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You've Got to be Efficient: Ambiguity, Misspecification and Variational Preferences
\title{You've Got to be Efficient: \\
Ambiguity, Misspecification and Variational Preferences}
\author{Karun Adusumilli$^\dagger$}
\begin{abstract}
This article introduces a framework for evaluating statistical decisions
under both prior ambiguity and likelihood misspecification. We begin
with an ambiguity set --- a frequentist model that pairs a possibly
misspecified likelihood with every possible prior --- and uniformly
expand it by a Kullback--Leibler radius to accommodate likelihood
misspecification. We show that optimal decisions under this framework
are equivalent to minimax decisions with an exponentially tilted loss
function. Misspecification manifests as an exponential tilting of
the loss, while ambiguity corresponds to a search for the least favorable
prior. This separation between ambiguity and misspecification enables
local asymptotic analysis under global misspecification, achieved
by localizing the priors alone. Remarkably, for both estimation and
treatment assignment, we show that optimal decisions coincide with
those under correct specification, regardless of the degree of misspecification.
These results extend to semi-parametric models. As a practical consequence,
our findings imply that practitioners should prefer maximum likelihood
over the simulated method of moments, and efficient GMM estimators
--- such as two-step GMM --- over diagonally weighted alternatives.
\end{abstract}
\thanks{\textit{This version}: \today{}\\
\thispagestyle{empty}I would like to thank Xu Cheng, Frank Diebold,
Wayne Gao, George Mailath, and seminar participants at the University
of Pennsylvania for valuable discussions and comments that substantially
improved this article.\\
$^\dagger$Department of Economics, University of Pennsylvania}
\maketitle
\section{Introduction \label{sec:Introduction}}
\citet{box1976science} famously observed that all models are wrong,
since they are necessarily approximations of reality. Any researcher
or decision-maker who relies on a statistical model to learn about
a parameter of interest must therefore contend with the possibility
that the likelihood is misspecified. At the same time, researchers
are often unable or unwilling to commit to a single prior over the
parameter. In practice, then, decision-makers confront both prior
ambiguity and likelihood misspecification.
This article introduces a framework for evaluating statistical decisions
under both sources of concern. Following Bayesian practice, we define
a statistical model as a joint distribution comprising a prior and
a likelihood. We argue that both components are necessary because
they capture fundamentally different types of uncertainty. The prior
encodes epistemic uncertainty --- subjective uncertainty arising
from incomplete knowledge about the parameter of interest --- while
the likelihood captures aleatoric uncertainty --- the objective randomness
inherent in any statistical experiment.
To account for prior ambiguity, we define an ambiguity set: a frequentist
model that pairs a possibly misspecified likelihood with every possible
prior. Following \citet{cerreia2026making}, we then uniformly expand
this set by a Kullback--Leibler radius to accommodate likelihood
misspecification. The optimal decision rule is defined as the one
that achieves the lowest expected loss under the worst-case model
from this expanded set.
We show that optimal decisions under this formulation are equivalent
to minimax decisions with an exponentially tilted loss function. Likelihood
misspecification manifests as an exponential tilting of the loss,
while prior ambiguity corresponds to a search for the least favorable
prior. Our framework thus enables a clean separation between ambiguity
and misspecification. Furthermore, when there is no fear misspecification,
the optimal decisions reduce to the standard \citet{wald1950statistical}
formulation of minimax decisions under ambiguity alone --- the formulation
underlying most of frequentist analysis.
This separation between ambiguity and misspecification also enables
us to develop a local asymptotic theory under global misspecification,
achieved by localizing the priors around a reference parameter. Under
mild conditions, the finite-sample likelihoods, which may themselves
be misspecified, can be replaced by a limit experiment involving the
Gaussian family as the reference likelihood. While a substantial literature
studies local asymptotics under local misspecification, where misspecification
typically manifests as an added bias in the Gaussian limit, our framework
permits the Gaussian family itself to be globally misspecified in
the limit experiment, thereby accommodating much richer classes of
misspecification.
Local asymptotics also simplifies the search for optimal decisions,
as these are considerably easier to characterize in the limit experiment.
Quite remarkably, we find that for estimation and treatment assignment
problems, optimal decisions coincide with those under correct specification,
regardless of the degree of global misspecification. For these problems,
it is therefore always optimal for the decision-maker to proceed as
if the likelihood were correctly specified and select the resulting
optimal decision rule. Intuitively, these results arise because misspecification
under our formulation is symmetric around the reference Gaussian likelihood.
Since the estimation and treatment-assignment loss functions are also
symmetric around the parameter of interest, any estimator that is
not efficient under correct specification would break this symmetry.
Because nature chooses the least favorable likelihood specification
given the decision-maker's choice of estimator, departures from symmetry
necessarily incur higher decision risk.
We extend our local asymptotic theory to semi-parametric models, and
show that the above results on optimal decisions apply to that setting
as well.
Our findings have a number of practical consequences. In applications,
researchers often employ inefficient estimators over efficient ones,
a practice frequently justified on the grounds that under misspecification,
no estimand recovers the precise parameter of economic interest (\citealp{andrews2025purpose}).
Our results, however, suggest that this reasoning is incomplete. While
the parameter of interest cannot be recovered with certainty under
misspecification, our decision-theoretic analysis shows that efficient
estimators under correct specification also deliver the lowest decision
risk under arbitrary misspecification. In the case of parametric models,
these results suggest that practitioners should prefer maximum likelihood
over the simulated method of moments, irrespective of the degree of
misspecification. Similarly, in the context of GMM, practitioners
should prefer efficient estimation methods, such as two-step GMM,
over diagonally weighted or inefficient alternatives. Misspecification
concerns alone cannot justify the use of the inefficient estimators
over parametrically or semi-parametrically efficient alternatives.
\subsection{Related literature}
This article relates to an extensive literature on ambiguity and misspecification
spanning economics, statistics, and computer science. A detailed comparison
of our approach with alternative decision-theoretic frameworks is
deferred to Section \ref{subsec:Alternative-approaches}. Here, we
restrict ourselves to a broad survey of the literature on ambiguity
and misspecification.
The analysis of optimal decisions under prior ambiguity originates
with \citet{wald1950statistical}. A substantial body of work in statistics
has extended this framework to local asymptotics; we refer to \citet{ibragimov1981problem,le1986asymptotic,van1996weak,van2000asymptotic}
for textbook treatments. A central result from this literature is
that semi-parametrically efficient estimators are asymptotically minimax
optimal under prior ambiguity.
The literature on misspecification is equally extensive. \citet{huber1964robust}
proposes a contamination model to address likelihood misspecification.
\citet{hansen2011robustness} develop an approach that involves selecting
a worst-case likelihood from an ambiguity set defined by surrounding
a reference or approximate likelihood with a Kullback--Leibler divergence
ball of finite radius. The related field of Distributionally Robust
Optimization (DRO) takes the reference distribution to be the empirical
distribution $\hat{\mathbb{P}}$ of the data in the experiment, and
employs more general measures of distance from $\hat{\mathbb{P}}$
to define ambiguity sets --- including $\phi$-divergence measures
(e.g., reverse KL divergence and total-variation distance), Wasserstein
distances and Levy-Prokhorov distances. We refer to \citet{Ben-Tal:RobustOptimization}
for a textbook treatment of DRO, and to \citet{rahimian2019distributionally},
\citet{kuhn2025distributionally} for recent surveys. These methods
do not account for prior ambiguity and, consequently, do not reduce
to the standard minimax formulation that underpins frequentist analysis
in the absence of misspecification concerns. Our approach instead
follows the recent work of \citet{cerreia2026making} by first defining
an ambiguity set to address prior ambiguity and then uniformly expanding
this entire set by a KL divergence radius to accommodate likelihood
misspecification. In contrast to the results from DRO, we find that
optimal estimation and treatment assignment are invariant to the degree
of misspecification.
This article adopts a decision-theoretic approach to ambiguity and
misspecification, which is closely related to the literature in economic
theory on variational preferences, as expounded in \citet{maccheroni2006ambiguity}
and \citet{cerreia2026making}. The econometrics literature has also
studied alternative, non-decision-theoretic approaches to misspecification,
and we refer to \citet{armstrong2025misspecification} for a recent
survey. For instance, \citet{white1982maximum} defines pseudo-parameters
as the probability limits of estimators, and views them as suitably
defined approximations to the underlying parameter of interest. The
partial identification approach of \citet{manski2003partial} proposes
set-identifying the parameter under misspecification, while \citet{masten2021salvaging}
develop methods for sensitivity analysis. A limitation of these approaches
relative to the decision-theoretic framework is that they do not directly
identify the optimal statistical decision that a decision-maker should
employ.
\section{Decision-Making Under Ambiguity and Misspecification\label{sec:Diffusion-asymptotics-and}}
\subsection{An illustrative example\label{subsec:An-illustrative-example}}
To illustrate our formalism, we introduce the following running example.
A decision-maker, Alice, is tasked with determining whether a drug
should be approved for use in the US population. She is therefore
interested in learning about the parameter $\theta\in\Theta$, defined
as the average population treatment effect. We assume binary outcomes,
so that the population outcome distribution is $\text{Bernoulli}(\theta)$.
To assess the drug's efficacy, the pharmaceutical company has conducted
a randomized controlled trial with $n$ observations. Given the observed
data $\bm{x}\in\mathcal{X}$ from the trial, Alice seeks to choose
a decision $\delta:\mathcal{X}\to\mathcal{A}$ so as to maximize her
utility $u_{n}(\theta,\delta)$, or equivalently, minimize the loss
function $l_{n}(\theta,\delta)=-u_{n}(\theta,\delta)$. Examples of
loss functions include the estimation loss,
\[
l_{n}(\theta,\delta)=\ell\bigl(\sqrt{n}(\theta-\delta)\bigr),\quad\delta\in\Theta,
\]
for some bowl-shaped function $\ell(\cdot)$, e.g., $\ell(z)=z^{2}$
for mean squared error, and the treatment-assignment loss,
\[
l_{n}(\theta,\delta)=\sqrt{n}\bigl(\theta\,\mathbf{1}\{\theta\geq0\}-\theta\,\delta\bigr),\quad\delta\in\{0,1\}.
\]
Under the estimation loss, the goal is to learn directly about the
parameter $\theta$, whereas under the treatment-assignment loss,
the goal is to either approve ($\delta=1$) or reject ($\delta=0$)
the drug for use in the entire population.
Unfortunately for Alice, the trial was conducted exclusively in the
state of Pennsylvania. Because the drug is novel, she has no formal
basis for judging whether, or to what degree, treatment effects observed
in Pennsylvania are representative of those in the broader US population.
This gives rise to model misspecification concerns. At the same time,
Alice also faces ambiguity concerns, as she is unable to form an initial
prior over $\theta$. We now describe a formalism that accommodates
both.
\subsection{Bayesian, Frequentist and Misspecified models\label{subsec:Bayesian,-Frequentist-and-misspecified}}
\subsubsection{(Bayesian) Statistical models}
We begin by formally defining the notion of a statistical model in
the absence of ambiguity or misspecification concerns.
Since the loss function takes the form $l_{n}(\theta,\delta(\bm{x}))$,
the payoff-relevant state of the world is given by $\omega=(\theta,\bm{x})$:
an oracle who knows $\omega$ would recover Alice's loss with certainty.
