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Curvature-Calibrated Quasi-Bayesian Updating for Moment-Restricted Models
\title{Curvature-Calibrated Quasi-Bayesian Updating for Moment-Restricted
Models}
\author{Masahiro Tanaka\thanks{Faculty of Economics, Fukuoka University, Fukuoka, Japan. Address:
8-19-1, Nanakuma, Jonan, Fukuoka, Japan 814-0180. E-mail: [email removed].}}
\maketitle
\begin{abstract}
Moment restrictions provide a flexible basis for quasi-Bayesian inference
when a full likelihood is unavailable, but the weighting matrix in
a quadratic moment criterion determines both the relative importance
of the moments and the information scale of posterior updating. We
propose curvature-calibrated quasi-Bayesian updating, which uses the
inverse of the covariance (or long-run covariance) of the moment conditions
evaluated at a self-consistent quasi-posterior center. The resulting
fixed-point procedure alternates between covariance estimation and
simulation from a fixed-weight quasi-posterior, thereby avoiding parameter-dependent
weighting during each simulation run. Under a Bernstein--von Mises
condition for the fixed-weight quasi-posterior at the efficient population
weight, we show that the calibration map is locally contractive, that
its fixed point is consistent at the standard parametric rate, and
that the Gaussian approximation continues to hold under the calibrated
data-dependent weight, with covariance given by the inverse Godambe
information matrix. Under a uniform fourth-moment condition, the scaled
quasi-posterior covariance converges to the same matrix, so quasi-posterior
and repeated-sampling uncertainty agree to first order. Simulations
show improved covariance calibration and interval coverage after a
few updates. An application to longitudinal binary-response data illustrates
the method with within-subject dependence and overidentified residual
moments.\\
\\
Keywords: Quasi-Bayesian inference; generalized method of moments;
Godambe information; moment restrictions; quasi-posterior calibration.
\end{abstract}
\section{Introduction}
\label{sec:intro}
Moment restrictions provide a useful basis for Bayesian-style inference
when a complete likelihood is unavailable, difficult to evaluate,
or deliberately left unspecified. A growing literature constructs
quasi-posteriors by combining a prior distribution with an empirical
criterion derived from moment conditions \citep{Kim2002,Chernozhukov2003,Yin2009,Liao2011,Li2016}.
Such methods retain familiar Bayesian computational and inferential
tools such as Markov chain Monte Carlo (MCMC), while allowing empirical
information to be specified through a collection of moments rather
than a full probability model. This flexibility is particularly attractive
in settings with complex dependence structures, latent variables,
or other features for which specifying and computing a full likelihood
can be burdensome.
An important issue arises, however, when a quadratic moment criterion
is used as a surrogate for a log likelihood. In an ordinary likelihood-based
posterior, the likelihood determines the local information scale on
which the evidence in the data is combined with the prior. A moment
criterion has no analogous intrinsic scale. For a quasi-posterior
based on a quadratic form in the sample moments, the weighting matrix
determines both the overall magnitude and the directional curvature
of the empirical criterion. Its relative structure controls the weighting
of different moment conditions, while its overall magnitude acts as
a learning rate. Consequently, two quasi-posteriors constructed from
the same moments and the same prior can represent quite different
balances between the prior and the data simply because different weighting
matrices are used.
This distinction has no direct counterpart in classical point estimation
by the generalized method of moments (GMM) \citep{Hansen1982}. In
particular, multiplying a GMM weighting matrix by a positive scalar
leaves the minimizer of the criterion unchanged. For quasi-Bayesian
updating, the same rescaling changes the dispersion of the quasi-posterior
and, in finite samples, can also change its center through its interaction
with the prior. More generally, an arbitrary weighting matrix can
cause the covariance represented by the quasi-posterior to differ
from the repeated-sampling covariance of its center. Related work
by \citet{Kleijn2012} shows that, under model misspecification, a
posterior may satisfy a Bernstein--von Mises approximation while
its local covariance fails to reproduce the repeated-sampling variability
of its center. Thus, posterior curvature and frequentist uncertainty
need not coincide once the likelihood-based information identity is
lost. This observation provides additional motivation for choosing
the weighting matrix not merely for point-estimation efficiency but
also to place the empirical component of the quasi-posterior on an
appropriate local information scale.
This paper proposes curvature-calibrated quasi-Bayesian (CCQB) updating
for this purpose. The basic idea is to use the inverse of the local
covariance, or long-run covariance, of the moment conditions as the
weighting matrix and to evaluate that covariance at a center determined
self-consistently by the resulting quasi-posterior. Starting from
a pilot quasi-posterior, CCQB alternates between estimating the covariance
of the moments at the current quasi-posterior center and simulating
a fixed-weight quasi-posterior using its inverse. The iteration terminates
when the center stabilizes, after which a final quasi-posterior is
simulated using the calibrated weight. Thus, CCQB calibrates the full
local curvature of the empirical criterion rather than only its scalar
magnitude. At the same time, the weighting matrix remains fixed within
each MCMC run, avoiding the potentially substantial cost of evaluating
a parameter-dependent covariance matrix and its determinant at every
MCMC iteration.
The calibration has a direct information-geometric interpretation.
Let $\boldsymbol{G}_{0}$ denote the population derivative of the
moments and $\boldsymbol{C}_{0}$ their long-run covariance at the
true parameter. For a generic fixed weighting matrix, the first-order
sampling covariance of the quasi-posterior center need not agree with
the covariance represented by the quasi-posterior itself. With the
covariance weight $\boldsymbol{C}_{0}^{-1}$, the asymptotic covariance
matrices of the corresponding root-$N$ scaled quantities both reduce
to $\left(\boldsymbol{G}_{0}^{\top}\boldsymbol{C}_{0}^{-1}\boldsymbol{G}_{0}\right)^{-1}$,
the inverse of the Godambe information matrix associated with the
specified moments. CCQB estimates this locally appropriate weight
while resolving the dependence of the covariance estimate on the unknown
parameter through a fixed-point condition. In this sense, the procedure
calibrates the empirical curvature before the final quasi-posterior
is formed, so that the prior is combined with a data component having
an interpretable local information scale.
We establish several large-sample properties of this procedure. Under
local regularity conditions, the ideal CCQB calibration map is, with
probability approaching one, a contraction in a neighborhood of the
true parameter, with a contraction modulus of order $N^{-1/2}$. Its
fixed point is root-$N$ consistent, and the corresponding weighting
matrix converges to $\boldsymbol{C}_{0}^{-1}$. The calibrated quasi-posterior
satisfies a Bernstein--von Mises approximation with asymptotic covariance
equal to the inverse Godambe information matrix, and the fixed-point
center has the same first-order sampling covariance. The contraction
result also shows that only a small number of calibration updates
is needed asymptotically: from any initial value in the local contraction
region, two ideal updates are first-order equivalent to the fixed
point, while one update suffices when the initial center is consistent.
We additionally give conditions under which a pilot MCMC quasi-posterior
provides such an initial center and characterize the Monte Carlo accuracy
required for the computed calibration iterates.
The finite-sample evidence follows the same distinction between covariance
weighting and self-consistent iteration. In linear instrumental-variables
and heteroskedastic and serially dependent regression designs, replacing
an identity weight by a covariance weight accounts for most of the
improvement in covariance calibration. In a nonlinear Poisson design,
iteration to the self-consistent fixed point provides additional gains
in covariance calibration and interval coverage. Across the designs,
calibration converges in only a few updates. The simulations also
show that calibration need not eliminate finite-sample displacement
of the quasi-posterior center from the corresponding unpenalized estimator;
rather, it puts the empirical criterion on a covariance-determined
scale against which the prior operates. We also illustrate CCQB with
longitudinal childhood wheeze data, where child-level moment covariances
accommodate within-child dependence without requiring a full joint
likelihood and the analysis incorporates informative priors.
The proposed framework is related to several strands of work. A complementary
strand of the literature develops Bayesian procedures for moment-restricted
models using likelihood-like constructions based on empirical likelihood
or exponential tilting, including Bayesian empirical likelihood \citep{Lazar2003},
Bayesian exponentially tilted empirical likelihood \citep{Schennach2005},
and the Bayesian semiparametric approach of \citet{Chib2018}. These
methods and CCQB share the goal of conducting Bayesian inference without
a fully specified parametric likelihood. CCQB differs in that it retains
a quadratic moment quasi-posterior and calibrates its weighting matrix,
and hence its local information scale.
CCQB is also related to calibration methods for generalized and Gibbs
posteriors. Generalized Bayesian procedures replace the negative log
likelihood by a loss function \citep{Zhang2006a,Zhang2006b,Jiang2008,Bissiri2016,Miller2021},
while several methods calibrate a scalar learning rate to control
the scale of the loss relative to the prior \citep{Gruenwald2017,Holmes2017,Lyddon2019,Syring2019}.
A scalar adjustment, however, cannot generally correct multidimensional
discrepancies in posterior curvature \citep{Miller2021,Tanaka2026a}.
More recently, \citet{Frazier2026} construct a covariance-weighted
quadratic criterion from the gradient of a generic loss, yielding
asymptotically calibrated uncertainty without scalar tuning. Like
their approach, CCQB uses covariance weighting to align posterior
and repeated-sampling uncertainty. However, their estimating-equation
representation corresponds to a just-identified system, whereas CCQB
directly accommodates both just-identified and overidentified moment
conditions. CCQB reduces the per-evaluation computational burden by
fixing the covariance weight within each simulation run, at the cost
of a small number of calibration runs.
CCQB further differs from probability-calibration methods and post-simulation
sandwich corrections. Probability calibration targets agreement between
posterior credible levels and repeated-sampling coverage probabilities
\citep{Dawid1982}, whereas CCQB targets agreement between local quasi-posterior
curvature and the Godambe information implied by the moment restrictions.
Likewise, sandwich adjustments such as \citet{Mueller2013} and \citet{Shaby2014}
modify uncertainty after a posterior or quasi-posterior has been constructed,
whereas CCQB calibrates the empirical criterion before the final quasi-posterior
is simulated. Although CCQB also resembles iterated GMM \citep{Hansen1996}
in updating a covariance-based weight, the scale of that weight is
irrelevant for classical GMM point estimation but determines the balance
between the empirical criterion and the prior in quasi-Bayesian inference.
The remainder of the paper is organized as follows. Section~\ref{sec:method}
formulates the calibration problem, introduces the CCQB fixed-point
algorithm, and establishes its large-sample properties. Section~\ref{sec:simulation}
reports the simulation study. Section~\ref{sec:application} presents
the empirical application. The final section concludes.
\section{Method}
\label{sec:method}
\subsection{Moment quasi-posteriors and the calibration problem}
\label{subsec:setup}
Let $\mathcal{D}_{N}=(D_{1},\dots,D_{N})$ denote the sample, let
$\boldsymbol{\theta}\in\Theta\subset\mathbb{R}^{J}$ be the parameter
of interest, and let $\boldsymbol{m}(D_{i},\boldsymbol{\theta})\in\mathbb{R}^{K}$,
with $K\geq J$, be a vector of moment functions. For notational convenience,
we suppress the dependence on $D_{i}$ and write $\boldsymbol{m}_{i}(\boldsymbol{\theta})=\boldsymbol{m}(D_{i},\boldsymbol{\theta})$.
