect to the natural parameter.
\begin{lemma}[Expectation-parameter posterior identity]
\label{lem:nef_expectation_parameter_identity}
For natural exponential family
$$
p(y|\theta)=h(y)\exp\{\theta y-\psi(\theta)\},
\qquad
\mu=\psi^\prime(\theta).
$$
Let $\pi(\theta)$ be a differentiable prior density on the natural parameter space. Suppose the boundary term
$\exp\{\theta y-\psi(\theta)\}\pi(\theta)$ vanishes at the endpoints of the natural parameter space. Then
$$
\mathbb{E}(\mu|Y=y)
=
y+\mathbb{E}\{\partial_\theta\log \pi(\theta)|Y=y\}.
$$
In particular, if $\pi(\theta)\propto\exp\{\tau\theta-n\psi(\theta)\}$, where $n>0$, then
$$
\mathbb{E}(\mu|Y=y)
=
\frac{\tau+y}{n+1}.
$$
\end{lemma}
\begin{proof}
The posterior kernel is $\exp\{\theta y-\psi(\theta)\}\pi(\theta)$. The boundary condition gives
$$0=\int \frac{d}{d\theta}\left[\exp\{\theta y-\psi(\theta)\}\pi(\theta)\right]d\theta.$$
Dividing by the posterior normalizing constant yields $0=y-\mathbb{E}\{\psi^\prime(\theta)|Y=y\}
+\mathbb{E}\{\partial_\theta\log\pi(\theta)|Y=y\}$. Since $\mu=\psi^\prime(\theta)$, the first identity follows. Finally, if
$\pi(\theta)\propto\exp\{\tau\theta-n\psi(\theta)\}$, then
$\partial_\theta\log\pi(\theta)=\tau-n\mu$, so
$\mathbb{E}(\mu|Y=y)=y+\tau-n\mathbb{E}(\mu|Y=y)$, which gives
$\mathbb{E}(\mu|Y=y)=(\tau+y)/(n+1)$.
\end{proof}
The two identities target different posterior means. Lemma
\ref{lem:tweedie_nef} gives a marginal-score representation for the posterior
mean of the natural parameter \(\theta\). Lemma
\ref{lem:nef_expectation_parameter_identity} gives an identity for the posterior
mean of the expectation parameter \(\mu=\psi'(\theta)\). The latter is the
parameterization in which inverse-Fisher scaling turns the conditional score
into the forecast error \(y-\mu\), and it is the object used in the exact
recursive filtering result below.
For discrete natural exponential families, such as the Poisson distribution, the same empirical-Bayes principle applies, but the derivative with respect to the observation is replaced by a finite-difference or probability-ratio identity. This qualification concerns the marginal Tweedie identity, not the conditional likelihood score with respect to a continuous parameter.
The implication for score-driven models is therefore precise. Lemmas
\ref{lem:tweedie_nef} and \ref{lem:nef_expectation_parameter_identity} show that
natural exponential families provide exact Bayesian identities for both natural
and expectation parameters. These identities should not be confused with the
conditional parameter score used in score-driven recursions. Tweedie's identity
differentiates a predictive marginal density with respect to the observation,
whereas score-driven models differentiate the conditional density with respect
to the time-varying parameter. The exact expectation-space filtering result in
Proposition \ref{prop:exact_nef_conjugate} and the local approximation results below
make this distinction explicit.
\section{Tweedie's Index and Variance-Function Scaling}
The preceding section used Tweedie's formula to explain why marginal scores arise in Bayesian signal extraction. We now turn to a second, distinct contribution by Tweedie: the classification of exponential dispersion models through their variance functions \citep{Tweedie:1984,Jorgensen:1987}. Tweedie's formula explains why scores appear as Bayesian posterior corrections. Tweedie's index helps explain how conditional scores normalize forecast errors in exponential dispersion models.
The distinction between normalization and scaling is important. The conditional likelihood score itself often normalizes raw forecast errors by the variance structure of the observation density. Score-driven models may then apply an additional scaling matrix, denoted $S_t$, often based on Fisher information. Thus, two operations should be kept conceptually separate: variance-function normalization inside the likelihood score, and score-driven scaling applied to that score in the recursion.
For expositional simplicity, this section uses scalar notation. The scalar case is sufficient for the equivalence developed below. Multivariate extensions require the corresponding gradient and information-matrix notation, with variance functions replaced by covariance structures, and are not pursued here.
For an exponential dispersion model, let $\mu$ denote the conditional mean and let $\phi$ denote the dispersion parameter. The conditional variance is
\[
\operatorname{var}(Y|\mu)=\phi V(\mu),
\]
where $V(\mu)$ is the variance function. The following lemma records the corresponding score normalization.
\begin{lemma}[Variance-function normalization]
\label{lem:variance_function_score}
For an exponential dispersion model with conditional mean $\mu$ and variance function $V(\mu)$,
$$
\partial_\mu\ell(y;\mu)=\frac{y-\mu}{\phi V(\mu)},
\qquad
\mathcal{I}_\mu(\mu)=\frac{1}{\phi V(\mu)}.
$$
Consequently, $\mathcal{I}_\mu(\mu)^{-1}\partial_\mu\ell(y;\mu)=y-\mu$.
\end{lemma}
\begin{proof}
In mean parameterization, we have $d\theta/d\mu=1/V(\mu)$. Since the natural-parameter score is $(y-\mu)/\phi$, the chain rule gives
$\partial_\mu\ell(y;\mu)=\frac{y-\mu}{\phi V(\mu)}$.
Taking the conditional variance of this score gives
$\mathcal{I}_\mu(\mu)
=
\frac{\operatorname{var}(Y|\mu)}{\phi^2V(\mu)^2}
=
\frac{1}{\phi V(\mu)}$.
The final identity follows immediately.