Following the framework of Savage or Anscombe--Aumann (\citealp{anscombe1963definition}),
we define a model $m\equiv m(\theta,\bm{x})$ as a probability distribution
over the payoff-relevant state $\omega=(\theta,\bm{x})$. This distribution
admits a natural decomposition into a prior and a likelihood:
\begin{equation}
m(\theta,\bm{x})=\pi(\theta)\otimes p_{\theta}(\bm{x}).\label{eq:Bayesian_model}
\end{equation}
Here, $\pi(\theta)$ denotes the posited prior, the marginal distribution
over $\theta$, while $p_{\theta}(\bm{x})=p(\bm{x}\mid\theta)$ denotes
the posited likelihood, the conditional distribution of $\bm{x}$
given $\theta$. Equation (\ref{eq:Bayesian_model}) is nothing more
than the definition of a Bayesian statistical model; see \citet[Definition 1.2.1]{robert2007bayesian}.
The decomposition of a model into a prior and a likelihood is a canonical
feature of Bayesian decision-making. Given the importance of this
decomposition for what follows, it is worth understanding why both
components are necessary. As we argue below, they capture fundamentally
different sources of uncertainty: epistemic and aleatoric. In the
terminology of \citet{anscombe1963definition}, these correspond to
the uncertainties involved in horse gambles and roulette wheels.
Epistemic uncertainty refers to uncertainty arising from a lack of
knowledge --- uncertainty that can, in principle, be reduced through
the acquisition of additional data or evidence. In the Anscombe--Aumann
framework, this is the uncertainty of a horse gamble. Because the
parameter $\theta$ enters Alice's loss function directly, it is natural
to regard it as a quantity that exists in principle but that Alice
does not know. The prior $\pi(\theta)$ thus encodes Alice's epistemic
uncertainty due to her imperfect knowledge of $\theta$. Crucially,
Alice can conceptualize $\theta$ independently of the likelihood.
She may, for instance, have access to prior information --- such
as data from related studies --- that enables her to form a prior
$\pi$ without reference to whatever experiment the pharmaceutical
company may have conducted.
Aleatoric uncertainty, by contrast, refers to inherent randomness,
which is implicit in the design of any statistical experiment. The
likelihood $p_{\theta}(\bm{x})$ captures precisely this source of
uncertainty. In conducting the trial, the pharmaceutical company presumably
drew a random sample of $n$ observations from the population of Pennsylvania.
This sampling procedure requires the use of an implicit or explicit
random number generator and therefore introduces genuine randomness;
in the Anscombe--Aumann framework, this is uncertainty generated
by roulette wheels. The likelihood thus describes the distribution
of the data $\bm{x}$ induced by this randomness, for any given value
of $\theta$.
Importantly, in our framework, the likelihood does not rise to the
status of a model. It provides only a mapping from the parameter $\theta$
to the distribution of $\bm{x}$. Because $\theta$ enters Alice's
loss function directly, knowledge of the correct likelihood would
not enable Alice to obtain a probabilistic forecast of her loss, as
she would still face epistemic uncertainty over $\theta$.
\subsubsection{Models with prior ambiguity, aka Frequentist models\label{subsec:Models-with-prior-ambiguity}}
We now incorporate prior ambiguity into our framework. Suppose that
Alice is unable to form a single prior, perhaps because she is ambiguity-averse
in the sense of \citet{maccheroni2006ambiguity}. Instead, she posits
a structured set of models,
\[
\mathcal{Q}:=\bigl\{\pi(\theta)\otimes p_{\theta}(\bm{x}):\pi\in\Delta(\Theta)\bigr\},
\]
where $\Delta(\Theta)$ denotes the set of all probability distributions
over $\theta$, while continuing to treat the likelihood $p_{\theta}(\bm{x})$
as correctly specified. In the spirit of \citet{wald1950statistical},
Alice could then choose the decision rule that performs best against
the worst-case model in $\mathcal{Q}$ --- effectively, the one associated
with the least favorable prior --- thereby guarding against prior
ambiguity:
\[
\delta_{n,f}^{*}:=\operatorname*{arg\,min}_{\delta}\left[\sup_{m\in\mathcal{Q}}\mathbb{E}_{m}\bigl[l_{n}(\theta,\delta)\bigr]\right].
\]
Because the Wald approach underpins much of frequentist analysis,
we refer to $\mathcal{Q}$ as a frequentist model and to $\delta_{n,f}^{*}$
as a frequentist (or minimax) decision rule.
\subsubsection{Models with prior ambiguity and likelihood misspecification\label{subsec:Ambiguity_and_miss}}
Now suppose that Alice entertains the possibility that the likelihood
$p_{\theta}(x)$ employed in her frequentist model $\mathcal{Q}$
may not be correctly specified. Likelihood misspecification can arise
in two distinct ways. The first is misspecification of functional
form, e.g., specifying a Gaussian likelihood when the true data-generating
process follows a $t$-distribution. In our running example, however,
the outcomes are Bernoulli, so the likelihood is necessarily a product
of Bernoulli distributions and functional-form misspecification is
not a concern. In fact, as we show in Section 2.3, functional-form
misspecification can always be ameliorated by making the likelihood
sufficiently flexible. The misspecification that Alice confronts here
is of the second kind: the link between the parameter of interest
$\theta$ and the distribution over $x$ may be incorrectly specified.
In the running example, the true outcome distribution is $Y\sim\text{Bernoulli}(\theta_{P})$,
where $\theta_{P}$ is the average treatment effect in Pennsylvania.
Likelihood misspecification arises because, in general, $\theta\neq\theta_{P}$.
To accommodate misspecification concerns, we follow \citet{cerreia2026making}
and place a protective belt around the frequentist model $\mathcal{Q}$.
Formally, we posit a set of unstructured models of the form
\[
\mathcal{M}=\Bigl\{ m\in\Delta(\Theta,\mathcal{X}):\min_{q\in\mathcal{Q}}\,R_{q}(m)\leq K\Bigr\},
\]
where $R_{q}(m)=\text{KL}(m\,\|\,q)$ denotes the Kullback--Leibler
divergence and $K$ is a constant reflecting Alice's degree of ambiguity
aversion. Alice could then choose the decision rule that performs
best against the worst-case model in $\mathcal{M}$, thereby guarding
against both prior ambiguity and misspecification:
\[
\delta_{n}^{*}:=\operatorname*{arg\,min}_{\delta}\left[\sup_{m\in\mathcal{M}}\mathbb{E}_{m}\bigl[l_{n}(\theta,\delta)\bigr]\right].
\]
For the remainder of this article, we decompose any generic model
$m$ as $m(\theta,\bm{x})=\pi(\theta)\otimes m_{\theta}(\bm{x})$,
and reserve the notation $p_{\theta}(\bm{x})$ for a reference likelihood
specification, which may itself be misspecified. Define $R_{\mathcal{Q}}(m):=\min_{q\in\mathcal{Q}}R_{q}(m)$.
Straightforward algebra yields
\begin{equation}
R_{\mathcal{Q}}(m)=\int\text{KL}\bigl(m_{\theta}(\cdot)\,\|\,p_{\theta}(\cdot)\bigr)\,d\pi(\theta),\label{eq:characterization_of_R_Q}
\end{equation}
so that the set of unstructured models can be equivalently written
as
\[
\mathcal{M}=\left\{ \pi(\theta)\otimes m_{\theta}(x)\;:\;\int\text{KL}\bigl(m_{\theta}(\cdot)\,\|\,p_{\theta}(\cdot)\bigr)\,d\pi(\theta)\leq K,\;\;\pi\in\Delta(\Theta)\right\} .
\]
The class of unstructured models thus comprises every possible prior
$\pi$, paired with all likelihoods $m_{\theta}(\cdot)$ satisfying
the integrated KL constraint.
Note that $\mathcal{M}$ expands $\mathcal{Q}$ by adding a protective
radius of KL divergence of size $K$. Because $\mathcal{Q}$ already
accommodates unrestricted prior ambiguity, this protective belt serves
entirely to guard against likelihood misspecification. Within the
class of alternative models, however, the prior and likelihood may
interact in subtle ways: the constraint permits $m_{\theta}(\cdot)$
to deviate substantially from $p_{\theta}(\cdot)$ for certain values
of $\theta$, provided the associated prior $\pi$ places low weight
on those values.
To further understand this interaction between the prior and likelihood,
it is instructive to see how likelihood misspecification would be
modeled in the absence of prior ambiguity. If Alice were able to commit
to a single prior $\pi$, then $\mathcal{Q}$ would be a singleton
and $\mathcal{M}$ would consist of all models $\pi(\theta)\otimes m_{\theta}(\bm{x})$
such that $m_{\theta}(\cdot)\in\mathcal{M}_{\bm{x}|\theta}(\pi)$,
where
\[
\mathcal{M}_{\bm{x}|\theta}(\pi):=\left\{ m_{\theta}(\cdot):\int\text{KL}\bigl(m_{\theta}(\cdot)\,\|\,p_{\theta}(\cdot)\bigr)\,d\pi(\theta)\leq K\right\} .
\]
When the prior is fixed, the class of candidate likelihoods thus depends
on $\pi$; the prior shapes which deviations from the reference likelihood
are admissible. In the general case with unrestricted priors, $\mathcal{M}$
can be interpreted as the union of $\pi(\theta)\otimes\mathcal{M}_{\bm{x}|\theta}(\pi)$
over all $\pi\in\Delta(\Theta)$.
\subsection{Nuisance and structural parameters\label{subsec:Nuisance-and-structural-parameters}}
A nuisance parameter is an unknown quantity that enters the likelihood
but does not affect the loss function. In our running example, if
the outcomes are distributed as $\mathcal{N}(\mu,\sigma^{2})$ but
Alice is solely interested in learning about the mean treatment effect
$\mu$, then $\sigma^{2}$ is a nuisance parameter. Nuisance parameters
make the likelihood more flexible and can be used to ameliorate functional-form
misspecification; indeed, one can allow for nonparametric specifications
by making the nuisance parameters infinite-dimensional. As noted earlier,
however, nuisance parameters cannot address the second type of misspecification
concerning the link between the parameter of interest and the data.
Even if Alice were to adopt a fully nonparametric specification of
the outcome distribution, she would still face misspecification concerns,
as treatment effects in Pennsylvania may differ fundamentally from
those in the broader US population.
In contrast, we refer to the parameters that enter the utility function
directly as structural parameters. In what follows, $\theta$ denotes
the full collection of unknown parameters, which may include both
structural and nuisance components. The structural parameters are
modeled as known functions $\mu(\theta)$ of $\theta$. With this
notation, the loss functions introduced earlier take the more general
form
\[
l_{n}(\theta,\delta)=\ell\bigl(\sqrt{n}\,(\mu(\theta)-\delta)\bigr),\quad\delta\in\mu(\Theta),
\]
for estimation loss, and
\[
l_{n}(\theta,\delta)=\sqrt{n}\bigl(\mu(\theta)\,\mathbf{1}\{\mu(\theta)\geq0\}-\mu(\theta)\,\delta\bigr),\quad\delta\in\{0,1\},
\]
for treatment-assignment loss.
The definitions of Bayesian, frequentist, and misspecified models
remain unchanged; the introduction of nuisance parameters affects
only the form of the loss functions.
\subsection{Alternative approaches to misspecification: A comparison\label{subsec:Alternative-approaches}}
\citet{andrews2025purpose} define an econometric model $(\theta,p_{\theta}(\cdot))$
as a combination of the parameter and the likelihood.\footnote{In the terminology of \citet{andrews2025purpose}, the likelihood
is referred to as a data-generating process.} Apart from the prior over $\theta$, this coincides with the definition
of a Bayesian statistical model. Introducing the prior allows us to
account for prior ambiguity, which, as we have seen, plays a key role
even in the standard frequentist approach.