Write the sample mean of $\boldsymbol{m}_{i}(\boldsymbol{\theta})$
as
\[
\bar{\boldsymbol{m}}_{N}(\boldsymbol{\theta})=\frac{1}{N}\sum_{i=1}^{N}\boldsymbol{m}_{i}(\boldsymbol{\theta}),
\]
and suppose that
\[
\mathbb{E}\{\boldsymbol{m}_{i}(\boldsymbol{\theta}_{0})\}=\boldsymbol{0}_{K},
\]
where $\boldsymbol{\theta}_{0}$ is the unique zero of the map $\boldsymbol{\theta}\mapsto\mathbb{E}\{\boldsymbol{m}_{i}(\boldsymbol{\theta})\}$
in a neighborhood of $\boldsymbol{\theta}_{0}$, and $\boldsymbol{0}_{K}$
denotes the $K$-dimensional zero vector.
Let $\mathbb{S}^{K}$ denote the vector space of symmetric $K\times K$
matrices, and let $\mathbb{S}_{++}^{K}$ denote its positive-definite
cone. All topological notions for subsets of $\mathbb{S}^{K}$ are
understood relative to $\mathbb{S}^{K}$. Throughout, $\lVert\cdot\rVert$
denotes the Euclidean norm for vectors and the induced spectral norm
for matrices. For a weakly stationary sequence, let
\[
\boldsymbol{\Gamma}_{h}(\boldsymbol{\theta})=\operatorname{Cov}\{\boldsymbol{m}_{i}(\boldsymbol{\theta}),\boldsymbol{m}_{i-h}(\boldsymbol{\theta})\},
\]
denote the lag-$h$ autocovariance matrix, and suppose that these
matrices are absolutely summable. Define the population long-run covariance
matrix by
\[
\boldsymbol{C}(\boldsymbol{\theta})=\sum_{h\in\mathbb{Z}}\boldsymbol{\Gamma}_{h}(\boldsymbol{\theta}).
\]
We assume below that $\boldsymbol{C}(\boldsymbol{\theta})$ is positive
definite in a neighborhood of $\boldsymbol{\theta}_{0}$. Under independence,
the long-run covariance reduces to the population covariance matrix:
\[
\boldsymbol{C}(\boldsymbol{\theta})=\operatorname{Var}\{\boldsymbol{m}_{i}(\boldsymbol{\theta})\}.
\]
Let $\boldsymbol{C}_{0}=\boldsymbol{C}(\boldsymbol{\theta}_{0})$
and $\widehat{\boldsymbol{C}}_{N}(\boldsymbol{\theta})$ denote an
estimator of $\boldsymbol{C}(\boldsymbol{\theta})$. Define the weighting
matrix by
\[
\boldsymbol{W}_{N}(\boldsymbol{\theta})=\widehat{\boldsymbol{C}}_{N}(\boldsymbol{\theta})^{-1}.
\]
Motivated by the central limit theorem
\[
\sqrt{N}\bar{\boldsymbol{m}}_{N}(\boldsymbol{\theta}_{0})\rightsquigarrow\mathcal{N}(\boldsymbol{0}_{K},\boldsymbol{C}_{0}),
\]
we use a Gaussian working likelihood for the sample moments, which
yields the quasi-posterior:
\begin{equation}
\Pi^{\boldsymbol{W}_{N}(\cdot)}(d\boldsymbol{\theta}\mid\mathcal{D}_{N})\propto\det\{\boldsymbol{W}_{N}(\boldsymbol{\theta})\}^{1/2}\exp\left\{ -\frac{N}{2}\bar{\boldsymbol{m}}_{N}(\boldsymbol{\theta})^{\top}\boldsymbol{W}_{N}(\boldsymbol{\theta})\bar{\boldsymbol{m}}_{N}(\boldsymbol{\theta})\right\} \pi(\boldsymbol{\theta})\,d\boldsymbol{\theta},\label{eq:quasi-posterior}
\end{equation}
where $\pi(\boldsymbol{\theta})$ denotes the prior density. Quasi-posterior
draws of $\boldsymbol{\theta}$ are simulated using standard Bayesian
computational methods. Below we use MCMC terminology for concreteness,
although alternatives such as slice sampling \citep{Neal2003} and
sequential Monte Carlo sampling \citep{DelMoral2006} are also feasible.
Evaluating $\boldsymbol{W}_{N}(\boldsymbol{\theta})$ and its determinant
at every MCMC iteration can, however, be computationally expensive
and numerically unstable Section\citep{Yin2011,Tanaka2025,Tanaka2026b}.
A common alternative is therefore to fix a positive definite matrix
$\boldsymbol{W}$ and simulate from
\begin{equation}
\Pi^{\boldsymbol{W}}(d\boldsymbol{\theta}\mid\mathcal{D}_{N})\propto\exp\left\{ -\frac{N}{2}\bar{\boldsymbol{m}}_{N}(\boldsymbol{\theta})^{\top}\boldsymbol{W}\bar{\boldsymbol{m}}_{N}(\boldsymbol{\theta})\right\} \pi(\boldsymbol{\theta})\,d\boldsymbol{\theta}.\label{eq:fixed-W-quasi-posterior}
\end{equation}
Unlike an ordinary likelihood, the quadratic moment criterion has
no intrinsic information scale. The matrix $\boldsymbol{W}$ controls
the local information metric: its scalar magnitude acts as a learning
rate, whereas its relative eigenstructure determines the relative
weighting of the moment conditions. Thus, different choices of $\boldsymbol{W}$
can produce different quasi-posteriors even when the moments and the
prior are unchanged.
To make this distinction precise, we discuss the asymptotic behavior
of the moment-based quasi-posterior. Let
\[
\boldsymbol{G}_{0}=\mathbb{E}\left\{ \frac{\partial\boldsymbol{m}_{i}(\boldsymbol{\theta}_{0})}{\partial\boldsymbol{\theta}^{\top}}\right\} \in\mathbb{R}^{K\times J}.
\]
Assume that $\boldsymbol{G}_{0}$ has full column rank. For a fixed
$\boldsymbol{W}\in\mathbb{S}_{++}^{K}$, define the quasi-posterior
mean
\[
\boldsymbol{\mu}_{N}(\boldsymbol{W})=\mathbb{E}_{\Pi^{\boldsymbol{W}}(\cdot\mid\mathcal{D}_{N})}(\boldsymbol{\theta}).
\]
Suppose that the corresponding fixed-weight quasi-posterior satisfies
the usual pointwise regularity conditions, including a local quadratic
expansion, a Bernstein--von Mises approximation, and posterior moment
conditions sufficient to transfer the local Gaussian approximation
to its first two moments \citep{Chernozhukov2003,Hong2021}. Then
\[
\sqrt{N}\{\boldsymbol{\mu}_{N}(\boldsymbol{W})-\boldsymbol{\theta}_{0}\}=-\boldsymbol{A}(\boldsymbol{W})\sqrt{N}\bar{\boldsymbol{m}}_{N}(\boldsymbol{\theta}_{0})+o_{p}(1),
\]
where
\[
\boldsymbol{A}(\boldsymbol{W})=(\boldsymbol{G}_{0}^{\top}\boldsymbol{W}\boldsymbol{G}_{0})^{-1}\boldsymbol{G}_{0}^{\top}\boldsymbol{W}.
\]
Consequently, if
\[
\sqrt{N}\bar{\boldsymbol{m}}_{N}(\boldsymbol{\theta}_{0})\rightsquigarrow\mathcal{N}(\boldsymbol{0}_{K},\boldsymbol{C}_{0}),
\]
then
\[
\sqrt{N}\{\boldsymbol{\mu}_{N}(\boldsymbol{W})-\boldsymbol{\theta}_{0}\}\rightsquigarrow\mathcal{N}\bigl(\boldsymbol{0}_{J},\boldsymbol{V}(\boldsymbol{W})\bigr),
\]
with
\[
\boldsymbol{V}(\boldsymbol{W})=\boldsymbol{A}(\boldsymbol{W})\boldsymbol{C}_{0}\boldsymbol{A}(\boldsymbol{W})^{\top}.
\]
By contrast, under the same pointwise conditions, the conditional
covariance of the fixed-weight quasi-posterior satisfies
\[
N\operatorname{Var}_{\Pi^{\boldsymbol{W}}(\cdot\mid\mathcal{D}_{N})}(\boldsymbol{\theta})\xrightarrow{p}(\boldsymbol{G}_{0}^{\top}\boldsymbol{W}\boldsymbol{G}_{0})^{-1}.
\]
For later use, define
\[
\begin{aligned}\boldsymbol{W}_{0} & =\boldsymbol{C}_{0}^{-1}, & \boldsymbol{A}_{0} & =\boldsymbol{A}(\boldsymbol{W}_{0}),\\
\boldsymbol{V}_{0} & =\boldsymbol{A}_{0}\boldsymbol{C}_{0}\boldsymbol{A}_{0}^{\top}=(\boldsymbol{G}_{0}^{\top}\boldsymbol{C}_{0}^{-1}\boldsymbol{G}_{0})^{-1}, & \boldsymbol{\xi}_{N} & =\sqrt{N}\bar{\boldsymbol{m}}_{N}(\boldsymbol{\theta}_{0}).
\end{aligned}
\]
Thus, the frequentist asymptotic covariance of the quasi-posterior
mean and the covariance represented by the quasi-posterior need not
agree. A canonical choice that makes the two covariance matrices coincide
is $\boldsymbol{W}=\boldsymbol{C}_{0}^{-1}$. In that case, both are
equal to $(\boldsymbol{G}_{0}^{\top}\boldsymbol{C}_{0}^{-1}\boldsymbol{G}_{0})^{-1}$.
Although every positive scalar multiple of $\boldsymbol{C}_{0}^{-1}$
yields the same efficient GMM point estimator, that scalar changes
the quasi-posterior dispersion and therefore cannot be left unspecified
here.
Before introducing CCQB, it is useful to separate calibration of the
quasi-posterior itself from a post-simulation covariance correction.
A sandwich-adjusted covariance estimator can be applied after simulation.
For a generic evaluation point $\boldsymbol{\theta}^{\dagger}\in\Theta$,
define
\[
\widehat{\boldsymbol{G}}_{N}(\boldsymbol{\theta}^{\dagger})=\left.\frac{1}{N}\sum_{i=1}^{N}\frac{\partial\boldsymbol{m}_{i}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}^{\top}}\right|_{\boldsymbol{\theta}=\boldsymbol{\theta}^{\dagger}}.
\]
Given retained draws $\boldsymbol{\theta}_{1},\dots,\boldsymbol{\theta}_{T}$
from $\Pi^{\boldsymbol{W}}(d\boldsymbol{\theta}\mid\mathcal{D}_{N})$,
let
\[
\widetilde{\boldsymbol{\mu}}_{N,T}(\boldsymbol{W})=\frac{1}{T}\sum_{t=1}^{T}\boldsymbol{\theta}_{t}
\]
denote the MCMC estimate of the target quasi-posterior mean $\boldsymbol{\mu}_{N}(\boldsymbol{W})$
and set $\boldsymbol{\theta}^{\dagger}=\widetilde{\boldsymbol{\mu}}_{N,T}(\boldsymbol{W})$.