\end{proof}
Thus, before any additional score-driven scaling is applied, the likelihood score already transforms the raw forecast error $y-\mu$ by the variance function of the observation density. Inverse-Fisher scaling reverses this normalization and recovers the raw innovation in expectation space.
The Tweedie class is the subclass of exponential dispersion models with variance function
$$
V(\mu)=\mu^p.
$$
The scalar $p$ is the Tweedie index. Lemma \ref{lem:variance_function_score} gives
$\partial_\mu\ell(y;\mu)=\frac{y-\mu}{\phi\mu^p}$ and $\mathcal{I}_\mu(\mu)=\frac{1}{\phi\mu^p}$.
If a score-driven recursion scales the mean score by $\mathcal{I}_\mu(\mu)^{-d}$, then
$$
\mathcal{I}_\mu(\mu)^{-d}\partial_\mu\ell(y;\mu)
=
\frac{y-\mu}{\phi^{1-d}\mu^{p(1-d)}}.
$$
Thus, the Tweedie index determines how the unscaled conditional mean score normalizes forecast errors as the conditional mean changes, while the scaling exponent $d$ determines how much of this variance-function normalization is preserved in the score-driven update. With $d=0$, the update uses the fully variance-normalized score. With $d=1/2$, it uses a standardized forecast error. With $d=1$, it uses the raw innovation $y-\mu$ in expectation space. The last case is the one that aligns with the exact conjugate-prior Bayesian update in Proposition \ref{prop:exact_nef_conjugate}.
The parameterization also matters. For a transformed parameter, say $\eta=\log\mu$, we have
$$
\partial_\eta\ell(y;\mu)
=
\partial_\mu\ell(y;\mu)\frac{\partial\mu}{\partial\eta}
=
\mu\frac{y-\mu}{\phi V(\mu)}.
$$
For the Tweedie class, this becomes
$\partial_\eta\ell(y;\mu)
=\frac{y-\mu}{\phi\mu^{p-1}}$. Thus, the variance-function logic remains, but the exact normalization depends on whether the update is written for the mean, the log mean, or another transformation.
The special role of inverse-Fisher scaling in expectation space should not be read as a universal prescription. When the information content of observations varies substantially over time, full inverse-Fisher scaling may amplify low-precision observations. For example, in a binomial model with logit parameter $\eta_t$, the Fisher information is $\mathcal{I}_t=N_tp_t(1-p_t)$, so the conditional variance of $\mathcal{I}_t^{-d}s_t$ is proportional to $\mathcal{I}_t^{1-2d}$. Hence $d=0$ preserves the information weighting supplied by the likelihood, $d=1/2$ standardizes the score, and $d=1$ amplifies observations with low information. In applications with declining risk sets or other sources of time-varying information, large scaling exponents can therefore make the recursion overly sensitive to noisy observations.
\section{Exact Expectation-Space Filtering under Conjugacy}\label{sec:ExactResults}
The preceding sections establish two complementary facts. First, Tweedie's
formula gives exact Bayesian identities for posterior means in static
signal-extraction problems. Second, in expectation parameterization, inverse
Fisher scaling converts the conditional score of an exponential dispersion model
into the raw forecast error. We now combine them in a recursive filtering
setting.
Consider the conditional observation density at time $t$,
$$
p_t(y_t|\theta_t)
=
h(y_t)\exp\{\theta_t y_t-\psi(\theta_t)\},
\qquad
\mu_t=\psi^\prime(\theta_t).
$$
Although conjugacy is naturally expressed in the natural parameter
$\theta_t$, the score representation of the exact Bayesian filtering update is
most transparent after transforming to the expectation parameter $\mu_t$.
Conjugate updating and discounting for dynamic generalized linear models are
classical; see \citet{WestHarrisonMigon:1985} and
\citet{WestHarrison:1989}. Proposition
\ref{prop:exact_nef_conjugate} is not to rederive this Bayesian recursion, but
to record its score-driven representation. The expectation-parameter identity
used below is the conjugate-prior special case of Lemma
\ref{lem:nef_expectation_parameter_identity}. The boundary condition rules out
integration-by-parts boundary terms and is satisfied in the usual conjugate
families used below.
\begin{proposition}[Exact expectation-space NEF score representation]
\label{prop:exact_nef_conjugate}
Let
$$
p_t(y_t|\theta_t)
=
h(y_t)\exp\{\theta_t y_t-\psi(\theta_t)\},
\qquad
\mu_t=\psi^\prime(\theta_t).
$$
Suppose the predictive prior density of $\theta_t|\mathcal F_{t-1}$ is
conjugate,
$$
\pi(\theta_t|\mathcal F_{t-1})
\propto
\exp\{\tau_{t|t-1}\theta_t-n_{t|t-1}\psi(\theta_t)\},
\qquad
n_{t|t-1}>0.
$$
Suppose the boundary condition in Lemma \ref{lem:nef_expectation_parameter_identity} holds. Then the filtered posterior is conjugate, with
$$
\tau_{t|t}=\tau_{t|t-1}+y_t,\qquad n_{t|t}=n_{t|t-1}+1.
$$
If the local precision-discounting condition $n_{t|t-1}=\delta n_{t|t}$ with $0<\delta<1$ holds, then $n_{t|t-1}=\delta/(1-\delta)$, and the exact Bayesian posterior mean in expectation space satisfies
$$
\mu_{t|t}
=
\mu_{t|t-1}
+
(1-\delta)
\mathcal I_\mu(\mu_{t|t-1})^{-1}
\left.
\partial_\mu\log p_t(y_t|\mu)
\right|_{\mu=\mu_{t|t-1}}.
$$
\end{proposition}
\begin{proof}
The same integration-by-parts argument as in Lemma
\ref{lem:nef_expectation_parameter_identity}, applied to the conjugate
predictive prior, gives $\mu_{t|t-1}
=\mathbb E(\mu_t|\mathcal F_{t-1})={\tau_{t|t-1}}/{n_{t|t-1}}$.