A growing recent literature has considered accounting for misspecification
through partial identification; see, e.g., \citet{ishihara2021evidence,yata2021optimal,christensen2022optimal,montiel2026decision}.
This literature supposes that the parameter of interest $\theta$
lies within a bounded distance, $d(\theta,\theta_{P})\leq L$, of
an identifiable parameter $\theta_{P}$. In our running example, this
would require Alice to assume that the population treatment effect
$\theta$ differs from the treatment effect in Pennsylvania $\theta_{P}$
by at most $L$. While bounds of this form can arise naturally in
a number of applications, the approach falls short as a general framework
for misspecification for several reasons.
First, unlike our formalism, bounding the parameters directly lacks
an axiomatic justification. Second, the approach is sensitive to the
choice of the distance measure $d(\cdot)$, which in turn makes $L$
difficult to calibrate. In the Bernoulli setting, for instance, it
would not be reasonable to use Euclidean distance $d(\theta,\theta_{P})=|\theta-\theta_{P}|$,
as it does not respect the constraint $\theta\in[0,1]$. Third, and
most importantly, imposing a uniform bound $d(\theta,\theta_{P})\leq L$
for all $\theta_{P}$ implies comparisons across different values
of $\theta_{P}$ that may be at odds with the decision-maker's actual
preferences over ambiguity and misspecification. To see why, note
that there is always epistemic uncertainty over the value of $\theta_{P}$.
There is no a priori reason to believe that the bound $d(\theta,\theta_{P})\leq L$
provides the same degree of protection against misspecification when
$\theta_{P}=0.9$ as when $\theta_{P}=0.1$. Depending on Alice's
preferences, e.g., her loss function, she may be less concerned about
misspecification at high values of $\theta_{P}$ (which suggests the
treatment is highly effective) than at low values. The constraint
$d(\theta,\theta_{P})\leq L$ is not directly linked to her attitudes
toward misspecification; it is a constraint on parameters, not on
payoff-relevant quantities.
In more closely related work, \citet{andrews2020informativeness},
\citet{bonhomme2022minimizing} and \citet{christensen2023counterfactual}
characterize misspecification through a statistical distance $d(m_{\theta}(\cdot),p_{\theta}(\cdot))$,
e.g., KL divergence, over likelihoods. The specific setups and goals
of these works differ substantially from our own.\footnote{\citet{andrews2020informativeness} study the relationship between
descriptive statistics and structural parameters. \citet{bonhomme2022minimizing}
analyze estimation under local misspecification, i.e., when $d(m_{\theta}(\cdot),p_{\theta}(\cdot))\to0$.
\citet{christensen2023counterfactual} study partial identification
of $\theta$. These goals are all distinct from ours: devising optimal
decisions under global ambiguity and misspecification.} Quite apart from this, however, a uniform bound on the KL divergence
over likelihoods, of the form $\sup_{\theta}d(m_{\theta}(\cdot),p_{\theta}(\cdot))\leq L$,
is subject to the same criticism as a bound $d(\theta,\theta_{P})\leq L$
over parameters: there is no a priori reason to believe that it provides
the same degree of protection against misspecification across different
values of $\theta$. As before, Alice may be less concerned about
misspecification at high values of $\theta$ than at low values. Furthermore,
quantities such as KL divergence are sensitive to the choice of the
reference measure $p_{\theta}(\cdot)$, and KL divergences evaluated
at different parameter values such as $R_{p_{\theta_{1}}}(m_{\theta_{1}})$
and $R_{p_{\theta_{2}}}(m_{\theta_{2}})$ are not directly comparable.
Our formulation avoids this problem because we first postulate an
infinite-dimensional ambiguity set $\mathcal{Q}$ and then uniformly
expand it by a KL radius $K$. As in \citet{cerreia2026making}, the
value of $K$ can be tied to the decision-maker's underlying preferences
over ambiguity and misspecification. But rather remarkably, as it
turns out, our optimal decisions are asymptotically independent of
the choice of $K$.
\section{Characterizing Optimal Decisions\label{sec:Characterizing-Optimal-Decisions}}
We now characterize optimal decisions under ambiguity and misspecification.
For reasons that will become apparent shortly, it is convenient to
start with utility maximization rather than loss minimization. Following
the framework described in Section \ref{subsec:Ambiguity_and_miss},
the optimal decision rule takes the form:
\[
\delta_{n}^{*}:=\operatorname*{arg\,max}_{\delta}\left[\inf_{m\in\mathcal{M}}\,\mathbb{E}_{m}\bigl[u_{n}(\theta,\delta)\bigr]\right]=\operatorname*{arg\,max}_{\delta}\,\inf_{m}\,\left\{ \mathbb{E}_{m}\bigl[u_{n}(\theta,\delta)\bigr]:R_{\mathcal{Q}}(m)\le K\right\} .
\]
The function $R_{\mathcal{Q}}(\cdot):\Delta(\Theta\times\mathcal{X})\to\mathbb{R}$
is strictly convex, so we may apply a minimax theorem to show that
for each $K$ there exists a unique multiplier $\lambda$ such that
\[
V_{n}(\delta)=\inf_{m}\left\{ \mathbb{E}_{m}\bigl[u_{n}(\theta,\delta)\bigr]+\lambda\,R_{\mathcal{Q}}(m)\right\} .
\]
Following \citet{cerreia2026making}, we refer to $V_{n}(\cdot)$
as the variational decision criterion.
Recalling the definition $R_{\mathcal{Q}}(m):=\min_{q\in\mathcal{Q}}R_{q}(m)$
and interchanging the the order of the $\min_{q\in\mathcal{Q}}$ and
$\inf_{m}$ operations, we can write
\[
V_{n}(\delta)=\min_{q\in\mathcal{Q}}\inf_{m}\left\{ \mathbb{E}_{m}\bigl[u_{n}(\theta,\delta)\bigr]+\lambda\,R_{q}(m)\right\} .
\]
The Donsker-Varadhan variational formula yields
\begin{equation}
\inf_{m}\left\{ \mathbb{E}_{m}\bigl[u_{n}(\theta,\delta)\bigr]+\lambda\,R_{q}(m)\right\} =-\lambda\ln\mathbb{E}_{q}\!\left[e^{-u_{n}(\theta,\delta)/\lambda}\right].\label{eq:DV formula}
\end{equation}
Converting back to the loss function via $l_{n}(\theta,\delta)=-u_{n}(\theta,\delta)$,
we obtain
\begin{equation}
V_{n}(\delta)=-\lambda\ln\left\{ \max_{q\in\mathcal{Q}}\,\mathbb{E}_{q}\left[e^{l_{n}(\theta,\delta)/\lambda}\right]\right\} ,\label{eq:decision-risk-eq}
\end{equation}
and consequently, the optimal decision can be characterized as
\begin{equation}
\delta_{n}^{*}=\operatorname*{arg\,min}_{\delta}\,\max_{q\in\mathcal{Q}}\,\mathbb{E}_{q}\!\left[e^{l_{n}(\theta,\delta)/\lambda}\right].\label{eq:optimal_decision_Q}
\end{equation}
So far, the calculations above follow \citet{cerreia2026making}.
However, due to the special structure of $\mathcal{Q}$ in our setting---comprising
all possible priors paired with the reference likelihood $p_{\theta}(x)$---we
can simplify (\ref{eq:optimal_decision_Q}) further:
\begin{equation}
\delta_{n}^{*}=\operatorname*{arg\,min}_{\delta}\,\max_{\pi\in\Delta(\Theta)}\int\mathbb{E}_{p(\bm{x}|\theta)}\!\left[e^{l_{n}(\theta,\delta)/\lambda}\right]d\pi(\theta).\label{eq:optimal_decision_pi}
\end{equation}
In this expression, the quantity $R_{n}(\theta,\delta):=\mathbb{E}_{p(\bm{x}|\theta)}\!\left[e^{l_{n}(\theta,\delta)/\lambda}\right]$
admits a natural interpretation as the frequentist risk of the decision
rule $\delta$ under the reference likelihood $p(x|\theta)$, evaluated
with respect to the exponentiated loss $e^{l_{n}(\theta,\delta)/\lambda}$.
Notice that (\ref{eq:optimal_decision_pi}) corresponds to a standard
minimax decision framework under exponentiated loss: we can interpret
the optimal decision as the result of a two-player game in which nature
chooses the least favorable prior while the decision-maker chooses
the optimal rule. Equation (\ref{eq:optimal_decision_pi}) is therefore
a key result of this article. It establishes that optimal decisions
under both ambiguity and misspecification are equivalent to optimal
decisions under ambiguity alone, but with an exponentiated loss function.
The result also reveals how the two sources of concern separate naturally.
The effect of misspecification is to transform the loss function into
an exponentiated version; intuitively, the decision-maker magnifies
the impact of large losses while attenuating the impact of small ones.
The effect of ambiguity, as in the setting without misspecification,
manifests in the search for the least favorable prior.
\section{Local Asymptotics with Global Misspecification}
It is rarely feasible to solve the minimax problem (\ref{eq:optimal_decision_pi})
exactly in finite samples. Instead, as is standard even in classical
frequentist (i.e., minimax) settings, we turn to local asymptotic
approximations.
Following the usual approach, we fix a reference parameter $\theta_{0}$
and consider local perturbations of the form $\theta_{0}+h/\sqrt{n}$.
Priors $\pi(\theta)$ over $\theta$ are then mapped to local priors
$\pi(h)$ over $h$.
In the case of treatment-assignment loss, $l_{n}(\theta,\delta)=n\bigl(\mu(\theta)\,\mathbf{1}\{\mu(\theta)\geq0\}-\mu(\theta)\,\delta\bigr)$,
local asymptotics arise naturally from the global minimax problem
(\ref{eq:optimal_decision_pi}). The key observation is that the least
favorable prior concentrates its mass on regions where the treatment
effect $\mu(\theta)$ is of order $1/\sqrt{n}$. When $\mu(\theta)$
is of a higher order of magnitude than $1/\sqrt{n}$, determining
the optimal assignment is asymptotically trivial. Conversely, when
$\mu(\theta)$ is of a lower order of magnitude than $1/\sqrt{n}$,
the difference between treatment and status quo is negligible, so
the loss is close to zero regardless of the choice of $\delta$. It
is therefore natural to choose a reference parameter $\theta_{0}$
satisfying $\mu(\theta_{0})=0$, since the least favorable prior would
concentrate around this value in any case. As \citet{hirano2009asymptotics}
showed, these same considerations apply to treatment-assignment problems
in the absence of misspecification as well.
But how should one choose a reference $\theta_{0}$ for estimation
loss, and what is the meaning of local asymptotics in this setting?
We offer two interpretations.
The first is that local asymptotics amounts to localizing prior ambiguity
around a reference parameter $\theta_{0}$ that the decision-maker
believes is close to the true value. In our running example, Alice
may have a priori reason to believe that the true treatment effect
lies near $\theta_{0}$, even if she is uncertain about its exact
value. It would then be natural to restrict her ambiguity set to a
$1/\sqrt{n}$ neighborhood of $\theta_{0}$. Because this prior information
is obtained independently of the data, likelihood misspecification
does not affect the choice of $\theta_{0}$. Consequently, we can
localize the priors, even as we can --- and do --- allow for global
misspecification of the likelihood.
Under the second interpretation, the choice of the reference $\theta_{0}$
is itself subject to adversarial optimization. The basic idea, following
\citet{ibragimov1981problem}, is to decompose the global minimax
problem into two stages: first, fix a reference $\theta_{0}$ and
evaluate the local minimax performance of a decision rule $\delta$
against local alternatives of the form $\theta_{0}+h/\sqrt{n}$; then,
in an outer step, select the least favorable reference $\theta_{0}$.