Define the sample analogue of $\boldsymbol{A}(\boldsymbol{W})$ by
\[
\widehat{\boldsymbol{A}}_{N}(\boldsymbol{\theta}^{\dagger},\boldsymbol{W})=\left\{ \widehat{\boldsymbol{G}}_{N}(\boldsymbol{\theta}^{\dagger})^{\top}\boldsymbol{W}\widehat{\boldsymbol{G}}_{N}(\boldsymbol{\theta}^{\dagger})\right\} ^{-1}\widehat{\boldsymbol{G}}_{N}(\boldsymbol{\theta}^{\dagger})^{\top}\boldsymbol{W}.
\]
The sandwich estimator is
\begin{equation}
\widehat{\boldsymbol{\Sigma}}_{\mathrm{adj}}(\boldsymbol{\theta}^{\dagger},\boldsymbol{W})=\frac{1}{N}\widehat{\boldsymbol{A}}_{N}(\boldsymbol{\theta}^{\dagger},\boldsymbol{W})\widehat{\boldsymbol{C}}_{N}(\boldsymbol{\theta}^{\dagger})\widehat{\boldsymbol{A}}_{N}(\boldsymbol{\theta}^{\dagger},\boldsymbol{W})^{\top}.\label{eq:sandwich-estimator}
\end{equation}
This estimator targets the repeated-sampling covariance of the quasi-posterior
mean $\boldsymbol{\mu}_{N}(\boldsymbol{W})$ to first order, while
leaving the simulated quasi-posterior and its center unchanged. In
particular, it does not correct the finite-sample balance between
the empirical criterion and the prior. This motivates calibrating
$\boldsymbol{W}$ before the final quasi-posterior is simulated.
\subsection{Curvature-calibrated updating}
\label{subsec:updating}
Throughout this subsection, $\boldsymbol{\theta}$ denotes either
the parameter coordinate of a quasi-posterior or an MCMC draw. The
symbol $\boldsymbol{\vartheta}$ denotes a candidate calibration center.
Exact data-dependent quantities carry the subscript $N$, and a tilde
denotes a numerical approximation computed by MCMC.
For a candidate calibration center $\boldsymbol{\vartheta}\in\Theta$,
define
\[
\boldsymbol{W}_{N}(\boldsymbol{\vartheta})=\widehat{\boldsymbol{C}}_{N}(\boldsymbol{\vartheta})^{-1}.
\]
Recall that $\boldsymbol{\mu}_{N}(\boldsymbol{W})$ denotes the exact
mean of the fixed-weight quasi-posterior conditional on the observed
data. Define the ideal calibration map by
\[
\mathcal{T}_{N}(\boldsymbol{\vartheta})=\boldsymbol{\mu}_{N}\{\boldsymbol{W}_{N}(\boldsymbol{\vartheta})\}=\mathbb{E}_{\Pi^{\boldsymbol{W}_{N}(\boldsymbol{\vartheta})}(\cdot\mid\mathcal{D}_{N})}(\boldsymbol{\theta}).
\]
The ideal fixed-point problem does not require a separate consistent
preliminary estimator. Let $\boldsymbol{\vartheta}_{N}^{(0)}$ be
an initial center in the local contraction region characterized below,
and define the ideal calibration iterates recursively by
\[
\boldsymbol{\vartheta}_{N}^{(s+1)}=\mathcal{T}_{N}(\boldsymbol{\vartheta}_{N}^{(s)}),\qquad s=0,1,\dots.
\]
We define the curvature-calibrated center as the local fixed point
$\boldsymbol{\vartheta}_{N}^{\star}=\mathcal{T}_{N}(\boldsymbol{\vartheta}_{N}^{\star})$.
This condition is self-consistent: the covariance of the moments is
evaluated at the center of the quasi-posterior generated by the resulting
weighting matrix.
A convenient way to construct an initial center without preliminary
optimization is to use a pilot fixed-weight quasi-posterior. Choose
a possibly data-dependent positive-definite pilot weighting matrix
$\boldsymbol{W}_{N}^{\mathrm{p}}$ that does not depend on the candidate
calibration center. A deterministic choice is included as a special
case; for example, $\boldsymbol{W}_{N}^{\mathrm{p}}=\boldsymbol{I}_{K}$
may be used after rescaling moments whose numerical magnitudes differ
substantially. Let $\boldsymbol{\theta}_{1}^{\mathrm{p}},\dots,\boldsymbol{\theta}_{T_{\mathrm{p}}}^{\mathrm{p}}$
denote retained draws from $\Pi^{\boldsymbol{W}_{N}^{\mathrm{p}}}(d\boldsymbol{\theta}\mid\mathcal{D}_{N})$
and set
\begin{equation}
\widetilde{\boldsymbol{\vartheta}}_{N}^{(0)}=\widetilde{\boldsymbol{\mu}}_{N,T_{\mathrm{p}}}^{\mathrm{p}}=\frac{1}{T_{\mathrm{p}}}\sum_{t=1}^{T_{\mathrm{p}}}\boldsymbol{\theta}_{t}^{\mathrm{p}}.\label{eq:pilot-initial-center}
\end{equation}
The pilot matrix need not be an estimate of $\boldsymbol{C}_{0}^{-1}$.
What is required is that the corresponding exact pilot quasi-posterior
mean localizes at $\boldsymbol{\theta}_{0}$ and that the pilot Monte
Carlo error is $o_{p}(1)$. The pilot Markov chain may be initialized
at any state in a class over which this Monte Carlo approximation
holds uniformly. In particular, its initial state is unrestricted
when the approximation is uniform over the whole state space. Formal
conditions are stated in Section~A of the Supplementary Material
\citep{Tanaka2026c}.
The map $\mathcal{T}_{N}$ is generally unavailable in closed form.
For a candidate center $\boldsymbol{\vartheta}$ at calibration run
$s$, let $\boldsymbol{\theta}_{1}^{(s)},\dots,\boldsymbol{\theta}_{T_{s}}^{(s)}$
denote retained draws from $\Pi^{\boldsymbol{W}_{N}(\boldsymbol{\vartheta})}(d\boldsymbol{\theta}\mid\mathcal{D}_{N})$.
Define the Monte Carlo approximation to the map by
\[
\widetilde{\mathcal{T}}_{N,T_{s}}^{(s)}(\boldsymbol{\vartheta})=\frac{1}{T_{s}}\sum_{t=1}^{T_{s}}\boldsymbol{\theta}_{t}^{(s)}.
\]
The computed iterates satisfy
\[
\widetilde{\boldsymbol{\vartheta}}_{N}^{(s+1)}=\widetilde{\mathcal{T}}_{N,T_{s}}^{(s)}(\widetilde{\boldsymbol{\vartheta}}_{N}^{(s)}).
\]
The fixed point can be approximated as follows.
\begin{enumerate}
\item Choose a positive-definite pilot weighting matrix $\boldsymbol{W}_{N}^{\mathrm{p}}$
that does not depend on the candidate calibration center, run the
pilot Markov chain, and set the initial center according to \eqref{eq:pilot-initial-center}.
Alternatively, any initial center that lies in the local contraction
region with probability approaching one may be used.
\item At iteration $s$, set
\[
\boldsymbol{W}_{N}^{(s)}=\widehat{\boldsymbol{C}}_{N}(\widetilde{\boldsymbol{\vartheta}}_{N}^{(s)})^{-1}.
\]
Simulate $\boldsymbol{\theta}_{1}^{(s)},\dots,\boldsymbol{\theta}_{T_{s}}^{(s)}$
from $\Pi^{\boldsymbol{W}_{N}^{(s)}}(d\boldsymbol{\theta}\mid\mathcal{D}_{N})$
and compute
\[
\widetilde{\boldsymbol{\vartheta}}_{N}^{(s+1)}=\frac{1}{T_{s}}\sum_{t=1}^{T_{s}}\boldsymbol{\theta}_{t}^{(s)}.
\]
\item Repeat Step 2 until
\[
\eta(\widetilde{\boldsymbol{\vartheta}}_{N}^{(s+1)},\widetilde{\boldsymbol{\vartheta}}_{N}^{(s)})\leq\tau,
\]
where $\eta(\cdot,\cdot)$ is a stopping criterion and $\tau>0$ is
a prespecified threshold.
\item Set $S=s+1$, define
\[
\boldsymbol{W}_{N}^{(S)}=\widehat{\boldsymbol{C}}_{N}(\widetilde{\boldsymbol{\vartheta}}_{N}^{(S)})^{-1},
\]
and run one final Markov chain targeting $\Pi^{\boldsymbol{W}_{N}^{(S)}}(d\boldsymbol{\theta}\mid\mathcal{D}_{N})$.
Retain the draws from this final run.
\end{enumerate}
Because the resulting weight calibrates the local quadratic curvature
of the criterion to the optimal GMM information matrix, we call the
procedure curvature-calibrated quasi-Bayesian (CCQB) updating. As
the weighting matrix is fixed within each MCMC run, the procedure
avoids repeated evaluation of $\widehat{\boldsymbol{C}}_{N}(\boldsymbol{\theta})$
and its determinant. The iteration is used only to determine the locally
calibrated weighting matrix.
This study uses a scale-free stopping criterion. Define the reference
covariance matrix as
\[
\widehat{\boldsymbol{\Sigma}}_{N}^{\mathrm{ref}}=\frac{1}{N}\left\{ \widehat{\boldsymbol{G}}_{N}(\widetilde{\boldsymbol{\vartheta}}_{N}^{(0)})^{\top}\widehat{\boldsymbol{C}}_{N}(\widetilde{\boldsymbol{\vartheta}}_{N}^{(0)})^{-1}\widehat{\boldsymbol{G}}_{N}(\widetilde{\boldsymbol{\vartheta}}_{N}^{(0)})\right\} ^{-1}.
\]
Then the stopping criterion is defined by
\[
\eta(\widetilde{\boldsymbol{\vartheta}}_{N}^{(s+1)},\widetilde{\boldsymbol{\vartheta}}_{N}^{(s)})=\left\{ \frac{1}{J}(\widetilde{\boldsymbol{\vartheta}}_{N}^{(s+1)}-\widetilde{\boldsymbol{\vartheta}}_{N}^{(s)})^{\top}(\widehat{\boldsymbol{\Sigma}}_{N}^{\mathrm{ref}})^{-1}(\widetilde{\boldsymbol{\vartheta}}_{N}^{(s+1)}-\widetilde{\boldsymbol{\vartheta}}_{N}^{(s)})\right\} ^{1/2}.
\]
When the pilot center is consistent, this criterion measures the root-mean-square
change in the calibration center in asymptotic standard-error units
and is invariant under nonsingular linear reparameterizations. More
generally, the stopping argument only requires the eigenvalues of
$N\widehat{\boldsymbol{\Sigma}}_{N}^{\mathrm{ref}}$ to be bounded
above and bounded away from zero in probability. The standardized
calibration tolerance then has a clear interpretation: calibration
stops when the Mahalanobis root-mean-square change, normalized by
dimension, is at most $\tau$.