After observing $y_t$, the likelihood contributes $\exp\{\theta_t y_t-\psi(\theta_t)\}$. Therefore the filtered density is again conjugate, with
$\tau_{t|t}=\tau_{t|t-1}+y_t$ and $n_{t|t}=n_{t|t-1}+1$. Applying the conjugate-prior special case of Lemma
\ref{lem:nef_expectation_parameter_identity} to the filtered density gives
$\mu_{t|t}={(\tau_{t|t-1}+y_t)}/{(n_{t|t-1}+1)}$.
Since $\tau_{t|t-1}=n_{t|t-1}\mu_{t|t-1}$,
$
\mu_{t|t}
=
\frac{n_{t|t-1}\mu_{t|t-1}+y_t}{n_{t|t-1}+1}
=
\mu_{t|t-1}
+
\frac{y_t-\mu_{t|t-1}}{n_{t|t-1}+1}$.
The local precision-discounting condition gives
$n_{t|t-1}=\delta n_{t|t}=\delta(n_{t|t-1}+1)$, and hence
$n_{t|t-1}=\delta/(1-\delta)$ and $(n_{t|t-1}+1)^{-1}=1-\delta$. Thus
$\mu_{t|t}=\mu_{t|t-1}+(1-\delta)(y_t-\mu_{t|t-1})$. Finally, by Lemma
\ref{lem:variance_function_score},
$\mathcal I_\mu(\mu)^{-1}\partial_\mu\log p_t(y_t|\mu)=y_t-\mu$. Evaluating at
$\mu=\mu_{t|t-1}$ gives the stated formula.
\end{proof}
Proposition \ref{prop:exact_nef_conjugate} is the recursive counterpart of the
expectation-parameter identity in Lemma
\ref{lem:nef_expectation_parameter_identity}. The conjugate prior is naturally
defined for the natural parameter \(\theta_t\), but the reported posterior mean
is that of the expectation parameter \(\mu_t=\psi'(\theta_t)\). Thus the result
should be read as a conjugate Bayesian update expressed in expectation space,
not as an arbitrary prior placed directly on \(\mu_t\).
The base-measure correction in Lemma \ref{lem:tweedie_nef} is not a missing term
in Proposition \ref{prop:exact_nef_conjugate}. It appears because Tweedie's
identity differentiates the predictive marginal density with respect to the
observation. By contrast, Proposition \ref{prop:exact_nef_conjugate} uses the
conditional likelihood score with respect to the expectation parameter, for
which the base measure drops out. The exact equivalence is obtained not by
adding the base-measure correction to a score-driven recursion, but by combining
conjugate updating with inverse-Fisher scaling in expectation space.
The proposition identifies a narrow setting in which a score-driven correction
is not merely a local Bayesian approximation but an exact Bayesian update. The
observation density must belong to a natural exponential family, the prior must
be conjugate, the prior strength must satisfy the local precision-discounting
condition, the correction must be written in expectation space, and inverse
Fisher information must be used as the score scaling. The equality $n_{t|t-1}
=\frac{\delta}{1-\delta}$ is the prior-strength value implied by
$n_{t|t-1}=
\delta n_{t|t}$, $n_{t|t}=n_{t|t-1}+1$,
and is not derived here from a separate stochastic transition equation. This
corresponds to the role of the discount factor in dynamic generalized linear
models, where \(\delta\) controls the rate at which past observations are
down-weighted; see \citet[Ch.~12]{WestHarrison:1989}. If the recursion is
written in the natural parameter, in a transformed parameter such as a log mean,
or with a different scaling rule, the exact algebraic equivalence generally
fails, although the resulting update may remain a useful local or
transformed-parameter approximation.
The timing distinction also remains important. The filtering correction in
Proposition \ref{prop:exact_nef_conjugate} concerns the posterior mean before
transition dynamics are imposed. Once an explicit score-driven recursion
combines the correction with intercepts, autoregressive terms, or other
propagation rules, it should be interpreted as a deterministic rule for forming
the next predictive parameter, not as the posterior mean of the current latent
parameter itself.
Proposition \ref{prop:exact_nef_conjugate} gives an exact benchmark: in natural
exponential families with conjugate priors, the posterior mean update in
expectation space is exactly an inverse-Fisher-scaled conditional score update
under local precision discounting. We next turn to local approximation results
for general conditional densities, where exact conjugate updating is generally
unavailable.
\section{Local Score-Driven Approximations to Bayesian Filtering}
The exact filtering result in Section~\ref{sec:ExactResults} relies on conjugate updating in
expectation space. Outside such conjugate cases, the posterior mean is generally
not available in closed form. We therefore turn to local approximations. The
first step is to record an exact parameter-space identity: under a regular
predictive law for $\theta_t$, the Bayesian posterior correction can be written
as a predictive-marginal score. The second step is to approximate this marginal
score by the conditional likelihood score evaluated at the predictive mean.
Consider a general dynamic model in which
$$
y_t|\theta_t\sim p_t(y_t|\theta_t),
$$
where $\theta_t$ is a time-varying parameter and
$\mathcal F_{t-1}$ denotes the information available before observing
$y_t$. In a parameter-driven formulation, $\theta_t$ is stochastic
conditional on $\mathcal F_{t-1}$. We write
$$
a_t
=
\mathbb E(\theta_t|\mathcal F_{t-1}),
\qquad
P_t
=
\operatorname{var}(\theta_t|\mathcal F_{t-1})
$$
for its predictive mean and covariance. Bayes' rule gives
$\pi(\theta_t|y_t,\mathcal F_{t-1})
\propto
p_t(y_t|\theta_t)\pi(\theta_t|\mathcal F_{t-1})$, and the filtered estimate is
$\mathbb E(\theta_t|y_t,\mathcal F_{t-1})$.