Here, we focus primarily on a theoretical development of the first
interpretation. The second interpretation is detailed in Section \ref{subsec:Non-local-priors},
while the theory is developed in Appendix \ref{sec:Local-Asymptotics-with-non-local-prior}.
\subsection{Parametric models: Setup}
We assume that the data consists of an i.i.d collection of outcomes
$\ensuremath{\bm{x}:=\{Y_{i}^ {}\}_{i=1}^{n}}$. Under the reference
likelihood, $Y_{i}$ is distributed as $P_{\theta}$. Let $\nu$ denote
a dominating measure for $\{P_{\theta}:\theta\in\mathbb{R}^{d}\}$,
and set $p_{\theta}^ {}:=dP_{\theta}^ {}/d\nu$. We require the reference
class of likelihoods, $\{P_{\theta}^ {}\}_{\theta}$, to be quadratic
mean differentiable (qmd):
\begin{asm1} The class $\{P_{\theta}^ {}:\theta\in\mathbb{R}^{d}\}$
is qmd around $\theta_{0}^ {}$, i.e., there exists a score function
$\psi_{}(\cdot)$ such that for each $h^ {}\in\mathbb{R}^{d},$
\[
\int\left[\sqrt{p_{\theta_{0}^ {}+h^ {}}}-\sqrt{p_{\theta_{0}}}-\frac{1}{2}h^{\intercal}\psi_{}\sqrt{p_{\theta_{0}^ {}}}\right]^{2}d\nu=o(\vert h^ {}\vert^{2}).
\]
Furthermore, the information matrix $I_{0}:=\mathbb{E}_{\theta_{0}}[\psi\psi_{}^{\intercal}]$
is invertible. \end{asm1}
In the illustrative example, the outcomes are modeled as Bernoulli,
so Assumption 1 holds with $\psi_{}(y)=\left(\theta_{0}^ {}(1-\theta_{0}^ {})\right)^{-1}(y-\theta_{0}^ {})$.
More broadly, this assumption is rather mild and satisfied for almost
all commonly used likelihood models, including the Normal, Cauchy,
Exponential, and Poisson distributions. It is important to bear in
mind that Assumption 1 constrains the reference class of likelihoods,
$p_{\theta}$, not the actual likelihoods, which are unknown.
We also assume that the function $\mu(\theta)$, which maps $\theta$
to structural parameters, satisfies a mild differentiability condition:
\begin{asm2} There exists $\dot{\mu}_{0}\in\mathbb{R}^{d}$ and $\epsilon_{n}\to0$
independent of $h$ such that $\sqrt{n}\left(\mu(\theta_{0}+h/\sqrt{n})-\mu(\theta_{0})\right)=\dot{\mu}_{0}^{\intercal}h+\epsilon_{n}\vert h\vert^{2}$
for all bounded $h$. \end{asm2}
Let $P_{n,h}$ denote the joint probability measure over the iid $Y_{1},\dots,Y_{n}$
when each $Y_{i}\sim P_{\theta_{0}+h/\sqrt{n}}$, and let $\mathbb{E}_{n,h}[\cdot]$
denote the corresponding expectation. Under local asymptotics, the
minimal risk attained under the minimax problem (\ref{eq:optimal_decision_pi})
can be written as:
\begin{equation}
V_{n}^{*}=\min_{\delta}\,\max_{\pi(h)}\int\mathbb{E}_{n,h}\left[e^{l_{n}(\theta_{0}+h/\sqrt{n},\delta)/\lambda}\right]d\pi(h).\label{eq:minimax_value_local_approx}
\end{equation}
\subsection{Limit approximations and the Gaussian limit experiment\label{subsec:Limit-approximations}}
Define the standardized score statistic as
\[
x_{n}=\frac{I_{0}^{-1/2}}{\sqrt{n}}\sum_{i=1}^{n}\psi(Y_{i}).
\]
It is well known, see e.g., \citet[Chapter 7]{van2000asymptotic},
that quadratic mean differentiability (Assumption 1) implies $\mathbb{E}_{n,0}[\psi(Y_{i})]=0$.
Then, by the central limit theorem,
\begin{equation}
x_{n}\xrightarrow[P_{n,0}]{d}x\sim\mathcal{N}(0,I).\label{eq:Convergence of score process}
\end{equation}
Assumption 1 also implies the important property of Local Asymptotic
Normality (LAN; \citealp[Chapter 7]{van2000asymptotic}):
\begin{equation}
\ln\frac{dP_{n,\theta_{0}+h/\sqrt{n}}}{dP_{n,\theta_{0}}}=h^{\intercal}I_{0}^{1/2}x_{n}-\frac{1}{2}h^{\intercal}I_{0}h+o_{P_{n,0}}(1),\ \textrm{uniformly over bounded }h.\label{eq:LAN property}
\end{equation}
Consider now a limit experiment in which the decision-maker observes
a $d$-dimensional signal $x$, posited to be drawn from a reference
Gaussian likelihood, $P_{h}(x)\sim\mathcal{N}(I_{0}^{-1/2}h,I)$.
By the properties of the Gaussian distribution,
\[
\ln\frac{dP_{h}}{dP_{0}}=h^{\intercal}I_{0}^{1/2}x_{}-\frac{1}{2}h^{\intercal}I_{0}h.
\]
It follows from (\ref{eq:Convergence of score process}) and (\ref{eq:LAN property})
that the reference likelihood ratios in the finite-sample experiment
converge to their counterparts in the limit experiment:
\[
\ln\frac{dP_{n,\theta_{0}+h/\sqrt{n}}}{dP_{n,\theta_{0}}}\xrightarrow[P_{n,0}]{d}\ln\frac{dP_{h}}{dP_{0}},\ \textrm{for each }h.
\]
Furthermore, Assumption 2 implies that the loss functions admit asymptotic
approximations. For estimation-loss, defining $\tilde{\delta}_{n}=\sqrt{n}\dot{\mu}_{0}^{\intercal}(\delta_{n}-\theta_{0})$
and assuming $\tilde{\delta}_{n}$ has a weak limit $\tilde{\delta}$,
we have
\begin{equation}
l_{n}(\theta_{0}+h/\sqrt{n},\delta_{n})\equiv\ell\left(\sqrt{n}\left(\mu(\theta_{0}+h/\sqrt{n})-\delta_{n}\right)\right)\rightsquigarrow\ell(\dot{\mu}_{0}^{\intercal}h-\tilde{\delta}),\label{eq:estimation_loss_approx}
\end{equation}
where $`$$\rightsquigarrow$' represents weak convergence. For treatment-assignment
loss, since the reference parameter satisfies $\mu(\theta_{0})=0$,
Assumption 2 implies
\begin{equation}
l_{n}(\theta_{0}+h/\sqrt{n},a)\to\dot{\mu}_{0}^{\intercal}h\,\mathbf{1}\{\dot{\mu}_{0}^{\intercal}h\geq0\}-(\dot{\mu}_{0}^{\intercal}h)a,\ \textrm{uniformly over }a\in\{0,1\}\textrm{ and bounded }h.\label{eq:treatment_loss_approx}
\end{equation}
Convergence of likelihood ratios implies asymptotic equivalence between
the actual and limit experiments in the sense of \citet{le1986asymptotic}.
Combined with the loss function approximations above, this suggests
that the minimax value $V_{n}^{*}$ in (\ref{eq:minimax_value_local_approx})
should converge to the minimax value $V^{*}$ in the limit experiment,
where
\begin{align}
V^{*} & :=\min_{\tilde{\delta}}\,\max_{\pi(h)}\int\mathbb{E}_{h}\left[e^{l(h,\tilde{\delta})/\lambda}\right]d\pi(h),\ \textrm{with}\label{eq:minimax_value_limit_experiment}\\
l(h,\tilde{\delta}) & =\begin{cases}
\ell(\dot{\mu}_{0}^{\intercal}h-\tilde{\delta}) & \textrm{for estimation loss},\\
\dot{\mu}_{0}^{\intercal}h\,\left\{ \mathbf{1}\{\dot{\mu}_{0}^{\intercal}h\geq0\}-\tilde{\delta}\right\} & \textrm{for treatment-assignment loss}.
\end{cases}\nonumber
\end{align}
Formal statements to this effect are provided in Section \ref{subsec:Formal-results-parametric}.
It is instructive to compare our asymptotic approach with the more
traditional analysis of local asymptotics under local misspecification.
In locally misspecified models, the KL divergence between the true
and reference likelihoods is assumed to decline at a $1/n$ rate.
Consequently, as highlighted in \citet{andrews2020informativeness},
\citet{bonhomme2022minimizing} and \citet{muller2024locally}, local
misspecification manifests as asymptotic bias in the Gaussian limit
experiment. Our framework differs fundamentally since it permits the
Gaussian likelihood approximation itself to be globally misspecified.
This is possible because our asymptotic theory approximates only the
reference finite-sample likelihood ratios with Gaussian likelihoods;
it makes no claim about the convergence of the true likelihood ratios.
Global misspecification consequently manifests not as bias in the
Gaussian limit, but as an exponential tilting of the loss function.
Equation (\ref{eq:minimax_value_limit_experiment}) suggests that
asymptotically optimal decision rules can be derived by solving the
minimax problem in the limit experiment and mapping the solutions
back to the finite-sample setting. Since optimal decisions are considerably
easier to characterize under Gaussian likelihoods, this reduction
illustrates the key benefit of the local asymptotic approach.
\subsection{Characterization of optimal decisions in the limit experiment\label{subsec:Characterization-of-optimal-decisions}}
We begin with estimation-loss. Since $\ell(\cdot)$ is bowl-shaped,
so is $e^{\ell(\cdot)/\lambda}$. It then follows from Anderson's
lemma, see e.g., \citet[Proposition 8.6]{van2000asymptotic}, that
the minimax-optimal estimator in the limit experiment --- the solution
to (\ref{eq:minimax_value_limit_experiment}) --- is simply
\[
\tilde{\delta}^{*}=\dot{\mu}_{0}^{\intercal}I_{0}^{-1/2}x.
\]
Remarkably, $\tilde{\delta}^{*}$ is independent of $\lambda$, which
governs the degree of misspecification. In fact, $\tilde{\delta}^{*}$
coincides with most efficient estimator, the maximum likelihood estimator,
under correct misspecification, which corresponds to $\lambda=\infty$.
In other words, the optimal estimator under ambiguity and misspecification
is identical to the optimal estimator under ambiguity alone.
For treatment-assignment loss, Anderson's lemma does not apply. Nevertheless,
as the following proposition shows, the optimal decision rule again
takes a simple form: it recommends treatment whenever the MLE of the
treatment effect under correct specification is positive.
\begin{prop}\label{prop1}The minimax-optimal decision rule in the
limit experiment under the treatment assignment loss is $\tilde{\delta}^{*}=\mathbf{1}\{\dot{\mu}_{0}^{\intercal}I_{0}^{-1/2}x\geq0\}$.
The corresponding least-favorable prior is a symmetric two-point prior
supported on $(-h^{*},h^{*})$, with $h^{*}:=\frac{\Delta^{*}}{\dot{\mu}^{\intercal}I_{0}^{-1}\dot{\mu}}I_{0}^{-1}\dot{\mu}$,
and
\[
\Delta^{*}=\arg\max_{\Delta\ge0}\left\{ \left(e^{\frac{\Delta}{\lambda}}-1\right)\Phi(-\Delta)\right\} .