Two distinctions from the fully parameter-dependent quasi-posterior
in \eqref{eq:quasi-posterior} are worth keeping in view. That target
contains the determinant term $\det\{\boldsymbol{W}_{N}(\boldsymbol{\theta})\}^{1/2}$
and allows the weighting matrix to vary with $\boldsymbol{\theta}$,
whereas \eqref{eq:fixed-W-quasi-posterior} fixes the weight and therefore
has a constant determinant. The calibrated fixed-weight quasi-posterior
matches the leading local quadratic term obtained by evaluating the
covariance weight at a self-consistent center, but it does not reproduce
variation in $\boldsymbol{W}_{N}(\boldsymbol{\theta})$ away from
that center or the nonconstant log-determinant contribution. Those
differences can affect higher-order and finite-sample features.
\subsection{Large-sample properties}
\label{subsec:theory}
The theorem below establishes three points needed for the proposed
workflow. First, the ideal calibration map is locally contractive
and therefore has a unique local fixed point. Second, only one or
two ideal updates are needed for first-order equivalence to that fixed
point, depending on the quality of the initial center. Third, a first-order
mean expansion and Bernstein--von Mises approximation established
at the efficient population weight $\boldsymbol{W}_{0}$ continue
to hold under the self-consistently calibrated random weight.
The result is deliberately local. It requires local identification,
smoothness and positive definiteness of $\boldsymbol{C}(\boldsymbol{\theta})$,
regularity of the estimated weighting map and fixed-weight quasi-posteriors,
and a prior that is positive near $\boldsymbol{\theta}_{0}$. It does
not require global identification or consistency of an arbitrary initial
center: the ideal iteration only needs to start in the local contraction
region with probability approaching one. A separate global concentration
condition justifies the pilot construction. Formal conditions and
the high-probability contraction event are stated in Section~A of
the Supplementary Material \citep{Tanaka2026c}. Stochastic orders
for exact data-dependent quantities refer to the sampling law; statements
involving MCMC output refer to the joint law of the data and simulation.
\begin{theorem}
\label{thm:main-thm}
Suppose that Assumption A.1 in Section A of the Supplementary Material
\citep{Tanaka2026c} holds. There exist a fixed $\delta>0$ and a
sequence of events $\mathcal{E}_{N}$ satisfying $\Pr(\mathcal{E}_{N})\to1$
such that, on $\mathcal{E}_{N}$, the calibration map
\[
\mathcal{T}_{N}(\boldsymbol{\vartheta})=\boldsymbol{\mu}_{N}\left\{ \boldsymbol{W}_{N}(\boldsymbol{\vartheta})\right\}
\]
maps
\[
\overline{\mathcal{B}}_{\delta}(\boldsymbol{\theta}_{0})=\left\{ \boldsymbol{\vartheta}\in\mathbb{R}^{J}:\left\lVert \boldsymbol{\vartheta}-\boldsymbol{\theta}_{0}\right\rVert \leq\delta\right\}
\]
into itself and is a contraction on this set. Its contraction modulus
may be chosen to satisfy
\[
\rho_{N}=O_{p}(N^{-1/2}).
\]
On $\mathcal{E}_{N}$, let $\boldsymbol{\vartheta}_{N}^{\star}$ denote
the unique fixed point. Extend it by setting
\[
\boldsymbol{\vartheta}_{N}^{\star}=\boldsymbol{\mu}_{N}(\boldsymbol{W}_{0})\quad\text{on }\mathcal{E}_{N}^{c},
\]
and define
\[
\boldsymbol{W}_{N}^{\star}=\begin{cases}
\boldsymbol{W}_{N}\left(\boldsymbol{\vartheta}_{N}^{\star}\right), & \text{on }\mathcal{E}_{N},\\
\boldsymbol{W}_{0}, & \text{on }\mathcal{E}_{N}^{c}.
\end{cases}
\]
The ideal iteration converges to $\boldsymbol{\vartheta}_{N}^{\star}$
from every starting value in the ball on $\mathcal{E}_{N}$. Hence,
for any possibly random starting sequence satisfying
\[
\Pr\left\{ \boldsymbol{\vartheta}_{N}^{(0)}\in\overline{\mathcal{B}}_{\delta}(\boldsymbol{\theta}_{0})\right\} \to1,
\]
the iteration converges to $\boldsymbol{\vartheta}_{N}^{\star}$ with
probability approaching one. Moreover,
\[
\boldsymbol{\vartheta}_{N}^{(1)}-\boldsymbol{\theta}_{0}=O_{p}(N^{-1/2}),\qquad\boldsymbol{\vartheta}_{N}^{(1)}-\boldsymbol{\vartheta}_{N}^{\star}=O_{p}(N^{-1/2}),
\]
and
\[
\boldsymbol{\vartheta}_{N}^{(2)}-\boldsymbol{\vartheta}_{N}^{\star}=O_{p}(N^{-1})=o_{p}(N^{-1/2}).
\]
If, in addition,
\[
\boldsymbol{\vartheta}_{N}^{(0)}\xrightarrow{p}\boldsymbol{\theta}_{0},
\]
then
\[
\boldsymbol{\vartheta}_{N}^{(1)}-\boldsymbol{\vartheta}_{N}^{\star}=o_{p}(N^{-1/2}).
\]
The fixed point and calibrated weight satisfy
\[
\boldsymbol{\vartheta}_{N}^{\star}-\boldsymbol{\theta}_{0}=O_{p}(N^{-1/2}),\qquad\boldsymbol{W}_{N}^{\star}\xrightarrow{p}\boldsymbol{W}_{0},
\]
and
\[
\sqrt{N}\left(\boldsymbol{\vartheta}_{N}^{\star}-\boldsymbol{\theta}_{0}\right)=-\boldsymbol{A}_{0}\boldsymbol{\xi}_{N}+o_{p}(1).
\]
Conditionally on the data,
\[
d_{\mathrm{BL}}\left[\mathcal{L}_{\Pi^{\boldsymbol{W}_{N}^{\star}}(\cdot\mid\mathcal{D}_{N})}\left\{ \sqrt{N}\left(\boldsymbol{\theta}-\boldsymbol{\vartheta}_{N}^{\star}\right)\right\} ,\;\mathcal{N}\left(\boldsymbol{0}_{J},\boldsymbol{V}_{0}\right)\right]\xrightarrow{p}0,
\]
where $d_{\mathrm{BL}}(\cdot,\cdot)$ denotes bounded-Lipschitz distance.
Moreover,
\[
N\operatorname{Var}_{\Pi^{\boldsymbol{W}_{N}^{\star}}(\cdot\mid\mathcal{D}_{N})}(\boldsymbol{\theta})\xrightarrow{p}\boldsymbol{V}_{0}.
\]
If, in addition,
\[
\sqrt{N}\bar{\boldsymbol{m}}_{N}(\boldsymbol{\theta}_{0})\rightsquigarrow\mathcal{N}\left(\boldsymbol{0}_{K},\boldsymbol{C}_{0}\right),
\]
then
\[
\sqrt{N}\left(\boldsymbol{\vartheta}_{N}^{\star}-\boldsymbol{\theta}_{0}\right)\rightsquigarrow\mathcal{N}\left(\boldsymbol{0}_{J},\boldsymbol{V}_{0}\right).
\]
\end{theorem}
The proof is given in Section~A of the Supplementary Material \citep{Tanaka2026c}.
The conditional Gaussian approximation and the limiting distribution
of $\boldsymbol{\vartheta}_{N}^{\star}$ have the same first-order
covariance matrix $\boldsymbol{V}_{0}$. Thus, at the level of the
first-order Gaussian approximation, the calibrated quasi-posterior
dispersion and the asymptotic sampling dispersion of its center are
governed by the inverse Godambe information matrix associated with
the specified moments.
The contraction modulus is $O_{p}(N^{-1/2})$. Therefore, the first
ideal update maps any initial center in the local contraction region
into an $O_{p}(N^{-1/2})$ neighborhood of $\boldsymbol{\theta}_{0}$,
and the second update is first-order equivalent to the fixed point.
When the initial center is consistent---as is the pilot MCMC center
under the conditions in Section~A of the Supplementary Material \citep{Tanaka2026c}---one
ideal calibration update is already first-order equivalent. The pilot
run itself only needs $o_{p}(1)$ Monte Carlo accuracy, whereas the
calibration runs require Monte Carlo error to be negligible relative
to $N^{-1/2}$ for first-order equivalence of the computed iterate.
The proposed procedure should not be viewed as a Bayesian reinterpretation
of iterated GMM \citep{Hansen1996}. The overall scale of $\boldsymbol{W}$
acts as a learning rate, whereas its normalized shape determines the
relative weighting of the moment conditions. The prior may still have
an important finite-sample effect, but the empirical component is
no longer placed on an arbitrary local scale.
\section{Simulation Study}
\label{sec:simulation}
\subsection{Questions addressed by the simulation}
\label{subsec:sim-purpose}
The simulation study examines three related implications of CCQB updating.
First, replacing an arbitrary weighting matrix by a covariance weight
should bring the covariance represented by the quasi-posterior closer
to the repeated-sampling covariance of its center. We assess this
implication through covariance discrepancies and the frequentist coverage
of nominal intervals. Second, conditional on using a covariance weight,
we examine whether iterating to the self-consistent CCQB fixed point
provides a material finite-sample improvement over a single covariance
update. Because the one-step and converged procedures have the same
first-order limit under the conditions developed in Section~\ref{sec:method},
differences between them reflect higher-order finite-sample effects
and need not be substantial in every design. Third, calibration should
not mechanically remove the finite-sample displacement of the quasi-posterior
center from the corresponding unpenalized weighted-criterion estimator.
We therefore also measure this displacement.
Accordingly, the primary evaluation criteria are covariance calibration,
coverage, interval length, and center-to-unpenalized displacement.
Bias and mean squared error are not used to rank the procedures because
the proposed method is intended to calibrate quasi-posterior uncertainty
rather than to dominate alternative point estimators.
\subsection{Designs and common settings}
\label{subsec:mc-design}
We consider three data-generating processes: an instrumental-variables
(IV) regression design that may be overidentified, a heteroskedastic
and serially dependent (HSD) linear regression design, and a Poisson
regression design with a nonlinear moment function. In every design,
the parameter dimension is $J=4$, the sample size is
\[
N\in\{50,100,200,500\},
\]
and the prior is
\[
\boldsymbol{\theta}\sim\mathcal{N}\left(\boldsymbol{0}_{J},\varrho^{2}\boldsymbol{I}_{J}\right),\qquad\varrho=5.
\]
Unless otherwise stated explicitly, primitive covariates and disturbances
are independent across observations and mutually independent.
The HSD and Poisson designs are exactly identified. When $\boldsymbol{G}_{0}$
is nonsingular,
\[
\boldsymbol{A}(\boldsymbol{W})=(\boldsymbol{G}_{0}^{\top}\boldsymbol{W}\boldsymbol{G}_{0})^{-1}\boldsymbol{G}_{0}^{\top}\boldsymbol{W}=\boldsymbol{G}_{0}^{-1}.