Proposition \ref{prop:parameter_space_tweedie} is a parameter-space analogue of Tweedie's formula. It allows for a general conditional density $p_t(y_t|\theta_t)$, but it uses natural exponential family for
$\theta_t|\mathcal F_{t-1}$.
\begin{proposition}[Parameter-space Tweedie identity]
\label{prop:parameter_space_tweedie}
Let $y_t|\theta_t\sim p_t(y_t|\theta_t)$, where $\theta_t\in\mathbb R^d$.
Suppose that, conditionally on $\mathcal F_{t-1}$, the predictive density of
$\theta_t$ belongs to a regular natural exponential family with identity
natural statistic:
$$
\pi(\theta|\mathcal F_{t-1};\eta_t)
=
h_t(\theta)\exp\{\eta_t^\prime\theta-A_t(\eta_t)\},
$$
where $a_t=\mathbb E(\theta_t|\mathcal F_{t-1};\eta_t)=\nabla_{\eta}A_t(\eta_t)$, and $P_t=\operatorname{var}(\theta_t|\mathcal F_{t-1};\eta_t)=\nabla_{\eta}^2A_t(\eta_t)$ is positive definite. Define the predictive marginal density by
$$
f_t(y_t;\eta_t)
=
\int p_t(y_t|\theta)\pi(\theta|\mathcal F_{t-1};\eta_t)d\theta .
$$
Suppose that $f_t(y_t;\eta_t)>0$ and that differentiation with respect to
$\eta_t$ may be passed under the integral sign. Then
$$
\nabla_{\eta_t}\log f_t(y_t;\eta_t)
=
\mathbb E(\theta_t|y_t,\mathcal F_{t-1})-a_t.
$$
Equivalently, under the mean parameterization,
$$
\mathbb E(\theta_t|y_t,\mathcal F_{t-1})-a_t
=
P_t\nabla_{a_t}\log f_t(y_t;a_t).
$$
\end{proposition}
\begin{proof}
Differentiating the predictive marginal density with respect to $\eta_t$
gives $
\nabla_{\eta_t}f_t(y;\eta_t)
=
\int p_t(y|\theta)
\{\theta-\nabla_\eta A_t(\eta_t)\}
\pi(\theta|\mathcal F_{t-1};\eta_t)d\theta$. Since $a_t=\nabla_\eta A_t(\eta_t)$, division by $f_t(y;\eta_t)$ yields $\nabla_{\eta_t}\log f_t(y;\eta_t)=\mathbb E(\theta_t|y,\mathcal F_{t-1})-a_t$.
Evaluating at $y=y_t$ proves the first identity. For the mean
parameterization, the chain rule gives $
\nabla_{\eta_t}\log f_t(y_t;\eta_t)
=
\frac{\partial a_t}{\partial \eta_t^\prime}
\nabla_{a_t}\log f_t(y_t;a_t)
=
P_t\nabla_{a_t}\log f_t(y_t;a_t)$.
Combining the two identities proves the result.
\end{proof}
The key distinction is between the exact Bayesian correction and the operational
score-driven correction. Proposition \ref{prop:parameter_space_tweedie}
expresses the exact correction through the predictive-marginal score
$\nabla_{a_t}\log f_t(y_t;a_t,P_t)$. This score generally requires differentiating
an integral over the predictive distribution of the latent state and is rarely
available in closed form. Score-driven models instead use the tractable
conditional likelihood score
$s_t(a_t)=\nabla_\theta\log p_t(y_t|\theta)|_{\theta=a_t}$.
In this sense, they replace the exact marginal-score correction by a plug-in
conditional-score correction.
This interpretation is consistent with the forecasting evidence of
\citet{KoopmanLucasScharth:2016}: score-based observation-driven models can have
predictive accuracy close to that of correctly specified parameter-driven
models, even when the latter generate the data. The results below provide a
Bayesian approximation argument for why such near-equivalence can arise.
The Gaussian predictive law used in the local approximation below is a special
case of Proposition \ref{prop:parameter_space_tweedie}. With fixed covariance
$P_t$, the natural parameter is $\eta_t=P_t^{-1}a_t$, and the mean-parameter
form gives $\mathbb{E}(\theta_t|y_t,\mathcal F_{t-1})-a_t=P_t\nabla_{a_t}\log f_t(y_t;a_t,P_t)$.
\begin{proposition}[Local score approximation to the posterior correction]
\label{prop:small_variance_score_expansion}
Suppose that the parameter-space Tweedie identity in Proposition
\ref{prop:parameter_space_tweedie} holds, and that
$$
\theta_t|\mathcal F_{t-1}\sim N(a_t,P_t),
$$
where $P_t$ is positive definite. Let
$\ell_t(\theta)=\log p_t(y_t|\theta)$ and
$s_t(a_t)=\nabla_\theta\ell_t(\theta)|_{\theta=a_t}$.
Assume that $\ell_t$ is sufficiently smooth in a neighborhood of $a_t$, with derivatives bounded so that the local expansion below is valid after integration with respect to the Gaussian predictive distribution. Then, as $\|P_t\|\to0$,
$$
P_t\nabla_{a_t}\log f_t(y_t;a_t,P_t)
=
P_t s_t(a_t)+O(\|P_t\|^2),
$$
where the gradient is taken with respect to $a_t$, holding $P_t$ fixed. Consequently,
$$
\mathbb E(\theta_t|y_t,\mathcal F_{t-1})
=
a_t+P_t s_t(a_t)+O(\|P_t\|^2).
$$
\end{proposition}
\begin{proof}
By the parameter-space Tweedie identity in Proposition \ref{prop:parameter_space_tweedie}, we have $\mathbb{E}(\theta_t|y_t,\mathcal{F}_{t-1})-a_t
=P_t\nabla_{a_t}\log f_t(y_t;a_t,P_t)$.