\]
\end{prop}
As with the optimal estimator, the optimal treatment-assignment rule
is independent of the degree of misspecification.
Intuitively, these results arise because misspecification under our
formulation is unstructured and therefore symmetric around the reference
Gaussian likelihood. Since the loss functions are also symmetric around
the reference $\theta_{0}$, any estimator that is not efficient under
correct specification would break this symmetry. Because nature chooses
the least favorable likelihood specification given the decision-maker's
choice of estimator, departures from symmetry would necessarily incur
higher decision risk. It is therefore always optimal for the decision-maker
to proceed as if the likelihood were correctly specified and select
the resulting optimal decision rule.
\subsection{Formal results\label{subsec:Formal-results-parametric}}
We now formally establish the asymptotic equivalence of experiments
through two results. First, we show that the minimax value $V^{*}$
in the limit experiment forms an asymptotic lower bound on the sequence
of optimal decision risks in the finite-sample experiments. Second,
we show that plug-in versions of the optimal limit-experiment decision
rules $\tilde{\delta}^{*}$ -- obtained by replacing $I_{0}^{-1/2}x$
with the finite sample MLE $\hat{\theta}_{\textrm{mle}}$ -- are
asymptotically optimal, in the sense that their decision-risks converge
to $V^{*}$. Specifically, we argue that the asymptotically decisions
are given by
\begin{equation}
\hat{\delta}_{n}^{*}=\begin{cases}
\mu(\hat{\theta}_{\textrm{mle}}) & \textrm{for estimation,}\\
\mathbf{1}\left\{ \mu(\hat{\theta}_{\textrm{mle}})\geq0\right\} & \textrm{for treatment assignment.}
\end{cases}\label{eq:asymptotically_optimal_decisions}
\end{equation}
As discussed at the beginning of this section our formal results require
localization of ambiguity. This involves restricting attention to
the set of compactly supported priors $\Delta_{M}(\mathcal{H})\equiv\{\pi(h):\textrm{supp}(\pi)\in[-M,M]\}$
for some $M<\infty$. This restriction is not needed for the lower
bound, but plays a role in establishing that the bound is attained
by the plug-in rules.
\begin{thm}\label{thm:lower_bound}(Lower bound) Suppose that Assumptions
1 and 2 hold. Then, under both the estimation and treatment-assignment
loss functions,
\[
\liminf_{n\to\infty}\,\min_{\delta}\,\max_{\pi(h)\in\Delta_{M}(\mathcal{H})}\int\mathbb{E}_{n,h}\left[e^{l_{n}(\theta_{0}+h/\sqrt{n},\delta)/\lambda}\right]d\pi(h)\ge V^{*}.
\]
\end{thm}
We place a mild regularity condition on the MLE:
\begin{asm3} The maximum-likelihood estimator $\hat{\theta}_{\textrm{mle}}$
admits a locally linear score-function approximation:
\[
\hat{\theta}_{\textrm{mle}}-\theta_{0}=I_{0}^{-1/2}x_{n}+o_{P_{n,0}}(1).
\]
\end{asm3}
To avoid technical issues relating to the existence of moments for
the estimation problem, our theory also requires that $\ell(\cdot)$
be bounded. We state this as an additional assumption.
\begin{asm4} The function $\ell(\cdot)$ is bounded.\end{asm4}
Assumption 4 implies the estimation loss $l_{n}(\theta,\delta)$ is
bounded. As for the treatment-assignment loss, since we work with
compact priors, it is automatically bounded.
\begin{thm}\label{thm:upper_bound}(Asymptotic optimality of plug-in
rules) Suppose that Assumptions 1-4 hold. Then, under both the estimation
and treatment-assignment loss functions,
\[
\lim_{M\to\infty}\limsup_{n\to\infty}\max_{\pi(h)\in\Delta_{M}(\mathcal{H})}\int\mathbb{E}_{n,h}\left[e^{l_{n}(\theta_{0}+h/\sqrt{n},\hat{\delta}_{n}^{*})/\lambda}\right]d\pi(h)=V^{*}.
\]
\end{thm}
In fact, the MLE can be replaced by any asymptotically efficient estimator
satisfying Assumption 3. All such estimators attain the same minimax
lower bound $V^{*}$; our theory therefore does not distinguish among
them.
The statements of Theorems \ref{thm:lower_bound} and \ref{thm:upper_bound}
are new even in the absence of misspecification. Standard local asymptotic
minimax theorems are typically stated in terms of a maximum over discrete
sets of $h$ values, rather than the $\max_{\pi(h)\in\Delta_{M}(\mathcal{H})}$
operation employed here, see, e.g., \citet[Proposition 8.11]{van2000asymptotic}.
This distinction is consequential for our setting, as the discrete
formulation does not lend itself to a natural interpretation of prior
ambiguity. Our results avoid this limitation through a different method
of proof that directly accommodates optimization over a compact space
of local priors.
\subsection{Non-local priors\label{subsec:Non-local-priors}}
The formal results above were derived under priors localized around
a reference parameter $\theta_{0}$. As noted earlier, with a slight
strengthening of the assumptions, we can allow for global priors over
a compact set $\Theta$; essentially, we require the assumptions to
be valid uniformly over $\theta_{0}\in\Theta$. In this case, the
lower bounds on decision risk correspond to the least favorable choice
of the reference.
Formally, we show that under both loss functions,
\begin{align*}
& \liminf_{n\to\infty}\,\min_{\delta}\,\max_{\pi(\theta)\in\Delta(\Theta)}\int\mathbb{E}_{n,\theta}\left[e^{l_{n}(\theta,\delta)/\lambda}\right]d\pi(\theta)\\
& \ge\begin{cases}
\sup_{\theta_{0}\in\Theta}V_{\theta_{0}}^{*} & \textrm{for estimation loss},\\
\sup_{\{\theta_{0}\in\Theta:\mu(\theta_{0})=0\}}V_{\theta_{0}}^{*} & \textrm{for treatment-assignment loss},
\end{cases}
\end{align*}
where $V_{\theta_{0}}^{*}$ represents the minimal decision-risk in
the limit experiment (as defined in Section \ref{subsec:Limit-approximations})
evaluated at a given reference parameter $\theta_{0}$. The formal
statement, together with the required assumptions, is provided in
Appendix \ref{sec:Local-Asymptotics-with-non-local-prior}.
In the same appendix, we show that the decisions in (\ref{eq:asymptotically_optimal_decisions})
are asymptotically optimal under global priors as well, in that they
attain this bound:
\begin{align*}
& \lim_{K\to\infty}\,\limsup_{n\to\infty}\,\max_{\pi(\theta)\in\Delta(\Theta)}\int\mathbb{E}_{n,\theta}\left[e^{l_{n,K}(\theta,\delta)/\lambda}\right]d\pi(\theta)\\
& =\begin{cases}
\sup_{\theta_{0}\in\Theta}V_{\theta_{0}}^{*} & \textrm{for estimation loss},\\
\sup_{\{\theta_{0}\in\Theta:\mu(\theta_{0})=0\}}V_{\theta_{0}}^{*} & \textrm{for treatment-assignment loss},
\end{cases}
\end{align*}
where $l_{n,K}(\cdot):=K\wedge l_{n}(\cdot)$ denotes the loss function
truncated at level $K$.
To the best of our knowledge, these results appear new even in the
context of no misspecification concerns. A number of authors, e.g.,
\citet[Section 3.2]{hirano2025wald}, have raised concerns about the
interpretation of asymptotic analysis under local reparameterizations.
Our results address these concerns by showing that global minimax
analysis is equivalent to local asymptotic analysis with a least favorable
choice of reference.
\subsection{Application: Maximum likelihood vs Simulated method of moments}
As noted by \citet{andrews2025purpose}, researchers in applied work
often prefer the simulated method of moments --- which involves subjective
selection of moment functions --- over maximum likelihood, despite
the latter's superior efficiency. This preference is frequently justified
by the argument that ``with misspecification concerns, moment estimators
are often more reliable\textquotedbl{} (\citealp{bordalo2020overreaction}).
Our results suggest, however, that this reasoning is incomplete. Under
our framework, there is no tradeoff between misspecification robustness
and efficiency. Efficient estimators are misspecification-robust to
an arbitrary degree, and inefficient estimators remain suboptimal
even in the presence of potential misspecification. Misspecification
concerns alone therefore cannot justify the use of the simulated method
of moments over maximum likelihood.
This is not to say that the simulated method of moments should never
be used. The selection of moment functions can often be understood
as a form of model selection. For instance, if the parameter of interest
is the treatment effect on a subgroup of the population, it may be
reasonable to selectively overweight the relevant portion of the sample.
This, however, is a problem of model selection rather than model misspecification
per se.
\section{Semi-parametric Models\label{sec:Semi-parametric-Models}}
The preceding sections focused on parametric models for the likelihood.
In many applications, however, the exact distributional form is left
unspecified, motivating the use of semi-parametric models. As discussed
in \ref{subsec:Nuisance-and-structural-parameters}, our methodology
extends naturally to the semi-parametric and nonparametric settings
by allowing for infinite-dimensional nuisance parameters.
In semi-parametric models, the structural parameter is typically a
regular functional $\mu:=\mu(P)$, of the unknown population distribution
$P$. Common examples of regular functionals include the mean, median,
and quantiles. For simplicity, we assume that $\mu$ is scalar-valued.
The loss functions are then
\[
l_{n}(P,\delta)=\ell\bigl(\sqrt{n}\,(\mu(P)-\delta)\bigr),
\]
for estimation loss, and
\[
l_{n}(P,\delta)=\sqrt{n}\bigl(\mu(P)\,\mathbf{1}\{\mu(P)\geq0\}-\mu(P)\,\delta\bigr),\quad\delta\in\{0,1\},
\]
for treatment-assignment loss. The population distribution $P$ plays
the same role as $\theta$ in the parametric analysis. While $P$
is unknown, we are only interested in its scalar functional $\mu(P)$;
the remaining features of $P$ are treated as an infinite dimensional
nuisance parameter.
As in Section \ref{subsec:Bayesian,-Frequentist-and-misspecified},
we consider a framework in which the decision-maker confronts both
prior ambiguity --- an inability to form a single prior over the
parameter $P$ --- and model misspecification --- the concern that
the population distribution $P$ may not coincide with the outcome
distribution $\hat{P}$ in the experimental sample. In our running
example, Alice may adopt a fully nonparametric specification of the
outcome distribution yet still face misspecification concerns, as
treatment effects in Pennsylvania could differ fundamentally from
those in the broader US population. More generally, misspecification
also arises when the semi-parametric model imposes restrictions that
are not satisfied by the combination of the true $\mu$ and the sample
outcome distribution $\hat{P}$. For instance, in the GMM framework,
the researcher specifies a moment condition $\mathbb{E}_{P}[m(Y_{i},\mu)]=0$,
where $m(\cdot)\in\mathbb{R}^{p}$ is a known vector of moment functions,
$\mu\in\mathbb{R}^{d}$ is the structural parameter of interest, and
$P$ is the unknown population distribution. Misspecification arises
if this restriction does not hold in the sample, i.e., $\mathbb{E}_{\hat{P}}[m(Y_{i},\mu)]\neq0$
for the true $\mu$. In our running example, which fits within the
GMM framework with $m(Y_{i},\mu)=Y_{i}-\mu$, Alice may be concerned
that $\mathbb{E}_{\hat{P}}[Y_{i}-\mu]\neq0$.