\]
Consequently, the weighting matrix does not affect the first-order
sampling distribution of the quasi-posterior center. It does, however,
affect the covariance represented by the quasi-posterior and may affect
its center through higher-order effects and its finite-sample interaction
with the prior. The exactly identified designs therefore isolate covariance
calibration and higher-order effects from first-order changes in the
sampling distribution of the center. By contrast, in the overidentified
IV designs the relative weighting of the moment conditions can also
affect the first-order sampling distribution of the center.
The main text focuses on two complementary designs: the overidentified
IV design with $K=8$ and the exactly identified Poisson design with
$K=J=4$. The IV design has a quadratic criterion and a Gaussian fixed-weight
quasi-posterior available in closed form. It therefore provides a
transparent benchmark in which the effect of changing the weighting
matrix can be studied without numerical simulation error. The Poisson
design instead combines exact identification with a nonlinear moment
function and a non-Gaussian quasi-posterior, and therefore provides
a setting in which higher-order finite-sample effects of self-consistent
calibration can be more pronounced. We additionally consider the more
highly overidentified IV design with $K=12$ and the HSD design. Section~B
of the Supplementary Material \citep{Tanaka2026c} gives the design
and computational details, and Section~C reports the additional results.
\subsection{Competing procedures}
\label{subsec:compet-proc}
We compare three procedures corresponding to different stages of CCQB
calibration. For notational convenience, let $a\in\{0,1,\star\}$
index the procedures $s=0,1,\star$, each paired with either its raw
quasi-posterior covariance or a sandwich-adjusted covariance. The
comparison between $s=0$ and $s=1$ isolates the effect of replacing
the identity weight by a covariance weight, whereas the comparison
between $s=1$ and $s=\star$ isolates the additional finite-sample
effect of imposing self-consistency.
For each replication, we first evaluate the identity-weight pilot
quasi-posterior, denoted by $s=0$, with
\[
\boldsymbol{W}^{p}=\boldsymbol{W}_{N}^{(0)}=\boldsymbol{I}_{K}.
\]
The one-step covariance-weighted procedure, denoted by $s=1$, then
fixes
\[
\boldsymbol{W}_{N}^{(1)}=\widehat{\boldsymbol{C}}_{N}(\widetilde{\boldsymbol{\vartheta}}_{N}^{(0)})^{-1}
\]
using the center of the pilot quasi-posterior. Finally, $s=\star$
denotes the converged CCQB procedure, for which the covariance weight
is repeatedly updated until the stopping criterion is satisfied. For
each weighting scheme, we report both the raw quasi-posterior uncertainty
and the corresponding sandwich-adjusted uncertainty. These are alternative
uncertainty summaries for the same quasi-posterior center rather than
separate point-estimation procedures. In particular, the sandwich
adjustment changes the reported covariance but leaves both the underlying
quasi-posterior and its center unchanged.
Each design uses $R=1{,}000$ Monte Carlo replications and $\tau=0.05$.
Let $\mathcal{R}$ denote the replications successfully completed
by procedure $s=\star$, where failure means that a required covariance
matrix is singular or not numerically positive definite, a required
numerical calculation fails, or the CCQB iteration fails to satisfy
the stopping criterion within 100 updates. All procedures are compared
over this common set of successful replications so that the reported
differences are based on the same simulated samples.
\subsection{Evaluation measures}
\label{subsec:eval-crit}
Let $\widehat{\boldsymbol{\mu}}_{a,r}$ denote the quasi-posterior
center and let $\widehat{\boldsymbol{\Sigma}}_{a,r}^{\mathrm{raw}}$
and $\widehat{\boldsymbol{\Sigma}}_{a,r}^{\mathrm{adj}}$ denote,
respectively, the quasi-posterior covariance and the sandwich covariance
estimator evaluated at the same center and weighting matrix. Hereafter,
``raw'' and ``adj'' refer to the unadjusted and sandwich-adjusted
uncertainty summaries, respectively. Let $q_{a,r,j}(p)$ denote the
$p$th marginal quasi-posterior quantile, computed analytically for
the Gaussian IV and HSD designs and empirically from the retained
MCMC draws for the Poisson design. The raw and sandwich-adjusted $90\%$
intervals are
\[
I_{a,r,j}^{\mathrm{raw}}=[q_{a,r,j}(0.05),q_{a,r,j}(0.95)]
\]
and
\[
I_{a,r,j}^{\mathrm{adj}}=\widehat{\mu}_{a,r,j}\pm\zeta_{0.95}\sqrt{(\widehat{\boldsymbol{\Sigma}}_{a,r}^{\mathrm{adj}})_{jj}},
\]
respectively, where $\zeta_{0.95}$ denotes the 0.95 quantile of the
standard normal distribution.
We report the Monte Carlo coverage probabilities and mean lengths
of these intervals over the common successful replication set. Interval
length is interpreted jointly with coverage rather than as a separate
measure of point-estimation efficiency. In the Gaussian IV and HSD
designs, the fixed-weight quasi-posteriors are Gaussian, so differences
between the raw and adjusted intervals primarily reflect differences
in the covariance used to construct them. In the Poisson design, however,
the raw intervals are quasi-posterior quantile intervals and need
not be symmetric, whereas the adjusted intervals are symmetric normal
intervals centered at the same quasi-posterior mean. The raw-versus-adjusted
coverage comparison for the Poisson design therefore reflects both
covariance adjustment and finite-sample quasi-posterior shape. We
consequently use the covariance-discrepancy measures below as a more
direct assessment of covariance calibration.
To evaluate covariance calibration, we define the following diagnostics.
The repeated-sampling covariance of the quasi-posterior center is
estimated by
\[
\widehat{\boldsymbol{V}}_{a}^{\mathrm{RS}}=\frac{1}{|\mathcal{R}|-1}\sum_{r\in\mathcal{R}}(\widehat{\boldsymbol{\mu}}_{a,r}-\bar{\boldsymbol{\mu}}_{a})(\widehat{\boldsymbol{\mu}}_{a,r}-\bar{\boldsymbol{\mu}}_{a})^{\top},
\]
with
\[
\bar{\boldsymbol{\mu}}_{a}=\frac{1}{|\mathcal{R}|}\sum_{r\in\mathcal{R}}\widehat{\boldsymbol{\mu}}_{a,r}.
\]
We use an affine-invariant covariance discrepancy. For $u\in\{\mathrm{raw},\mathrm{adj}\}$,
define
\[
\boldsymbol{\Upsilon}_{a,r}^{u}=(\widehat{\boldsymbol{V}}_{a}^{\mathrm{RS}})^{-1/2}\widehat{\boldsymbol{\Sigma}}_{a,r}^{u}(\widehat{\boldsymbol{V}}_{a}^{\mathrm{RS}})^{-1/2},
\]
where $\boldsymbol{B}^{-1/2}$ denotes the symmetric inverse square
root of a matrix $\boldsymbol{B}$. Let $\lambda_{a,r,1}^{u},\dots,\lambda_{a,r,J}^{u}$
denote the eigenvalues of $\boldsymbol{\Upsilon}_{a,r}^{u}$. These
are the generalized eigenvalues of the covariance pair $(\widehat{\boldsymbol{\Sigma}}_{a,r}^{u},\widehat{\boldsymbol{V}}_{a}^{\mathrm{RS}})$.
We measure covariance discrepancy by
\[
d_{\mathrm{cov},a,r}^{u}=\left\{ \frac{1}{J}\sum_{j=1}^{J}(\log\lambda_{a,r,j}^{u})^{2}\right\} ^{1/2}
\]
and report its median
\[
D_{\mathrm{cov},a}^{u}=\operatorname{median}(d_{\mathrm{cov},a,r}^{u}).
\]
The generalized eigenvalues of $(\widehat{\boldsymbol{\Sigma}}_{a,r}^{u},\widehat{\boldsymbol{V}}_{a}^{\mathrm{RS}})$
can be interpreted as directional variance ratios of the reported
covariance relative to the repeated-sampling covariance of the quasi-posterior
center. Hence, $D_{\mathrm{cov},a}^{u}$ provides an affine-invariant
summary of overall covariance miscalibration, with zero corresponding
to exact agreement.
To assess how calibration changes the finite-sample location of the
quasi-posterior relative to the corresponding unpenalized analysis,
we compare the quasi-posterior center with the unpenalized minimizer
of the same weighted criterion. Define
\[
\widehat{\boldsymbol{\theta}}_{a,r}^{\natural}=\arg\min_{\boldsymbol{\theta}}\bar{\boldsymbol{m}}_{N,r}(\boldsymbol{\theta})^{\top}\boldsymbol{W}_{a,r}\bar{\boldsymbol{m}}_{N,r}(\boldsymbol{\theta}),
\]
where $\boldsymbol{W}_{a,r}$ is the weighting matrix used in the
final quasi-posterior for method $a$ in replication $r$. To summarize
finite-sample displacement of the quasi-posterior center from the
corresponding unpenalized weighted-criterion minimizer, define
\[
\Delta_{a,r}=\left[\frac{1}{J}(\widehat{\boldsymbol{\mu}}_{a,r}-\widehat{\boldsymbol{\theta}}_{a,r}^{\natural})^{\top}(\widehat{\boldsymbol{\Sigma}}_{N,r}^{\mathrm{ref}})^{-1}(\widehat{\boldsymbol{\mu}}_{a,r}-\widehat{\boldsymbol{\theta}}_{a,r}^{\natural})\right]^{1/2}.
\]
We report the median of $\Delta_{a,r}$ over the common successful
replication set. The normalization expresses the displacement in asymptotic
standard-error units and makes it comparable across sample sizes and
procedures. In the quadratic IV and HSD designs, the quasi-posterior
is Gaussian and this center-to-unpenalized displacement is attributable
to the finite-sample influence of the prior. In the nonlinear Poisson
design, the quasi-posterior mean may differ from the unpenalized criterion
minimizer even apart from prior shrinkage because of finite-sample
asymmetry and nonlinear curvature. We therefore interpret $\Delta_{a,r}$
more generally as a center-to-unpenalized displacement in that design.
Because the Poisson quasi-posteriors and calibration maps are evaluated
by MCMC rather than analytically, we additionally monitor the Monte
Carlo error in the estimated quasi-posterior centers. Section~B of
the Supplementary Material \citep{Tanaka2026c} gives the construction
of the standardized Monte Carlo error measure. For each sample size,
we report, across methods and replications, the 95th percentile of
the largest standardized Monte Carlo standard error across the calibration
and final chains. Values close to zero indicate that MCMC error is
small relative to the statistical scale on which the calibration updates
and quasi-posterior centers vary.
\subsection{Simulation results}
\label{subsec:sim-results}
The results separate two effects. The comparison of $s=0$ with $s=1$
measures the gain from replacing the identity weight by a covariance
weight, while the comparison of $s=1$ with $s=\star$ measures the
additional finite-sample effect of enforcing self-consistency. Because
the pilot center is consistent under the maintained conditions, the
latter two procedures have the same first-order limiting weight, $\boldsymbol{C}_{0}^{-1}$;
differences between them are therefore higher-order. We first report
numerical convergence and failures, then coverage and covariance calibration,
and finally interval length and center-to-unpenalized displacement.
The identity-weight procedure $s=0$ is retained as a pilot baseline.