Set $u_t=\theta_t-a_t$. Bayes' rule gives
$$
P_t\nabla_{a_t}\log f_t(y_t;a_t,P_t)
=
\frac{
\mathbb{E}\{u_t\exp[\ell_t(a_t+u_t)]|\mathcal{F}_{t-1}\}
}{
\mathbb{E}\{\exp[\ell_t(a_t+u_t)]|\mathcal{F}_{t-1}\}
}.
$$
Write $s_t=s_t(a_t)$ and $H_t=\nabla_\theta^2\ell_t(\theta)|_{\theta=a_t}$. A local expansion gives
$$
\exp[\ell_t(a_t+u_t)]
=
\exp[\ell_t(a_t)]
\left\{
1+s_t^\prime u_t
+\frac12 u_t^\prime(H_t+s_ts_t^\prime)u_t
+O(\|u_t\|^3)
\right\}.
$$
Since $u_t|\mathcal{F}_{t-1}\sim N(0,P_t)$, the numerator is
$\exp[\ell_t(a_t)]\{P_ts_t+O(\|P_t\|^2)\}$, while the denominator is
$\exp[\ell_t(a_t)]\{1+O(\|P_t\|)\}$. Dividing gives
$$
P_t\nabla_{a_t}\log f_t(y_t;a_t,P_t)
=
P_ts_t(a_t)+O(\|P_t\|^2).
$$
The posterior-mean expansion follows from the parameter-space Tweedie identity.
\end{proof}
Proposition \ref{prop:small_variance_score_expansion} turns the preceding
distinction into a local approximation. When the predictive distribution of
\(\theta_t\) is concentrated around \(a_t\), the exact Tweedie correction is
approximated by $P_t s_t(a_t)$, with remaining terms of order \(O(\|P_t\|^2)\). Thus the conditional likelihood score enters as the leading term of the Bayesian posterior correction.
When the time-varying state is the expectation parameter, so that
\(\theta_t=\mu_t\) and \(a_t=\mu_{t|t-1}\), this leading correction connects
directly to the exact benchmark in Proposition
\ref{prop:exact_nef_conjugate}. For an exponential dispersion model, Lemma
\ref{lem:variance_function_score} gives
$\mathcal I_\mu(a_t)^{-1}s_t(a_t)
=
y_t-a_t$. Thus, if $
P_t=\kappa_P\mathcal I_\mu(a_t)^{-1}$, the leading posterior correction becomes
$P_t s_t(a_t)=\kappa_P(y_t-a_t)$. This is the local analogue of the exact conjugate update, with \(\kappa_P\) interpreted as the predictive-covariance scale.
Proposition \ref{prop:small_variance_score_expansion} shows why the conditional
score appears in the leading Bayesian correction. To recover the inverse-Fisher scaling used in score-driven models, it remains
to characterize when the predictive covariance multiplying this score is locally
proportional to \(\mathcal I_t(a_t)^{-1}\). The next result shows that this
covariance specification is equivalent to a local precision-discounting
condition.
\begin{proposition}[Inverse-Fisher correction under local precision discounting]
\label{prop:information_matched_covariance}
Suppose that the small-variance conclusion of Proposition
\ref{prop:small_variance_score_expansion} holds, so that
\[
\mathbb E(\theta_t\mid y_t,\mathcal F_{t-1})
=
a_t+P_t s_t(a_t)+O(\|P_t\|^2).
\]
Suppose in addition that the filtering step is locally approximated by the
Fisher-scoring quadratic surrogate around \(a_t\),
\[
\tilde \ell_t(\theta)
=
\ell_t(a_t)
+
s_t(a_t)'(\theta-a_t)
-
\tfrac12(\theta-a_t)'\mathcal I_t(a_t)(\theta-a_t),
\]
where \(\mathcal I_t(a_t)\) is positive definite. Under the local Gaussian
covariance approximation, let \(P_{t|t}\) denote the filtered covariance after
observing \(y_t\). Then
\[
P_{t|t}^{-1}
=
P_t^{-1}+\mathcal I_t(a_t).
\]
Suppose further that the local predictive precision is obtained by discounting
the filtered precision,
\[
P_t^{-1}
=
\delta P_{t|t}^{-1},
\qquad
0<\delta<1.
\]
Then $P_t=\kappa_P\mathcal I_t(a_t)^{-1}$, where $\kappa_P=\frac{1-\delta}{\delta}$. If \(\|\mathcal I_t(a_t)^{-1}\|\) is bounded as \(\kappa_P\to0\), then
\[
\mathbb E(\theta_t\mid y_t,\mathcal F_{t-1})
=
a_t+
\kappa_P\mathcal I_t(a_t)^{-1}s_t(a_t)
+
O(\kappa_P^2).
\]
\end{proposition}
\begin{proof}
Under the Gaussian covariance approximation and the Fisher-scoring quadratic
surrogate, the surrogate posterior is proportional to $\exp\{\tilde\ell_t(\theta_t)\}\phi_{P_t}(\theta_t-a_t)$. Thus its log density is, up to a term independent of \(\theta_t\), $\tilde\ell_t(\theta_t)-
\frac12(\theta_t-a_t)'P_t^{-1}(\theta_t-a_t)$.
Substituting the surrogate likelihood gives
$
s_t(a_t)'(\theta_t-a_t)
-
\frac12(\theta_t-a_t)'
\{P_t^{-1}+\mathcal I_t(a_t)\}
(\theta_t-a_t)
+\mathrm{const}$. This is the log kernel of a Gaussian density. Hence the filtered covariance
under the local Gaussian surrogate satisfies $P_{t|t}^{-1}=P_t^{-1}+\mathcal I_t(a_t)$.