The central message of our semi-parametric results is that the sample
analogs of the optimal decisions characterized in Section \ref{subsec:Characterization-of-optimal-decisions}
remain optimal in the semi-parametric setting. In essence, one replaces
the score statistic from the parametric setting with an efficient
influence function process associated with the functional of interest.
For example, if the goal is to conduct inference on the mean, one
replaces $x_{n}$ with the cumulative sum of outcomes $n^{-1/2}\sum_{i=1}^{n}(Y_{i}/\sigma)$,
where $\sigma^{2}:=\textrm{Var}[Y_{i}]$.
\subsection{Local asymptotics for semi-parametric models\label{subsec:local_asymptotics_nonparametrics}}
It is easiest to discuss ambiguity and misspecification in semi-parametric
settings using local asymptotics. Our local asymptotic analysis employs
the formalism of \citet[Section 25.3]{van2000asymptotic}. Let $\mathcal{P}$
denote the class of candidate population distributions with bounded
variance, dominated by some measure $\nu$. We fix a reference distribution
$P_{0}\in\mathcal{P}$, and surround it with various smooth one-dimensional
parametric sub-models, $\{P_{s,h}:s\le\zeta\}$ for some $\zeta>0$,
whose score function is $h(\cdot)$, and that pass through $P_{0}$
at $s=0$ (i.e., $P_{0,h}=P_{0}$). Formally, these sub-models satisfy
\begin{equation}
\int\left[\frac{dP_{s,h}^{1/2}-dP_{0}^{1/2}}{s}-\frac{1}{2}hdP_{0}^{1/2}\right]^{2}d\nu\to0\ \textrm{as}\ s\to0.\label{eq:qmd non-parametrics}
\end{equation}
As in the parametric setting, for the treatment-assignment problem,
we choose $P_{0}$ such that $\mu(P_{0})=0$.
By \citet[Lemma 25.14]{van2000asymptotic}, condition (\ref{eq:qmd non-parametrics})
implies $\int hdP_{0}=0$ and $\int h^{2}dP_{0}<\infty$. The set
of all such functions $h$ is termed the tangent space $T(P_{0})$,
which is a subset of the Hilbert space $L^{2}(P_{0})$ endowed with
the inner product $\left\langle f,g\right\rangle =\mathbb{E}_{P_{0}}[fg]$
and norm $\left\Vert f\right\Vert =\mathbb{E}_{P_{0}}[f^{2}]^{1/2}$.
For any $h\in T(P_{0})$, let $P_{n,h}$ denote the joint probability
measure over $Y_{1},\dots,Y_{n}$, when each $Y_{i}$ is an iid draw
from $P_{1/\sqrt{n},h}$, and let $\mathbb{E}_{n,h}[\cdot]$ denote
the corresponding expectation. An important consequence of (\ref{eq:qmd non-parametrics})
is the LAN property:
\begin{align}
\sum_{i=1}^{n}\ln\frac{dP_{1/\sqrt{n},h}}{dP_{0}}(Y_{i}) & =\frac{1}{\sqrt{n}}\sum_{i=1}^{n}h(Y_{i})-\frac{1}{2}\left\Vert h\right\Vert ^{2}+o_{P_{n,0}}(1),\textrm{ uniformly over bounded }\left\Vert h\right\Vert .\label{eq:SLAN nonparametric setting}
\end{align}
Let $\psi\in T(P_{0})$ denote the efficient influence function corresponding
to $\mu$, defined by the property that for any $h\in T(P_{0})$,
\begin{equation}
\frac{\mu(P_{s,h})-\mu(P_{0})}{s}-\left\langle \psi,h\right\rangle =o(s).\label{eq:influence function}
\end{equation}
Set $\sigma^{2}=\mathbb{E}_{P_{0}}[\psi^{2}]$. The semi-parametric
analogue of the score statistic in the semi-parametric setting is
the standardized efficient influence function process
\[
x_{n}:=\frac{\sigma^{-1}}{\sqrt{n}}\sum_{i=1}^{n}\psi(Y_{i}).
\]
Any element $h\in T(P_{0})$ admits the orthogonal decomposition $h=\left\langle \psi/\sigma,h\right\rangle \psi/\sigma+\tilde{h}$,
where $\tilde{h}$ is orthogonal to $\psi$ (i.e., $\left\langle \psi,\tilde{h}\right\rangle =0$).
The component $\mu=\left\langle \psi,h\right\rangle $ represents
the structural parameter, while $\tilde{h}$ represents an infinite
dimensional nuisance parameter. Although the full perturbation direction
$h$ is unknown, only the projection onto the efficient influence
function is relevant for learning about $\mu$.
For each $h\in T(P_{0})$, define $\mu_{n}(h):=\mu(P_{1/\sqrt{n},h})$,
and let $\bm{x}:=(Y_{1},\dots,Y_{n})$ denote the collection of outcomes.
We can rewrite the loss functions in terms of $h$ as
\[
l_{n}(h,\delta)=\begin{cases}
\ell\bigl(\sqrt{n}\,(\mu_{n}(h)-\delta)\bigr) & \textrm{for estimation loss},\\
\sqrt{n}\bigl(\mu_{n}(h)\,\mathbf{1}\{\mu_{n}(h)\geq0\}-\mu_{n}(h)\,\delta\bigr) & \textrm{for treatment-assignment loss}.
\end{cases}
\]
In the semi-parametric setting, a Bayesian statistical model, $m(h,\bm{x})$,
is defined as a joint probability distribution over both $h,\bm{x}$.
As in Section \ref{subsec:Bayesian,-Frequentist-and-misspecified},
it admits the decomposition
\[
m(h,\bm{x})=\pi(h)\otimes p_{n,h}(\bm{x}),
\]
where $\pi\in\Delta(T(P_{0}))$ denotes a prior over the tangent space
$T(P_{0})$, and $p_{n,h}(\bm{x}):=\prod_{i}dP_{1/\sqrt{n},h}(Y_{i})/d\nu$
represents the likelihood, a parametric sub-model. Prior ambiguity
is incorporated by defining a frequentist model, a structured set
of models, as
\[
\mathcal{Q}:=\bigl\{\pi(h)\otimes p_{n,h}(\bm{x}):\pi\in\Delta(T(P_{0}))\bigr\}.
\]
Finally, misspecification concerns are addressed by placing a protective
belt around $\mathcal{Q}$, yielding the set of unstructured models
\[
\mathcal{M}=\Bigl\{ m\in\Delta(T(P_{0}),\mathcal{X}):\min_{q\in\mathcal{Q}}\,R_{q}(m)\leq K\Bigr\}.
\]
Expanding the set of structured models allows us to account for the
possibility that the true distribution of the experimental data $\bm{x}$
is not captured by $p_{n,h}(\bm{x})$ for any $h$.
The decision-maker chooses the decision rule that performs best against
the worst-case model in $\mathcal{M}$, thereby guarding against both
prior ambiguity and misspecification:
\[
\delta_{n}^{*}:=\operatorname*{arg\,min}_{\delta}\left[\sup_{m\in\mathcal{M}}\mathbb{E}_{m}\bigl[l_{n}(h,\delta)\bigr]\right].
\]
As in Section \ref{sec:Characterizing-Optimal-Decisions}, applying
a Lagrangian formulation and the same sequence of calculations yields
the following characterization of minimal decision risk:
\begin{equation}
V_{n}^{*}=\min_{\delta}\,\max_{\pi\in\Delta(T(P_{0}))}\int\mathbb{E}_{n,h}\!\left[e^{l_{n}(h,\delta)/\lambda}\right]d\pi(h).\label{eq:minimax_value_non_parametrics}
\end{equation}
\subsection{Formal results: Semi-parametric models}
We impose the following regularity conditions throughout this section:
\begin{asm5}The sub-models $\{P_{s,h};h\in T(P_{0})\}$ satisfy (\ref{eq:qmd non-parametrics}).
Furthermore, they admit an efficient influence function, $\psi(\cdot)$,
for $\mu(P)$ such that
\[
\sqrt{n}\left(\mu(P_{1/\sqrt{n},h})-\mu_{0}\right)=\left\langle \psi,h\right\rangle +\epsilon_{n}\left\Vert h\right\Vert ^{2},
\]
where $\mu_{0}:=\mu(P_{0})$, and $\epsilon_{n}$ is independent
of $h$ for bounded $\left\Vert h\right\Vert $. \end{asm5}
The first part of Assumption 5 simply states the definition of parametric
sub-models. The second part of Assumption 5 slightly strengthens (\ref{eq:influence function}).
Since $L^{2}(P_{0})$ is a Hilbert space, it is possible to select
$\{\phi_{1},\phi_{2},\dots\}\in L^{2}(P_{0}^ {})$ in such a manner
that $\{\psi/\sigma,\phi_{1},\phi_{2},\dots\}$ is a set of orthonormal
basis functions for the closure of $T(P_{0}^ {})$; the division by
$\sigma_{}$ in the first component ensures $\left\Vert \psi/\sigma_{}\right\Vert ^{2}=1$.
We can also choose these bases so they lie in $T(P_{0}^ {})$, i.e.,
$\mathbb{E}_{P_{0}}[\phi_{j}]=0$ for all $j$. By the Hilbert space
isometry, each $h_{}\in T(P_{0}^ {})$ is then associated with an
element from the $l_{2}$ space of square integrable sequences, $(\mu/\sigma,\gamma_{1},\gamma_{2},\dots)$,
where $\mu=\left\langle \psi,h\right\rangle $ and $\gamma_{k}=\left\langle \phi_{k},h\right\rangle _{}$
for all $k\neq0$. Consequently, any prior $\pi(h)$ over $T(P_{0})$
can be represented as a prior over $l_{2}$.
As in Section \ref{subsec:Formal-results-parametric}, our formal
results require localization of ambiguity. This involves restricting
attention to priors that are supported on a compact subset, $K_{M}$,
of $l_{2}$, defined as
\[
K_{M}\equiv\left\{ h=\left(\mu/\sigma,\gamma_{1},\dots\right):\left\Vert h\right\Vert \le M,\ \lim_{J\to\infty}\sup_{(\gamma_{1},\gamma_{2}\dots)}\sum_{j=J}^{\infty}\vert\gamma_{j}\vert^{2}=0\right\} .
\]
The compactness condition essentially requires the set of candidate
$h$ to be sufficiently smooth.
\subsubsection{Lower bounds\label{subsec:Lower-bounds-nonparametrics}}
As in the parametric setting, we show that the minimax value $V^{*}$
in the limit experiment also forms an asymptotic lower bound on the
sequence of optimal decision risks, $V_{n}^{*}$, in the finite sample
semi-parametric experiments. However, since the previous definition
of the limit experiment used a different interpretation of $h$, we
will need to modify the construction slightly.
Specifically, we now consider a limit experiment where we observe
a one dimensional signal $x$, posited to be drawn from a reference
Gaussian likelihood, $P_{\mu}(x)\sim\mathcal{N}(\mu/\sigma,1)$. Let
$\mathbb{E}_{\mu}[\cdot]$ denote the expectation corresponding to
$P_{\mu}$. The minimax value $V^{*}$ in this limit experiment is
then defined as
\begin{align}
V^{*} & :=\min_{\tilde{\delta}}\,\max_{\rho(\mu)\in\Delta(\mathbb{R})}\int\mathbb{E}_{\mu}\left[e^{l(\mu,\tilde{\delta})/\lambda}\right]d\rho(\mu),\ \textrm{with}\label{eq:minimax_value_limit_experiment_nonparametrics}\\
l(\mu,\tilde{\delta}) & =\begin{cases}
\ell(\mu-\tilde{\delta}) & \textrm{for estimation loss},\\
\mu\,\left\{ \mathbf{1}\{\mu\ge0\}-\tilde{\delta}\right\} & \textrm{for treatment-assignment loss}.