Table~\ref{tab:updates} shows that the CCQB iteration converges
rapidly and reliably in both designs. There are no failed runs, and
only a few calibration updates are typically required. In the IV design,
the median number of updates decreases from three at $N=50$ to two
for $N\geq100$; in the Poisson design, the median is three throughout,
while the maximum decreases from seven to four. Thus, the computational
cost of self-consistent calibration is modest.
\begin{table}
\caption{Number of CCQB calibration updates}
\label{tab:updates}
\medskip{}
\centering{}
\begin{tabular}{lrrrrr}
\hline
DGP & $N$ & \multicolumn{3}{c}{\begin{cellvarwidth}[t]
\raggedleft
Number of CCQB
calibration updates
\end{cellvarwidth}} & \multirow{2}{*}{\begin{cellvarwidth}[t]
\raggedleft
Number of
failed runs
\end{cellvarwidth}}\tabularnewline
\cline{3-5}
& & Min. & Median & Max. & \tabularnewline
\hline
\multirow{4}{*}{IV, $K=8$} & 50 & 1 & 3 & 8 & 0\tabularnewline
& 100 & 1 & 2 & 4 & 0\tabularnewline
& 200 & 1 & 2 & 3 & 0\tabularnewline
& 500 & 1 & 2 & 2 & 0\tabularnewline
\hline
\multirow{4}{*}{Poisson} & 50 & 2 & 3 & 7 & 0\tabularnewline
& 100 & 2 & 3 & 7 & 0\tabularnewline
& 200 & 2 & 3 & 6 & 0\tabularnewline
& 500 & 2 & 3 & 4 & 0\tabularnewline
\hline
\end{tabular}
\end{table}
Table~\ref{tab:coverage} contrasts the coverage results for $\theta_{1}$
in the IV and Poisson designs. In the IV design, the raw identity-weight
quasi-posterior severely undercovers, whereas the sandwich adjustment
largely corrects this distortion. Both covariance-weighted procedures
deliver substantially better raw coverage, approaching the nominal
$90\%$ level as $N$ increases; their raw and adjusted coverages
are nearly identical, and further iteration beyond the first covariance
update has little effect. The Poisson design displays a different
finite-sample pattern. The identity-weight procedure strongly overcovers,
and the one-step covariance-weighted procedure also exhibits appreciable
raw overcoverage. Iterating to the CCQB fixed point brings raw coverage
much closer to the nominal level. Under $s=\star$, raw coverage ranges
from $0.870$ at $N=50$ to $0.906$ at $N=500$, with the adjusted
coverage likewise remaining close to $90\%$. Thus, iteration matters
little for coverage in the linear IV design but provides a visible
finite-sample improvement in the nonlinear Poisson design.
\begin{table}
\caption{$90\%$ nominal coverage of $\theta_{1}$}
\label{tab:coverage}
\medskip{}
\centering{}
\begin{tabular}{lccrrrr}
\hline
DGP & $s$ & Cov. & \multicolumn{4}{c}{$N$}\tabularnewline
\cline{4-7}
& & & 50 & 100 & 200 & 500\tabularnewline
\hline
\multirow{6}{*}{IV, $K=8$} & \multirow{2}{*}{$0$} & raw & 0.286 & 0.279 & 0.289 & 0.303\tabularnewline
& & adj & 0.859 & 0.846 & 0.879 & 0.911\tabularnewline
\cline{2-7}
& \multirow{2}{*}{$1$} & raw & 0.804 & 0.828 & 0.867 & 0.903\tabularnewline
& & adj & 0.805 & 0.827 & 0.869 & 0.904\tabularnewline
\cline{2-7}
& \multirow{2}{*}{$\star$} & raw & 0.799 & 0.827 & 0.869 & 0.903\tabularnewline
& & adj & 0.804 & 0.828 & 0.869 & 0.904\tabularnewline
\hline
\multirow{6}{*}{Poisson} & \multirow{2}{*}{$0$} & raw & 1.000 & 1.000 & 1.000 & 1.000\tabularnewline
& & adj & 1.000 & 0.998 & 0.979 & 0.849\tabularnewline
\cline{2-7}
& \multirow{2}{*}{$1$} & raw & 0.976 & 0.976 & 0.962 & 0.965\tabularnewline
& & adj & 0.953 & 0.942 & 0.911 & 0.909\tabularnewline
\cline{2-7}
& \multirow{2}{*}{$\star$} & raw & 0.870 & 0.880 & 0.897 & 0.906\tabularnewline
& & adj & 0.897 & 0.885 & 0.901 & 0.907\tabularnewline
\hline
\end{tabular}
\end{table}
Tables~C.1 and C.3 in Section~C of the Supplementary Material \citep{Tanaka2026c}
report parameter-wise coverage for the IV and Poisson designs, respectively.
They broadly reinforce the conclusions for $\theta_{1}$: in the IV
design, covariance weighting largely eliminates the severe undercoverage
of the raw identity-weight procedure, with little difference between
$s=1$ and $s=\star$. In the Poisson design, the benefits of further
iteration are again more apparent, although $\theta_{4}$ remains
relatively difficult to calibrate in finite samples.
Table~\ref{tab:cov-calib} provides a more direct assessment of covariance
calibration. In the IV design, the raw identity-weight covariance
is severely miscalibrated, whereas covariance weighting reduces the
discrepancy substantially; for $s=1$ and $s=\star$, the raw and
adjusted discrepancies are nearly identical and decline with $N$.
Moreover, there is little difference between the one-step and converged
procedures. The Poisson design shows a stronger role for iteration:
although $s=1$ greatly improves on the identity-weight procedure,
$s=\star$ further reduces the raw covariance discrepancy, especially
at larger sample sizes. By $N=500$, its raw covariance is essentially
as well calibrated as the corresponding sandwich-adjusted covariance.
\begin{table}
\caption{Covariance calibration}
\label{tab:cov-calib}
\medskip{}
\centering{}
\begin{tabular}{lccrrrr}
\hline
DGP & $s$ & Cov. & \multicolumn{4}{c}{$N$}\tabularnewline
\cline{4-7}
& & & 50 & 100 & 200 & 500\tabularnewline
\hline
\multirow{6}{*}{IV, $K=8$} & \multirow{2}{*}{$0$} & raw & 3.141 & 3.026 & 2.984 & 2.884\tabularnewline
& & adj & 0.657 & 0.439 & 0.310 & 0.195\tabularnewline
\cline{2-7}
& \multirow{2}{*}{$1$} & raw & 0.738 & 0.470 & 0.320 & 0.193\tabularnewline
& & adj & 0.743 & 0.472 & 0.318 & 0.192\tabularnewline
\cline{2-7}
& \multirow{2}{*}{$\star$} & raw & 0.780 & 0.480 & 0.320 & 0.192\tabularnewline
& & adj & 0.781 & 0.478 & 0.320 & 0.192\tabularnewline
\hline
\multirow{6}{*}{Poisson} & \multirow{2}{*}{$0$} & raw & 4.217 & 4.186 & 4.089 & 4.026\tabularnewline
& & adj & 3.268 & 2.710 & 2.117 & 1.291\tabularnewline
\cline{2-7}
& \multirow{2}{*}{$1$} & raw & 0.965 & 0.920 & 0.851 & 0.688\tabularnewline
& & adj & 0.884 & 0.618 & 0.442 & 0.310\tabularnewline
\cline{2-7}
& \multirow{2}{*}{$\star$} & raw & 0.863 & 0.639 & 0.441 & 0.307\tabularnewline
& & adj & 0.941 & 0.663 & 0.452 & 0.310\tabularnewline
\hline
\end{tabular}
\end{table}
Table~\ref{tab:length} reports the mean lengths of the nominal $90\%$
intervals for $\theta_{1}$. In the IV design, the raw intervals under
$s=0$ are much too short, whereas for $s=1$ and $s=\star$ the raw
and adjusted lengths are nearly identical, with little additional
effect from iteration. The Poisson design exhibits the opposite distortion
under $s=0$, with excessively long intervals. Covariance weighting
substantially shortens these intervals, and iteration from $s=1$
to $s=\star$ yields a further reduction, particularly for the raw
intervals. Under $s=\star$, the raw and adjusted lengths become increasingly
similar as $N$ grows, consistent with the improved covariance calibration
in Table~\ref{tab:cov-calib}. These length comparisons should be
interpreted jointly with the coverage results in Table~\ref{tab:coverage}.
\begin{table}
\caption{Mean lengths of nominal $90\%$ intervals for $\theta_{1}$}
\label{tab:length}
\medskip{}
\centering{}
\begin{tabular}{lccrrrr}
\hline
DGP & $s$ & Cov. & \multicolumn{4}{c}{$N$}\tabularnewline
\cline{4-7}
& & & 50 & 100 & 200 & 500\tabularnewline
\hline
\multirow{6}{*}{IV, $K=8$} & \multirow{2}{*}{$0$} & raw & 0.418 & 0.280 & 0.195 & 0.122\tabularnewline
& & adj & 1.689 & 1.173 & 0.833 & 0.526\tabularnewline
\cline{2-7}
& \multirow{2}{*}{$1$} & raw & 1.499 & 1.104 & 0.807 & 0.520\tabularnewline
& & adj & 1.507 & 1.107 & 0.807 & 0.520\tabularnewline
\cline{2-7}
& \multirow{2}{*}{$\star$} & raw & 1.487 & 1.101 & 0.805 & 0.520\tabularnewline
& & adj & 1.502 & 1.106 & 0.807 & 0.520\tabularnewline
\hline
\multirow{6}{*}{Poisson} & \multirow{2}{*}{$0$} & raw & 13.613 & 12.469 & 11.151 & 9.112\tabularnewline
& & adj & 8.182 & 5.631 & 3.800 & 2.082\tabularnewline
\cline{2-7}
& \multirow{2}{*}{$1$} & raw & 5.647 & 4.098 & 2.843 & 1.645\tabularnewline
& & adj & 5.491 & 3.471 & 2.229 & 1.332\tabularnewline
\cline{2-7}
& \multirow{2}{*}{$\star$} & raw & 4.199 & 2.931 & 2.053 & 1.301\tabularnewline
& & adj & 4.693 & 3.082 & 2.092 & 1.310\tabularnewline
\hline
\end{tabular}
\end{table}
Table~\ref{tab:displace} reports the displacement of the quasi-posterior
center from the unpenalized minimizer of the same weighted criterion.
In the IV design, covariance weighting increases the displacement
relative to the identity weight, although it decreases steadily with
$N$; the results for $s=1$ and $s=\star$ are essentially identical.