By the local precision-discounting condition, $
P_t^{-1}
=
\delta P_{t|t}^{-1}
=
\delta\{P_t^{-1}+\mathcal I_t(a_t)\}$.
Therefore,
$
P_t
=
\frac{1-\delta}{\delta}\mathcal I_t(a_t)^{-1}
=
\kappa_P\mathcal I_t(a_t)^{-1}$. Since \(\|\mathcal I_t(a_t)^{-1}\|\) is bounded,
we have
$\|P_t\|=O(\kappa_P)$, and
$O(\|P_t\|^2)=O(\kappa_P^2)$.
Combining this with Proposition
\ref{prop:small_variance_score_expansion} gives the stated formula.
\end{proof}
Together, Propositions \ref{prop:small_variance_score_expansion} and
\ref{prop:information_matched_covariance} show how inverse-Fisher score scaling
arises locally: Proposition \ref{prop:small_variance_score_expansion} gives the
leading Bayesian correction \(P_t s_t(a_t)\), while Proposition
\ref{prop:information_matched_covariance} gives a precision-discounting
condition under which \(P_t=\kappa_P\mathcal I_t(a_t)^{-1}\).
\begin{remark}[Predictive and posterior learning scales]
The predictive covariance scale in Proposition
\ref{prop:information_matched_covariance}, $
\kappa_P=\frac{1-\delta}{\delta}$, should be distinguished from the posterior learning rate \(1-\delta\) in
Proposition \ref{prop:exact_nef_conjugate}. In Proposition
\ref{prop:information_matched_covariance}, local precision discounting implies
$P_t=\kappa_P\mathcal I_t(a_t)^{-1}$, so \(\kappa_P\) scales the predictive covariance. Under the same local Gaussian
surrogate, the filtered covariance is
$ P_{t|t}=
(P_t^{-1}+\mathcal I_t(a_t))^{-1}
=
(1-\delta)\mathcal I_t(a_t)^{-1}$. Thus \(1-\delta\) is the posterior learning scale. Proposition
\ref{prop:exact_nef_conjugate} is an exact conjugate posterior-mean update and
therefore involves this posterior learning scale. The two coefficients are
related by
$ 1-\delta
=
\frac{\kappa_P}{1+\kappa_P}$, and are first-order equivalent as \(\delta\to1\).
\end{remark}
This local approximation is closely related to approximate nonlinear filtering,
score-driven filtering, and optimization-based interpretations. Approximate
non-Gaussian state-space filtering and dynamic generalized linear model
filtering have long relied on local, Gaussian, or simulation-based
approximations; see, for example, \citet{Shephard:1994},
\citet{Fahrmeir:1992}, and \citet{FahrmeirTutz:2001}. In related
approximate-filtering work, \citet{BuccheriBormettiCorsiLillo:2023} derive
robust recursive filtering and smoothing recursions by expanding the observation
density around the predictive state. More recently,
\citet{LangeVanOsVanDijk:2026} develop implicit score-driven filters as
proximal, penalized-likelihood updates and obtain the usual explicit recursion
by linearization around the prediction. \citet{CataniaDInnocenzoLuati:2026}
provide a related bridge between unobserved-component models with finite-state
parameter uncertainty and score-driven filters. From a stochastic-optimization
perspective, \citet{DonkerVanHeelLangeVanOsVanDijk:2026} analyze explicit and
implicit gradient filters under misspecification. From an information-theoretic
perspective, \citet{dePunderDimitriadisLange:2026} characterize score-driven
updates through expected Kullback-Leibler improvement. The expansion above
provides a complementary Bayesian interpretation: the conditional score is the
leading term in the exact predictive-marginal score correction.
The approximation is not restricted to location, conditional-mean, or variance
parameters. It applies to any differentiable time-varying parameter of the
conditional density. In such cases there is generally no observation-space
Tweedie identity representing the posterior mean as a marginal score of \(y_t\),
but a Gaussian predictive approximation still yields an update in the direction
of the conditional likelihood score, scaled by the local predictive covariance.
With the inverse-Fisher choice of covariance, this becomes the usual
information-scaled score correction.
The approximation concerns the filtered estimate of the current state after
observing \(y_t\). In a parameter-driven filter, this estimate is then propagated
through a transition equation, for example
\[
\theta_{t+1|t}
=
G_t(\theta_{t|t}).
\]
An explicit score-driven model combines the filtering correction and propagation
directly in a recursion for the next predictive parameter,
$$
\theta_{t+1}
=
\omega+\beta\theta_t+\alpha S_t s_t,
\qquad
s_t=\nabla_\theta\log p_t(y_t|\theta_t).
$$
In light of Proposition \ref{prop:small_variance_score_expansion}, the natural
Bayesian-Fisher scaling is
\[
S_t
=
\mathcal I_t(\theta_t)^{-1},
\]
while \(\alpha\) plays the role of a learning rate or predictive-variance scale.
Thus the scaled score has a Bayesian-Fisher interpretation as a tractable local
posterior correction, complementing existing approximate-filtering and
optimization-based foundations of score-driven models.
\section{Examples}
The examples illustrate three implications of the preceding results. In the linear-Gaussian model, Tweedie's formula reproduces the Kalman update. In natural-exponential-family models, Proposition \ref{prop:exact_nef_conjugate} identifies the conditions under which the inverse-Fisher-scaled score is the exact posterior-mean correction in expectation space. Once this correction is embedded in a transition equation, it yields familiar observation-driven recursions, but these recursions are no longer literal posterior means.
\subsection{Gaussian location}
Consider $Y_t|\mu_t\sim N(\mu_t,\sigma^2)$ with $\mu_t|\mathcal{F}_{t-1}\sim N(\mu_{t|t-1},P_{t|t-1})$.