\end{cases}\nonumber
\end{align}
It is straightforward to verify that the value of $V^{*}$ in (\ref{eq:minimax_value_limit_experiment_nonparametrics})
is, in fact, the same as that in (\ref{eq:minimax_value_limit_experiment})
when $\theta=\mu$ and $I_{0}=1/\sigma^{2}$.
\begin{thm}\label{thm:lower_bound-nonparametric}Suppose that Assumption
5 holds. Then, under both the estimation and treatment-assignment
loss functions,
\[
\liminf_{n\to\infty}\min_{\delta}\,\max_{\pi(h)\in\Delta(K_{M})}\int\mathbb{E}_{n,h}\left[e^{l_{n}(h,\delta)/\lambda}\right]d\pi(h)\ge V^{*}.
\]
\end{thm}
\subsubsection{Asymptotically optimal decisions}
As in the parametric setting, asymptotically optimal decisions under
ambiguity and misspecification are the same as those under ambiguity
along. Let $\hat{\mu}_{n}$ denote any semi-parametrically efficient
estimator for $\mu$, understood as satisfying the following assumption:
\begin{asm6} The estimator $\hat{\mu}$ attains the semi-parametric
efficiency bound, in that it admits a locally linear influence-function
approximation:
\[
\hat{\mu}_{n}-\mu_{0}=\sigma x_{n}+o_{P_{n,0}}(1).
\]
\end{asm6}
As we show below, asymptotically optimal decisions are then given
by
\[
\hat{\delta}_{n}^{*}=\begin{cases}
\hat{\mu}_{n} & \textrm{for estimation,}\\
\mathbf{1}\left\{ \hat{\mu_{n}}\geq0\right\} & \textrm{for treatment assignment.}
\end{cases}
\]
\begin{thm}\label{thm:upper_bound-non-parametric}Suppose that Assumptions
4-6 hold. Then, under both the estimation and treatment-assignment
loss functions,
\[
\lim_{M\to\infty}\limsup_{n\to\infty}\max_{\pi(h)\in\Delta(K_{M})}\int\mathbb{E}_{n,h}\left[e^{l_{n}(h,\hat{\delta}_{n}^{*})/\lambda}\right]d\pi(h)=V^{*}.
\]
\end{thm}
\subsection{Application: Optimal GMM estimators under misspecification}
The Generalized Method of Moments (GMM) is an example of a semi-parametric
model that is widely used in economic applications. Recall that in
the GMM framework, the researcher specifies a moment condition $\mathbb{E}_{P}[m(Y_{i},\mu)]=0$,
where $m(\cdot)\in\mathbb{R}^{p}$ is a known vector of moments, $\mu\in\mathbb{R}^{d}$
is the structural parameter, and $P$ is the population distribution.
When $p>d$, the GMM model is said to be over-identified. In this
setting, the efficient influence function is given by
\[
\psi(Y_{i})=G_{0}^{\intercal}\Omega_{0}^{-1}m(Y_{i},\mu_{0}),
\]
where $\mu_{0}:=\mu(P_{0})$ is the unique solution to $\mathbb{E}_{P_{0}}[m(Y_{i},\mu)]=0$
under the reference distribution $P_{0}$, $G_{0}:=\mathbb{E}_{P_{0}}[\nabla_{\mu}m(Y_{i},\mu)]$
and $\Omega_{0}:=\mathbb{E}_{P_{0}}[m(Y_{i},\mu_{0})m(Y_{i},\mu_{0})^{\intercal}]$.
In the absence of misspecification concerns, it is well known that
several estimators that are asymptotically efficient, including 2-step
GMM and continuously updated GMM (CUGMM), among others. In practice,
however, researchers are often concerned that the model may be misspecified,
i.e., $\mathbb{E}_{\hat{P}}[m(Y_{i},\mu)]\neq0$ under the true structural
parameter $\mu^ {}$ and the sample outcome distribution $\hat{P}$.
Under misspecification, different estimators converge to different
limits under the distribution of sample outcomes $\hat{P}$, and researchers
often resort to inefficient estimators employing a weighing matrix
other than the optimal $W^{*}=G_{0}^{\intercal}\Omega_{0}^{-1}$.
This practice is frequently justified on the grounds that ``...under
misspecification the two-step GMM estimator is no more efficient than
any other estimator... each weighting leads us to recover a different
parameter'' (\citealp{andrews2025purpose}).
Our results suggest, however, that this reasoning is incomplete. When
the decision-maker confronts both prior ambiguity and model misspecification
as in our framework, the optimal estimator coincides with that under
prior ambiguity alone. Consequently, 2-step GMM remains superior to
diagonally-weighted GMM, even under an arbitrary degree of misspecification.
Researchers who wish to employ inefficient weighting should therefore
provide explicit justification for doing so. If, for instance, the
researcher believes that misspecification is not unstructured but
that certain forms or directions of misspecification are more likely
than others, this could in principle lead to different estimation
strategies. Even so, it would be difficult to justify use of identify
or diagonal weighting on the basis of directional misspecification
alone, since such weighting typically preserves symmetry across directions.
Apart from requiring efficiency, our results do not distinguish among
different efficient estimators. Under local asymptotics, all efficient
estimators attain the same decision risk $V^{*}$, so additional criteria
must be employed to select among them. For instance, imposing invariance
would lead one to prefer CUGMM over two-step GMM.
\section{Extensions}
In this section, we discuss several variations and extensions of our
framework.
\subsection{Alternative discrepancy measures}
So far, we have used relative entropy $R_{q}(m)$ to measure the discrepancy
between statistical models. \citet[Chapter 1.8]{hansen2011robustness}
provide two important reasons for using relative entropy. First, it
leads to a tractable characterization of minimal decision risk and
optimal decisions, as demonstrated in Section \ref{sec:Characterizing-Optimal-Decisions}.
Second, as discussed in \citet[Chapter 9]{hansen2011robustness},
it can be linked to risk-sensitivity adjustment through the theory
of large deviations.
It is, however, possible to employ alternative discrepancy measures.
Relative entropy, is but a special case of the $\phi$-divergence
class of discrepancies, which take the general form
\[
D_{\phi}(m\|q)=\int\phi\left(\frac{dm}{dq}\right)dq,
\]
where $\phi:[0,\infty)\to(-\infty,\infty]$ is a convex function satisfying
$\phi(x)<\infty$ for all $x>0$, $\phi(1)=0$ and $\phi(0)=\lim_{x\to0}\phi(x)$.
Setting $\phi(x)=x\ln x$ recovers KL divergence.
Under this more general class of discrepancies, the set of unstructured
models takes the form
\[
\mathcal{M}=\Bigl\{ m\in\Delta(\Theta,\mathcal{X}):\min_{q\in\mathcal{Q}}\,D_{\phi}(m\|q)\leq K\Bigr\},
\]
and following arguments analogous to those in Section \ref{sec:Characterizing-Optimal-Decisions},
the decision-risk of a rule $\delta$ can be characterized as
\[
V_{n,\phi}(\delta)=\min_{q\in\mathcal{Q}}\inf_{m}\left\{ \mathbb{E}_{m}\bigl[u_{n}(\theta,\delta)\bigr]+\lambda\,D_{\phi}(m\|q)\right\} .
\]
Let $\phi^{*}(\cdot)$ denote the convex conjugate of $\phi(\cdot)$.
The decision risk $V_{n,\phi}(\delta)$ then admits the variational
representation
\begin{align*}
V_{n,\phi}(\delta) & =\lambda\min_{q\in\mathcal{Q}}\sup_{\eta\in\mathbb{R}}\left\{ \eta-\int\phi^{*}\left(\eta-\frac{u_{n}(\theta,\delta)}{\lambda}\right)dq\right\} \\
& =\lambda\min_{\pi\in\Delta(\Theta)}\sup_{\eta\in\mathbb{R}}\left\{ \eta-\int\mathbb{E}_{p(\bm{x}|\theta)}\left[\phi^{*}\left(\eta+\frac{l_{n}(\theta,\delta)}{\lambda}\right)\right]d\pi(\theta)\right\} ,
\end{align*}
where the first equality is due to \citet[Proposition 1]{cerreia2026making}
and the second equality exploits the specific structure of $\mathcal{Q}$
in our setting.
By standard properties of convex conjugates, $\phi^{*}(x)<\infty$
for all bounded $x$ if and only if $\lim_{x\to\infty}\phi(x)/x=\infty$.
Since our loss functions $l_{n}(\theta,\delta)$ are generally unbounded,
$V_{n,\phi}(\delta)$ is therefore $\infty$ for any $\phi$-divergence
measure satisfying $\lim_{x\to\infty}\phi(x)/x<\infty$, unless sharp
support restrictions are imposed on the class of priors. In other
words, the decision risk is trivially infinite for these divergence
measures whenever the class of priors is sufficiently rich. We therefore
argue that such discrepancies are not well suited for a framework
that accommodates both ambiguity and misspecification. Notable examples
in this category include total variation $(\phi(x)=\vert x-1\vert/2$),
squared Hellinger distance ($\phi(x)=(\sqrt{x}-1)^{2}/2)$ and Pearson
$\chi^{2}$ divergence ($\phi(x)=(x-1)^{2}/x$).
The underlying issue with these divergence measures is that the misspecification
they permit is too broad. Consider, for instance, the total-variation
metric: it is possible to have $dm/dq=\infty$, meaning $q\in\mathcal{Q}$
may not share the same support as the true model, even as total variation
remains finite. This would imply that $\mathcal{Q}$ is blatantly
misspecified, in the sense that any specification test would surely
reject it almost surely. In contrast, as \citet[Chapter 9]{hansen2011robustness}
argue, when $R_{\mathcal{Q}}(m)<\infty$, the approximating model
can still be regarded as plausibly correct, since it would not be
rejected with probability one.
Among the $\phi$-divergence measures satisfying $\lim_{x\to\infty}\phi(x)/x=\infty$,
the only commonly used divergence apart from KL divergence is the
Neyman $\chi^{2}$ divergence ($\phi(x)=(x-1)^{2}$). The Neyman $\chi^{2}$
divergence is a stronger discrepancy measure than KL divergence: it
is possible to have $\chi^{2}(m\|q)<\infty$ while $R_{q}(m)=\infty$.
Consequently, a KL-based a misspecification set $\mathcal{M}=\Bigl\{ m\in\Delta(\Theta,\mathcal{X}):\min_{q\in\mathcal{Q}}\,D_{q}(m)\leq K\Bigr\}$
is strictly larger than the corresponding $\chi^{2}$-based set $\mathcal{M}_{\chi^{2}}=\Bigl\{ m\in\Delta(\Theta,\mathcal{X}):\min_{q\in\mathcal{Q}}\,D_{\chi^{2}}(m\|q)\leq K\Bigr\}$.
Based on this insight, we conjecture that our results on the optimality
of efficient decisions continue to hold for the Neyman $\chi^{2}$
divergence as well, though we leave the formal analysis to future
work.
\subsection{Asymmetric loss functions}
The estimation and treatment assignment loss functions considered
thus far share a crucial property: they are symmetric in the sense
that overestimating $\theta$ by a given amount incurs the same loss
as underestimating it by that amount. This symmetry is crucial to
the result that optimal decisions do not depend on the degree of misspecification.