In contrast, in the Poisson design covariance weighting substantially
reduces the displacement, and iteration to $s=\star$ produces a further
sizable reduction relative to $s=1$. Thus, as with the preceding
calibration results, iteration has little effect in the IV design
but is more consequential in the nonlinear Poisson design.
\begin{table}
\caption{Center-to-unpenalized displacement}
\label{tab:displace}
\medskip{}
\centering{}
\begin{tabular}{lcrrrr}
\hline
DGP & $s$ & \multicolumn{4}{c}{$N$}\tabularnewline
\cline{3-6}
& & 50 & 100 & 200 & 500\tabularnewline
\hline
\multirow{3}{*}{IV, $K=8$} & $0$ & 0.012 & 0.008 & 0.005 & 0.003\tabularnewline
& $1$ & 0.117 & 0.087 & 0.065 & 0.042\tabularnewline
& $\star$ & 0.117 & 0.087 & 0.065 & 0.042\tabularnewline
\hline
\multirow{3}{*}{Poisson} & $0$ & 1.317 & 1.599 & 1.920 & 2.505\tabularnewline
& $1$ & 0.357 & 0.283 & 0.208 & 0.138\tabularnewline
& $\star$ & 0.184 & 0.116 & 0.078 & 0.056\tabularnewline
\hline
\end{tabular}
\end{table}
Table~\ref{tab:pois-MC-error} shows that Monte Carlo error in the
Poisson quasi-posterior means is small relative to the statistical
scale. The 95th percentile of the worst-chain standardized MCSE ranges
from $0.020$ to $0.037$ across sample sizes, indicating that simulation
error is unlikely to materially affect the reported calibration results.
\begin{table}
\caption{Monte Carlo error in the quasi-posterior mean for Poisson regression}
\label{tab:pois-MC-error}
\medskip{}
\centering{}
\begin{tabular}{rr}
\hline
$N$ & \begin{cellvarwidth}[t]
\raggedleft
95th percentile of worst-chain
standardized MCSE
\end{cellvarwidth}\tabularnewline
\hline
50 & 0.020\tabularnewline
100 & 0.022\tabularnewline
200 & 0.026\tabularnewline
500 & 0.037\tabularnewline
\hline
\end{tabular}
\end{table}
Section~C of the Supplementary Material \citep{Tanaka2026c} reports
additional results for the IV design with $K=12$ and for the HSD
regression design. Across these designs, covariance weighting substantially
improves the calibration of the raw quasi-posterior relative to the
identity weight, and the discrepancy decreases as $N$ increases.
The one-step and converged procedures generally yield very similar
coverage, covariance calibration, interval lengths, and center-to-unpenalized
displacement, indicating little additional finite-sample gain from
iteration in these settings. As shown in Section~C of the Supplementary
Material \citep{Tanaka2026c}, CCQB converges rapidly and without
failures in all of these designs.
\section{Real Data Application}
\label{sec:application}
\subsection{Data and empirical design}
\label{subsec:wheeze-setup}
We study the age-specific association between baseline maternal smoking
status and childhood wheeze using the Ohio Children Wheeze Status
data from the Six Cities study \citep{Fitzmaurice1993}, available
in the R package \texttt{geepack}. The data contain four repeated
binary wheeze measurements for each of $537$ children, observed annually
from ages $7$ to $10$. We treat children as the independent sampling
units, so that $N=537$. Of the $537$ children, $187$ had mothers
classified as smokers in the first study year (age $7$); the observed
proportion with wheeze was higher in this group at each of the four
ages.
Let $Y_{it}$ indicate whether child $i$ wheezes at visit $t$, let
$\texttt{age}_{it}$ denote age centered at $9$ years, and let $\texttt{smoke}_{i}$
indicate maternal smoking status at baseline. We posit the marginal
mean model
\begin{align*}
p_{it}(\boldsymbol{\theta}) & =\Pr(Y_{it}=1\mid\texttt{age}_{it},\texttt{smoke}_{i})\\
& =\operatorname{logit}^{-1}(\theta_{1}+\theta_{2}\texttt{age}_{it}+\theta_{3}\texttt{smoke}_{i}+\theta_{4}\texttt{age}_{it}\texttt{smoke}_{i}),
\end{align*}
where $\boldsymbol{\theta}=(\theta_{1},\theta_{2},\theta_{3},\theta_{4})^{\top}$.
Our primary estimand is the age-specific smoking-wheeze odds ratio
\[
\operatorname{OR}(a)=\exp(\theta_{3}+\theta_{4}(a-9)),\qquad a\in\{7,8,9,10\}.
\]
Thus, $\exp(\theta_{3})$ is the smoking odds ratio at age $9$, while
$\exp(\theta_{4})$ is the multiplicative change in this odds ratio
for each one-year increase in age.
We use eight age-by-smoking residual moments. Let
\[
\mathcal{A}=\{-2,-1,0,1\},
\]
corresponding to ages 7 through 10. For $g\in\{0,1\}$ and $r\in\mathcal{A}$,
define
\[
m_{i,(g,r)}(\boldsymbol{\theta})=\mathbf{1}(\texttt{smoke}_{i}=g)\sum_{t=1}^{4}\mathbf{1}(\texttt{age}_{it}=r)\{Y_{it}-p_{it}(\boldsymbol{\theta})\}.
\]
Stacking these moments gives
\[
\boldsymbol{m}_{i}(\boldsymbol{\theta})=\{m_{i,(g,r)}(\boldsymbol{\theta}):g\in\{0,1\},r\in\mathcal{A}\}^{\top}\in\mathbb{R}^{8}.
\]
Hence $J=4$ and $K=8$, yielding an overidentified moment model that
fits a four-parameter marginal trajectory to the eight age-by-smoking
cell means. For a candidate calibration center $\boldsymbol{\vartheta}$,
$\widehat{\boldsymbol{C}}_{N}(\boldsymbol{\vartheta})$ is the empirical
covariance of the child-level moment vectors, and
\[
\boldsymbol{W}_{N}(\boldsymbol{\vartheta})=\widehat{\boldsymbol{C}}_{N}(\boldsymbol{\vartheta})^{-1}.
\]
Because the covariance is estimated across children using the full
child-level moment vector, within-child dependence among the four
repeated measurements is accommodated without specifying a working
correlation structure.
We place informative priors on the smoking association and its age
modification while leaving the baseline trajectory relatively diffuse.
Independently, we specify
\[
\theta_{3}\sim\mathcal{N}(0,0.5^{2}),
\]
which assigns approximately 95\% prior probability to
\[
0.38<\exp(\theta_{3})<2.66.
\]
For the interaction, we elicit the prior in terms of the change in
the smoking odds ratio over the observed age range:
\[
\theta_{4}\sim\mathcal{N}\left(0,\left\{ \frac{\log2}{3(1.96)}\right\} ^{2}\right).
\]
Since
\[
\frac{\operatorname{OR}(10)}{\operatorname{OR}(7)}=\exp(3\theta_{4}),
\]
this prior assigns approximately 95\% probability to
\[
\frac{1}{2}<\frac{\operatorname{OR}(10)}{\operatorname{OR}(7)}<2.
\]
Thus, the prior favors gradual age variation and places approximately
$95\%$ prior probability on changes in the smoking odds ratio of
no more than twofold in either direction between ages $7$ and $10$.
Finally, we use weakly informative proper priors
\[
\theta_{1},\theta_{2}\sim\mathcal{N}(0,5^{2}).
\]
We use the same MCMC implementation as for the Poisson regression
design in Section~\ref{sec:simulation}; computational details are
given in Section~B of the Supplementary Material \citep{Tanaka2026c}.
We first run an identity-weight pilot quasi-posterior, initializing
the MCMC chain at a draw from the prior. Each subsequent calibration
chain is initialized at the estimated quasi-posterior mean from the
preceding chain, with the covariance weight updated at that mean.
Each MCMC run uses $30{,}000$ iterations, of which the first $10{,}000$
are discarded as warm-up, and calibration is stopped using $\tau=0.05$.
\subsection{Results}
\label{subsec:wheeze-results}
Table~\ref{tab:wheeze-theta} reports the quasi-posterior summaries
for $\theta_{j}$ across the CCQB calibration stages. Replacing the
identity weight at $s=0$ by the covariance weight at $s=1$ changes
the quasi-posterior center most visibly for $\theta_{1}$, $\theta_{3}$,
and $\theta_{4}$, while the estimate of $\theta_{2}$ is comparatively
stable. The first update also substantially changes the reported uncertainty
for several parameters. In contrast, the estimates at $s=1$ and $s=\star$
are nearly indistinguishable: the largest change in a quasi-posterior
mean is only $0.007$, for $\theta_{3}$, and the corresponding intervals
change little. Thus, most of the effect of covariance weighting occurs
in the first calibration update; the second update produces only a
small further change while bringing the weighting matrix and quasi-posterior
center to within the prescribed tolerance of the self-consistent CCQB
solution.
\begin{table}
\caption{Quasi-posterior summaries for $\theta_{j}$ across CCQB calibration
stages}
\label{tab:wheeze-theta}
\medskip{}
\centering{}
\begin{tabular}{clccc}
\hline
Param. & & \multicolumn{3}{c}{\begin{cellvarwidth}[t]
\centering
Quasi-posterior mean
{[}$90\%$ interval{]}
\end{cellvarwidth}}\tabularnewline
\cline{3-5}
& & $s=0$ & $s=1$ & $s=\star$\tabularnewline
\hline
\multirow{3}{*}{$\theta_{1}$} & & -2.040 & -1.916 & -1.918\tabularnewline
& raw & {[}-2.732, -1.517{]} & {[}-2.109, -1.739{]} & {[}-2.116, -1.736{]}\tabularnewline
& adj & {[}-2.258, -1.821{]} & {[}-2.113, -1.719{]} & {[}-2.114, -1.721{]}\tabularnewline
\hline
\multirow{3}{*}{$\theta_{2}$} & & -0.141 & -0.135 & -0.135\tabularnewline
& raw & {[}-0.577, 0.261{]} & {[}-0.224, -0.048{]} & {[}-0.224, -0.047{]}\tabularnewline
& adj & {[}-0.248, -0.033{]} & {[}-0.232, -0.039{]} & {[}-0.232, -0.038{]}\tabularnewline
\hline
\multirow{3}{*}{$\theta_{3}$} & & 0.080 & 0.216 & 0.223\tabularnewline
& raw & {[}-0.590, 0.712{]} & {[}-0.089, 0.508{]} & {[}-0.081, 0.519{]}\tabularnewline
& adj & {[}-0.308, 0.468{]} & {[}-0.107, 0.539{]} & {[}-0.099, 0.545{]}\tabularnewline
\hline
\multirow{3}{*}{$\theta_{4}$} & & 0.001 & 0.046 & 0.047\tabularnewline
& raw & {[}-0.188, 0.189{]} & {[}-0.073, 0.164{]} & {[}-0.070, 0.164{]}\tabularnewline
& adj & {[}-0.177, 0.178{]} & {[}-0.106, 0.198{]} & {[}-0.104, 0.199{]}\tabularnewline
\hline
\end{tabular}
\end{table}
The raw and sandwich-adjusted intervals generally give similar uncertainty
summaries after calibration, although some differences remain, most
notably for $\theta_{4}$. These finite-sample differences can reflect
both residual differences between the quasi-posterior and sandwich
covariance estimates and differences in interval construction: the
raw intervals are quasi-posterior quantile intervals, whereas the
sandwich-adjusted intervals use a symmetric normal approximation.
Such differences are compatible with the first-order nature of the
calibration and sandwich approximation.