The predictive marginal density is Gaussian with mean $\mu_{t|t-1}$ and variance $P_{t|t-1}+\sigma^2$. Tweedie's formula therefore gives the Kalman update
$$
\mathbb{E}[\mu_t|Y_t=y_t,\mathcal{F}_{t-1}]
=
\mu_{t|t-1}
+
\frac{P_{t|t-1}}{P_{t|t-1}+\sigma^2}
(y_t-\mu_{t|t-1}).
$$
The conditional likelihood score at the predictive mean satisfies
$$
s_t(\mu_{t|t-1})
=
\frac{y_t-\mu_{t|t-1}}{\sigma^2},
\qquad
\mathcal{I}_\mu=\frac{1}{\sigma^2},
\qquad
\mathcal{I}_\mu^{-1}s_t(\mu_{t|t-1})
=
y_t-\mu_{t|t-1}.
$$
Thus the Kalman correction is an inverse-Fisher-scaled conditional-score correction. If $P_{t|t-1}=\sigma^2/n_{t|t-1}$, the Kalman gain is $1/(n_{t|t-1}+1)$. Under the local precision-discounting condition
\(n_{t|t-1}=\delta n_{t|t}\), equivalently
\(n_{t|t-1}=\delta/(1-\delta)\),
$$
\mathbb{E}[\mu_t|Y_t=y_t,\mathcal{F}_{t-1}]
=
\mu_{t|t-1}
+
(1-\delta)\mathcal{I}_\mu^{-1}s_t(\mu_{t|t-1}).
$$
Hence, in the Gaussian location model, the Tweedie, Kalman, and
inverse-Fisher-scaled score updates coincide under the local
precision-discounted conjugate-prior parameterization.
\subsection{Gaussian variance and GARCH}
Consider the zero-mean Gaussian variance model
$Y_t|h_t\sim N(0,h_t)$.
The conditional log likelihood satisfies
$$
\partial_h\ell_t(h_t)=\frac{y_t^2-h_t}{2h_t^2},
\qquad
\mathcal{I}_h(h_t)=\frac{1}{2h_t^2}.
$$
Hence the inverse-Fisher-scaled score is
$\mathcal{I}_h(h_t)^{-1}\partial_h\ell_t(h_t)=y_t^2-h_t$.
The relevant sufficient statistic for variance learning is $X_t=Y_t^2$. Since $X_t|h_t$ is Gamma with shape $1/2$, mean $h_t$, and variance \(2h_t^2\), it is a natural-exponential-family observation with expectation parameter $\mathbb{E}[X_t|h_t]=h_t$. Thus, in applying Proposition \ref{prop:exact_nef_conjugate}, the observation is $X_t$, not $Y_t$ itself. Proposition \ref{prop:exact_nef_conjugate} therefore gives the martingale conjugate-prior update
$$
h_{t|t}
=
h_{t|t-1}
+
(1-\delta)(y_t^2-h_{t|t-1}).
$$
A standard observation-driven variance recursion embeds the same innovation in a transition equation,
$$
h_{t+1}
=
\omega+\beta h_t+\alpha(y_t^2-h_t)
=
\omega+\alpha y_t^2+(\beta-\alpha)h_t.
$$
This is the GARCH(1,1) recursion of \citet{Bollerslev:1986} after reparameterization, with $\alpha_G=\alpha$ and $\beta_G=\beta-\alpha$. Hence the usual nonnegative GARCH coefficients require $\alpha\geq0$ and $\beta\geq\alpha$, in addition to $\omega>0$. The martingale update is the exact conjugate-prior Bayesian correction; the GARCH recursion is the observation-driven extension that combines this correction with autoregressive propagation.
The parameterization also matters. If $\eta_t=\log h_t$, then
$\partial_\eta\ell_t=\frac{1}{2}\left(\frac{y_t^2}{h_t}-1\right)$ and $\mathcal{I}_\eta=\frac{1}{2}$,
so inverse-Fisher scaling gives the relative variance innovation
$$
\mathcal{I}_\eta^{-1}\partial_\eta\ell_t
=
\frac{y_t^2}{h_t}-1.
$$
Thus variance-space updating has the exact expectation-space Bayesian interpretation, whereas log-variance updating uses a scale-free innovation. The latter is often attractive in volatility applications because $y_t^2-h_t$ has conditional variance $2h_t^2$, while $y_t^2/h_t-1$ has conditional variance $2$ under the Gaussian model.
\subsection{Poisson and Gamma observation models}
The same expectation-space logic applies to other natural exponential families. For a Poisson model,
$Y_t|\mu_t\sim\operatorname{Poisson}(\mu_t)$,
the variance function is $V(\mu_t)=\mu_t$. Lemma \ref{lem:variance_function_score} gives
$$
\partial_\mu\ell_t(\mu_t)=\frac{y_t-\mu_t}{\mu_t},
\qquad
\mathcal{I}_\mu(\mu_t)=\frac{1}{\mu_t},
\qquad
\mathcal{I}_\mu(\mu_t)^{-1}\partial_\mu\ell_t(\mu_t)=y_t-\mu_t.
$$
Hence, under the conjugate-prior and local precision-discounting conditions of
Proposition \ref{prop:exact_nef_conjugate},
$$
\mu_{t|t}
=
\mu_{t|t-1}
+
(1-\delta)(y_t-\mu_{t|t-1}).
$$
For discrete observations, the marginal Tweedie identity itself is expressed through probability ratios rather than ordinary derivatives. This affects the marginal observation-score representation, not the conditional parameter score above.