Table~\ref{tab:wheeze-or} translates the parameter estimates into
age-specific smoking-wheeze odds ratios. For the odds-ratio summaries,
raw intervals are empirical quantile intervals obtained by transforming
the retained quasi-posterior draws, whereas sandwich-adjusted intervals
use the delta-method variance based on $\widehat{\boldsymbol{\Sigma}}_{\mathrm{adj}}$.
Under the final calibrated quasi-posterior, the quasi-posterior mean
of $\operatorname{OR}(\texttt{age})$ increases from $1.160$ at age
$7$ to $1.340$ at age $10$, with intermediate estimates of $1.211$
and $1.271$ at ages $8$ and $9$, respectively. This increasing
pattern is more apparent after CCQB calibration: under the identity-weight
pilot, the estimated odds ratios are nearly constant across ages,
whereas the $s=1$ and $s=\star$ estimates are very similar and increase
steadily with age. Thus, as for the underlying parameters in Table~\ref{tab:wheeze-theta},
most of the change occurs in the first covariance-weight update, with
little additional change at the final update.
\begin{table}
\caption{Quasi-posterior summaries for $\operatorname{OR}(a)$ across CCQB
calibration stages}
\label{tab:wheeze-or}
\medskip{}
\centering{}
\begin{tabular}{clccc}
\hline
Age & & \multicolumn{3}{c}{\begin{cellvarwidth}[t]
\centering
Quasi-posterior mean
{[}$90\%$ interval{]}
\end{cellvarwidth}}\tabularnewline
\cline{3-5}
& & $s=0$ & $s=1$ & $s=\star$\tabularnewline
\hline
\multirow{3}{*}{7} & & 1.191 & 1.155 & 1.160\tabularnewline
& raw & {[}0.517, 2.194{]} & {[}0.806, 1.569{]} & {[}0.815, 1.573{]}\tabularnewline
& adj & {[}0.757, 1.624{]} & {[}0.747, 1.563{]} & {[}0.750, 1.570{]}\tabularnewline
\hline
\multirow{3}{*}{8} & & 1.173 & 1.204 & 1.211\tabularnewline
& raw & {[}0.551, 2.062{]} & {[}0.879, 1.578{]} & {[}0.888, 1.590{]}\tabularnewline
& adj & {[}0.792, 1.554{]} & {[}0.840, 1.568{]} & {[}0.846, 1.576{]}\tabularnewline
\hline
\multirow{3}{*}{9} & & 1.171 & 1.262 & 1.271\tabularnewline
& raw & {[}0.554, 2.038{]} & {[}0.915, 1.661{]} & {[}0.922, 1.681{]}\tabularnewline
& adj & {[}0.750, 1.591{]} & {[}0.861, 1.662{]} & {[}0.868, 1.673{]}\tabularnewline
\hline
\multirow{3}{*}{10} & & 1.183 & 1.329 & 1.340\tabularnewline
& raw & {[}0.535, 2.126{]} & {[}0.911, 1.827{]} & {[}0.920, 1.846{]}\tabularnewline
& adj & {[}0.653, 1.714{]} & {[}0.808, 1.849{]} & {[}0.816, 1.864{]}\tabularnewline
\hline
\end{tabular}
\end{table}
The raw and sandwich-adjusted intervals are broadly similar after
calibration, with the adjusted intervals somewhat wider, particularly
at the older ages. All 90\% intervals nevertheless include one. Thus,
although the point estimates suggest an association that increases
with age, neither uncertainty summary excludes an odds ratio of one
at any of the four ages.
From a computational perspective, the stopping criterion was satisfied
after two calibration updates, consistent with the rapid convergence
seen in the theory and simulations. The maximum standardized MCSE
across the calibration and final chains was $0.007$, substantially
smaller than the stopping criterion $\tau=0.05$.
Once the arbitrary pilot weighting is replaced by an empirically appropriate
covariance weight, self-consistency is inexpensive to enforce and,
in regular applications, may require only a small correction. The
first covariance-weight update accounts for most of the finite-sample
correction in this application, and only one additional calibration
update is required to reach the self-consistency criterion. Thus,
the application illustrates both the inferential importance of replacing
an arbitrary pilot weight and the rapid numerical stabilization of
CCQB.
\subsection{Sensitivity to moment rescaling}
\label{subsec:wheeze-ablation}
The application also provides a direct check of an invariance property
of covariance weighting. We deliberately rescale the last moment condition
by considering
\[
\boldsymbol{D}_{\varphi}=\operatorname{diag}(1,\dots,1,\varphi),\qquad\boldsymbol{m}_{i}^{(\varphi)}(\boldsymbol{\theta})=\boldsymbol{D}_{\varphi}\boldsymbol{m}_{i}(\boldsymbol{\theta}),\qquad\varphi\in\{0.01,1,100\}.
\]
All other aspects of the analysis are unchanged. With a nonsingular
diagonal rescaling $\boldsymbol{D}$, it follows that
\[
\widehat{\boldsymbol{C}}_{N}^{(\varphi)}(\boldsymbol{\vartheta})=\boldsymbol{D}_{\varphi}\widehat{\boldsymbol{C}}_{N}(\boldsymbol{\vartheta})\boldsymbol{D}_{\varphi}^{\top},
\]
so that
\[
\left\{ \widehat{\boldsymbol{C}}_{N}^{(\varphi)}(\boldsymbol{\vartheta})\right\} ^{-1}=\boldsymbol{D}_{\varphi}^{-\top}\widehat{\boldsymbol{C}}_{N}(\boldsymbol{\vartheta})^{-1}\boldsymbol{D}_{\varphi}^{-1}.
\]
Consequently,
\[
\bar{\boldsymbol{m}}_{N}^{(\varphi)}(\boldsymbol{\theta})^{\top}\left\{ \widehat{\boldsymbol{C}}_{N}^{(\varphi)}(\boldsymbol{\vartheta})\right\} ^{-1}\bar{\boldsymbol{m}}_{N}^{(\varphi)}(\boldsymbol{\theta})=\bar{\boldsymbol{m}}_{N}(\boldsymbol{\theta})^{\top}\widehat{\boldsymbol{C}}_{N}(\boldsymbol{\vartheta})^{-1}\bar{\boldsymbol{m}}_{N}(\boldsymbol{\theta}).
\]
This identity holds exactly for every candidate center $\boldsymbol{\vartheta}$.
Consequently, the entire ideal calibration map $\mathcal{T}_{N}(\boldsymbol{\vartheta})$
is invariant under such rescaling. Different intermediate iterates
can arise only because the identity-weight pilot produces a different
starting point; the exact fixed point itself is invariant, apart from
MCMC and numerical error. By contrast, a fixed weight that does not
adjust for the rescaling, such as the identity weight used in the
pilot, changes the relative contribution of the moment conditions
and can therefore affect the quasi-posterior, particularly in the
presence of an informative prior. Because the pilot center itself
depends on $\varphi$, intermediate CCQB iterates need not be exactly
identical across scalings. This ablation study examines how rapidly
CCQB removes this initialization-induced dependence.
Table~\ref{tab:wheeze-ablation} reports the results for $\theta_{2}$.
At the pilot stage $s=0$, the quasi-posterior depends noticeably
on the scaling of the last moment condition: in particular, the mean
shifts from about $-0.14$ for $\varphi\in\{0.01,1\}$ to $-0.066$
for $\varphi=100$, with corresponding changes in the reported intervals.
After the first covariance-weight update, however, both the estimates
and intervals are nearly identical across the three values of $\varphi$,
and this invariance persists at $s=\star$.
\begin{table}
\caption{Quasi-posterior summaries for $\theta_{2}$ across CCQB calibration
stages}
\label{tab:wheeze-ablation}
\medskip{}
\begin{centering}
\begin{tabular}{clccc}
\hline
$\varphi$ & & \multicolumn{3}{c}{\begin{cellvarwidth}[t]
\centering
Quasi-posterior mean
{[}$90\%$ interval{]}
\end{cellvarwidth}}\tabularnewline
\cline{3-5}
& & $s=0$ & $s=1$ & $s=\star$\tabularnewline
\hline
\multirow{3}{*}{0.01} & & -0.138 & -0.135 & -0.135\tabularnewline
& raw & {[}-0.606, 0.288{]} & {[}-0.221, -0.051{]} & {[}-0.223, -0.048{]}\tabularnewline
& adj & {[}-0.247, -0.030{]} & {[}-0.232, -0.038{]} & {[}-0.231, -0.038{]}\tabularnewline
\hline
\multirow{3}{*}{1} & & -0.141 & -0.135 & -0.135\tabularnewline
& raw & {[}-0.577, 0.261{]} & {[}-0.224, -0.048{]} & {[}-0.224, -0.047{]}\tabularnewline
& adj & {[}-0.248, -0.033{]} & {[}-0.232, -0.039{]} & {[}-0.232, -0.038{]}\tabularnewline
\hline
\multirow{3}{*}{100} & & -0.066 & -0.136 & -0.135\tabularnewline
& raw & {[}-0.338, 0.250{]} & {[}-0.226, -0.049{]} & {[}-0.223, -0.047{]}\tabularnewline
& adj & {[}-0.165, 0.033{]} & {[}-0.233, -0.039{]} & {[}-0.231, -0.038{]}\tabularnewline
\hline
\end{tabular}
\par\end{centering}
\end{table}
The results for the remaining parameters, reported in Table~D.1 in
Section~D of the Supplementary Material \citep{Tanaka2026c}, show
the same general pattern. The arbitrary scaling affects the pilot
quasi-posterior to varying degrees across parameters, including substantial
changes for $\theta_{3}$ and $\theta_{4}$, whereas the results at
$s=1$ and $s=\star$ are essentially unchanged across $\varphi$.
Thus, after one covariance-weight update, CCQB has already removed
nearly all of the dependence induced by the arbitrary scaling of an
individual moment condition; the converged solution is effectively
invariant under that scaling.
\section{Conclusion}
\label{sec:conclusion}
This paper introduced curvature-calibrated quasi-Bayesian (CCQB) updating
for inference based on moment conditions. The central idea is that
the weighting matrix in a quadratic moment quasi-posterior determines
not only the relative weighting of the moments but also the information
scale on which the data are combined with the prior. CCQB addresses
this issue by estimating the covariance of the moment conditions at
a self-consistent quasi-posterior center and using its inverse as
the fixed weight in the final quasi-posterior.
Under regularity conditions, the calibration map is locally contractive,
the calibrated weight converges to the inverse long-run covariance
of the moments, and the resulting quasi-posterior and its center share
the same first-order covariance given by the inverse Godambe information
matrix. The simulations show that covariance weighting can substantially
improve uncertainty calibration, while typically requiring only a
small number of updates to reach self-consistency. The application
illustrates how the proposed procedure can be used with real data.
These results suggest that curvature calibration provides a practical
way to retain prior-based quasi-Bayesian inference while placing moment-based
empirical information on an appropriate local scale.
Several directions for future research remain. On the theoretical
side, it would be useful to extend the analysis to settings with misspecified
moment conditions and to singular systems of moments, neither of which
is covered by the present theory. On the applied side, further exploration
of machine-learning applications appears particularly promising, since
black-box models are naturally compatible with a moment-based approach
that does not require an explicit likelihood.
\section*{Funding}
This work was supported by JSPS KAKENHI Grant Number 25K21168.