For a Gamma observation model with fixed dispersion $\phi$ and time-varying mean $\mu_t$, the variance function is $V(\mu_t)=\mu_t^2$. Lemma \ref{lem:variance_function_score} gives
$$
\partial_\mu\ell_t(\mu_t)=\frac{y_t-\mu_t}{\phi\mu_t^2},
\qquad
\mathcal{I}_\mu(\mu_t)=\frac{1}{\phi\mu_t^2},
\qquad
\mathcal{I}_\mu(\mu_t)^{-1}\partial_\mu\ell_t(\mu_t)=y_t-\mu_t.
$$
For fixed \(\phi\), this is a one-parameter natural exponential family after absorbing
\(\phi\) into the canonical parameter and log-partition function; the conjugate
prior is understood in that fixed-\(\phi\) parameterization. Thus the exact conjugate-prior posterior-mean update in expectation space has the same form,
$\mu_{t|t}=\mu_{t|t-1}
+(1-\delta)(y_t-\mu_{t|t-1})$.
These examples clarify the role of Proposition \ref{prop:exact_nef_conjugate}. In expectation space, inverse-Fisher scaling converts the conditional score into the raw innovation $y_t-\mu_t$ for natural exponential-family models. With conjugate priors and local precision discounting, this innovation is
exactly the Bayesian posterior-mean correction. Transformed-parameter updates and recursions with intercepts or autoregressive terms preserve the same score-correction logic, but they are no longer exact Bayesian posterior updates.
The examples also illustrate that exact Bayesian equivalence is a benchmark, not a universal design rule. Variance-space GARCH has the clean expectation-space interpretation, whereas log-variance updating uses a scale-free innovation that may be empirically more stable. Similarly, inverse-Fisher scaling is exact in the conjugate expectation-space benchmark, but alternative scalings may be preferable when information varies substantially over time. Thus the Tweedie perspective identifies the Bayesian reference point while leaving room for parameterization and scaling choices in empirical score-driven models.
\section{Conclusion}
This paper connects Tweedie's formula with score-driven updating. The main point is that scores enter Bayesian signal extraction in a precise way: in canonical empirical-Bayes problems, posterior corrections are marginal-score objects. In Gaussian signal extraction, Tweedie's formula gives the posterior mean of the latent signal as a scaled marginal score. In natural exponential families, Lemma \ref{lem:tweedie_nef} gives the corresponding identity for the natural parameter, including the base-measure adjustment, while Lemma \ref{lem:nef_expectation_parameter_identity} gives a companion identity for the expectation parameter under differentiable priors and a simple closed form under conjugacy. Separately, Tweedie's variance-function index explains how conditional scores normalize forecast errors through the variance structure of exponential dispersion models.
Score-driven models use a different score. They update time-varying parameters with conditional likelihood scores evaluated at the current predictive parameter, rather than with the generally unavailable predictive-marginal score appearing
in the corresponding Tweedie correction. This distinction is essential. It explains why score-driven recursions are not, in general, exact Bayesian filters, but also why they can be viewed as tractable plug-in analogues of Tweedie-style posterior correction.
The paper separates exact expectation-space filtering from local approximations. Proposition \ref{prop:exact_nef_conjugate} gives an exact conjugate-prior benchmark in which the posterior mean update for the expectation parameter is an inverse-Fisher-scaled conditional-score correction under local precision discounting. Proposition \ref{prop:parameter_space_tweedie} gives an exact parameter-space Tweedie identity for natural-exponential-family predictive laws. Propositions \ref{prop:small_variance_score_expansion} and \ref{prop:information_matched_covariance} then show how, outside exact conjugate cases, conditional scores arise as local covariance-scaled approximations and how local precision discounting yields inverse-Fisher scaling.
This Bayesian interpretation is distinct from the Kullback-Leibler optimality
results of \citet{BlasquesKoopmanLucas:2015},
\citet{GorgiLauriaLuati:2024}, and
\citet{dePunderDimitriadisLange:2026}. Those results justify the score
direction through information-theoretic improvement of the conditional model,
most recently in expected Kullback-Leibler terms. The present interpretation is
also distinct from, but complementary to, misspecification and quasi-likelihood
perspectives, where likelihood-based recursions target pseudo-true parameters;
see, for example, \citet{White:1994}. Here the focus is instead on settings in
which score corrections have a Bayesian signal-extraction interpretation:
exactly through Tweedie's formula and conjugate expectation-space updating, and
locally through Propositions \ref{prop:small_variance_score_expansion} and
\ref{prop:information_matched_covariance}.
The resulting interpretation is hierarchical. In Gaussian signal-extraction
problems and in natural exponential families with conjugate priors, score
corrections can be exact Bayesian posterior corrections in the appropriate
parameterization. For general conditional densities, the exact
predictive-marginal score is generally unavailable, but a local expansion yields
the conditional likelihood score as the leading covariance-scaled correction.
With a local precision-discounting covariance specification, this becomes the
inverse-Fisher-scaled update used in score-driven models.
This perspective also clarifies the role of scaling. In exponential dispersion
models, the conditional score already reflects the variance function.
Inverse-Fisher scaling removes this normalization and recovers the raw innovation
in expectation space. This is exactly appropriate in the conjugate benchmark, but
it is not a universal prescription. Different parameterizations and scaling
choices may be preferable in empirical applications, especially when the
information content of observations varies over time.
Overall, the paper provides a Bayesian signal-extraction interpretation of
score-driven updating without claiming that observation-driven recursions are
generally exact filters. Exact equivalence requires special structure, such as
Gaussian signal extraction or natural-exponential-family updating with conjugate
priors in expectation space. Outside such cases, score-driven recursions are best
understood as tractable local approximations to Bayesian posterior corrections.
\section*{Acknowledgement}
The authors are grateful to Siem Jan Koopman for helpful discussions and comments that improved the paper. Chen Tong acknowledges financial support from the Youth Fund of the National Natural Science Foundation of China (72301227) and the Fujian Provincial Natural Science Foundation of China (2025J08008).
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