There are, however, many loss functions that lack this symmetry. A
prominent example is the linex loss $l(\theta,\delta)=e^{(\mu(\theta)-\delta)}-(\mu(\theta)-\delta)-1$
which penalizes positive errors much more than negative errors. Under
such losses, the optimal estimator is biased even in the absence of
misspecification. Incorporating misspecification concerns introduces
additional bias whose magnitude depends on the misspecification parameter
$\lambda$; see Appendix \ref{sec:Asymmetric-Loss-Functions} for
details in the context of linex loss.\footnote{Under misspecification, the linex loss function must be truncated
to keep the minimax risk finite. The optimal estimator therefore also
depends on the level of truncation. However, for any level of truncation,
the bias of the optimal estimator decreases in $\lambda$ (note that
$\lambda=0$ corresponds to no misspecification risk). } Intuitively, misspecification entails an exponential tilting of the
loss function, which further exacerbates any asymmetry already present
in the loss. Consequently, for asymmetric losses, optimal decisions
under ambiguity and misspecification may not coincide with those under
ambiguity alone.
\subsection{Relaxing caution: Smooth ambiguity aversion and hierarchical Bayes\label{subsec:Relaxing-caution}}
Our framework employs the Waldian approach to ambiguity by selecting
the worst-case model within the ambiguity set -- the structured class
of models $\mathcal{Q}$. \citet{cerreia2026making} term the preference
axiom underlying this approach as `caution'. An alternative is to
adopt smooth ambiguity aversion, as in \citet{klibanoff2005smooth},
by introducing a probability distribution $\varrho_{Q}(\cdot)$ over
$\mathcal{Q}$. Intuitively, smooth ambiguity aversion corresponds
to making the decision-maker less cautious: rather than guarding against
the worst case, she averages over models according to $\varrho_{Q}(\cdot)$.
Since $\mathcal{Q}$ pairs a candidate likelihood with every possible
prior, a distribution over $\mathcal{Q}$ is equivalent to a distribution
$\varrho(\pi)$ over the space of priors $\Delta(\Theta)$: the latter
can simply be interpreted as a hyperprior in a Bayesian hierarchical
model.
\citet[Section 6.1]{cerreia2026making} show that under smooth ambiguity
aversion, the decision risk takes the form
\begin{align*}
\tilde{V}_{n}(\delta) & =\int_{\mathcal{Q}}\phi_{Q}\left(\inf_{m}\left\{ \mathbb{E}_{m}\bigl[u_{n}(\theta,\delta)\bigr]+\lambda\,R_{q}(m)\right\} \right)d\varrho_{\mathcal{Q}}(q)\\
& =\int_{\mathcal{Q}}\phi_{Q}\left(-\lambda\ln\mathbb{E}_{q}\!\left[e^{-u_{n}(\theta,\delta)/\lambda}\right]\right)d\varrho_{Q}(q),
\end{align*}
where $\phi_{Q}(\cdot)$ is a monotone function and the second equality
uses (\ref{eq:DV formula}). Setting $\phi_{Q}(t)=-e^{-t/\lambda}$
and converting $\varrho_{Q}(q)$ to a prior $\varrho(\pi)$ over $\Delta(\Theta)$
yields
\begin{align*}
\tilde{V}_{n}(\delta) & =-\int_{\Delta(\Theta)}\left(\int\mathbb{E}_{p(\bm{x}|\theta)}\!\left[e^{l_{n}(\theta,\delta)/\lambda}\right]d\pi(\theta)\right)d\varrho(\pi)\\
& =-\int\mathbb{E}_{p(\bm{x}|\theta)}\!\left[e^{l_{n}(\theta,\delta)/\lambda}\right]d\bar{\pi}(\theta),
\end{align*}
where $\bar{\pi}(\theta)$ is the effective prior induced by the hyperprior
$\varrho(\cdot)$ over $\Delta(\Theta)$. Hence, under this specification
of $\phi_{Q}(\cdot)$, misspecification combined with smooth ambiguity
aversion with $\phi_{Q}(t)=-e^{-t/\lambda}$ is equivalent to misspecification
with a single hierarchical prior.
The choice of $\phi_{Q}(t)=-e^{-t/\lambda}$, however, uses the same
parameter $\lambda$ to govern both aversion to prior ambiguity and
sensitivity to model misspecification. It may therefore be more natural
to set $\phi_{Q}(t)=-e^{-t/\xi}$, where $\xi$ captures aversion
to prior uncertainty separately from the misspecification parameter
$\lambda$. With this choice, the decision risk becomes
\begin{align}
\tilde{V}_{n}(\delta) & =-\int_{\mathcal{Q}}\left(\mathbb{E}_{q}\!\left[e^{l_{n}(\theta,\delta)/\lambda}\right]\right)^{\lambda/\xi}d\varrho_{Q}(q)\nonumber \\
& =-\int_{\Delta(\Theta)}\left(\int\mathbb{E}_{p(\bm{x}|\theta)}\!\left[e^{l_{n}(\theta,\delta)/\lambda}\right]d\pi\right)^{\lambda/\xi}d\varrho(\pi).\label{eq:decision-risk-general}
\end{align}
As $\xi\to0$, aversion to prior ambiguity grows without bound and
the objective $\tilde{V}_{n}(\delta)$ reduces to the decision risk
$V_{n}(\delta)$ from (\ref{eq:decision-risk-eq}) --- the Waldian
formulation involving the least favorable prior, and the primary focus
of this article. For $\xi\in(0,\infty)$, the decision-maker exhibits
less aversion to prior ambiguity, but this comes at the cost of a
nonlinear objective when $\lambda\neq\xi$, which complicates the
analysis. A formal treatment of the more general criterion (\ref{eq:decision-risk-general})
when $\xi\notin\{\lambda,\infty\}$ is therefore left for future research.
\subsection{Model selection\label{subsec:Model-selection}}
While we have so far focused on misspecification of a single likelihood,
practitioners are often interested in selecting among multiple competing
likelihood specifications, each potentially subject to varying degrees
of misspecification concern.
Let $p_{1,\theta}(\bm{x})$ and $p_{2,\theta}(\bm{x})$ denote two
candidate likelihoods, and let $\mathcal{Q}_{1},\mathcal{Q}_{2}$
denote the corresponding frequentist models, where, as in Section
\ref{subsec:Models-with-prior-ambiguity},
\[
\mathcal{Q}_{a}:=\bigl\{\pi(\theta)\otimes p_{a,\theta}(\bm{x}):\pi\in\Delta(\Theta)\bigr\},\quad a\in\{1,2\}.
\]
Suppose that, treating each likelihood in isolation, Alice contemplates
a misspecification set for each of the form
\[
\mathcal{M}_{a}=\Bigl\{ m\in\Delta(\Theta,\mathcal{X}):\min_{q\in\mathcal{Q}_{a}}R_{q_{a}}(m)\leq K_{a}\Bigr\},\quad a\in\{1,2\}.
\]
Here, $K_{a}$ quantifies the decision-maker's misspecification concern
for each model: $K_{1}<K_{2}$ implies that Alice has greater concern
about likelihood 2 being misspecified than about likelihood 1.
Rather than treating the models in isolation, however, Alice may wish
to combine them. The overall set of misspecified models under consideration
can then be taken to be the intersection $\mathcal{M}=\mathcal{M}_{1}\cap\mathcal{M}_{2}.$
With this choice, the decision risk becomes
\begin{align*}
V_{n}(\delta) & :=\inf_{m\in\mathcal{M}}\mathbb{E}_{m}\bigl[l_{n}(\theta,\delta)\bigr]\\
& =\inf_{m}\left\{ \mathbb{E}_{m}\bigl[u_{n}(\theta,\delta)\bigr]:R_{\mathcal{Q}_{1}}(m)\leq K_{1},\;R_{\mathcal{Q}_{2}}(m)\leq K_{2}\right\} .
\end{align*}
As $R_{\mathcal{Q}_{a}}(\cdot)$ is strictly convex, standard duality
arguments yield the Lagrangian form
\[
V_{n}(\delta)=\inf_{m}\left\{ \mathbb{E}_{m}\bigl[u_{n}(\theta,\delta)\bigr]+\lambda\left(\alpha\,R_{\mathcal{Q}_{1}}(m)+(1-\alpha)\,R_{\mathcal{Q}_{2}}(m)\right)\right\} ,
\]
for some $\alpha\in[0,1]$ and $\lambda\geq0$ that depend on $K_{1},K_{2}$.
In particular, $\alpha>1/2$ whenever $K_{1}<K_{2}$: the decision
risk places greater weight on the likelihood that is less likely to
be misspecified.
Recalling the decomposition $m=\pi(\theta)\otimes m_{\theta}(\bm{x})$
and applying (\ref{eq:characterization_of_R_Q}) yields
\begin{align*}
V_{n}(\delta) & =\inf_{\pi(\theta)\otimes m_{\theta}(\bm{x})}\int\left\{ \mathbb{E}_{m_{\theta}(\bm{x})}\bigl[u_{n}(\theta,\delta)\bigr]+\lambda\left(\alpha\,\text{KL}(m_{\theta}\|p_{1,\theta})+(1-\alpha)\,\text{KL}(m_{\theta}\|p_{2,\theta})\right)\right\} d\pi(\theta)\\
& =\inf_{\pi(\theta)\otimes m_{\theta}(\bm{x})}\,\int\left\{ \mathbb{E}_{m_{\theta}(\bm{x})}\left[u_{n}(\theta,\delta)\right]+\lambda\text{KL}\bigl(m_{\theta}\|p_{1,\theta}^{\alpha}\cdot p_{2,\theta}^{1-\alpha}\bigr)\right\} d\pi(\theta),
\end{align*}
where the last step makes use of the fact that the weighted sum of
KL divergences can be expressed as a single KL divergence against
a geometric mixture. Applying the Donsker--Varadhan variational formula
again then gives
\begin{align*}
V_{n}(\delta) & =\inf_{\pi(\theta)}\,\int\inf_{m_{\theta}(\bm{x})}\left\{ \mathbb{E}_{m_{\theta}(\bm{x})}\left[u_{n}(\theta,\delta)\right]+\lambda\text{KL}\bigl(m_{\theta}\|p_{1,\theta}^{\alpha}\cdot p_{2,\theta}^{1-\alpha}\bigr)\right\} d\pi(\theta)\\
& =\inf_{\pi(\theta)}-\lambda\ln\left\{ \int\mathbb{E}_{p_{\alpha}(\bm{x}|\theta)}\!\left[e^{l_{n}(\theta,\delta)/\lambda}\right]d\pi(\theta)\right\} ,
\end{align*}
where $p_{\alpha}(\bm{x}|\theta):=p_{1}^{\alpha}(\bm{x}|\theta)\cdot p_{2}^{1-\alpha}(\bm{x}|\theta)$
is the geometric mixture likelihood that combines $p_{1}(\bm{x}|\theta)$
and $p_{2}(\bm{x}|\theta)$ with mixing weights $\alpha$ and $1-\alpha$,
respectively.
The optimal decision therefore solves
\[
\delta_{n}^{*}=\operatorname*{arg\,min}_{\delta}\,\max_{\pi\in\Delta(\Theta)}\int\mathbb{E}_{p_{\alpha}(\bm{x}|\theta)}\!\left[e^{l_{n}(\theta,\delta)/\lambda}\right]d\pi(\theta).
\]
Optimal decisions under multiple possibly misspecified candidate likelihoods
are thus equivalent to minimax decisions with an exponentiated loss
function and a single mixture likelihood $p_{\alpha}(\bm{x}|\theta)$.
All of our theoretical results therefore continue to apply upon reinterpreting
$p_{\alpha}(\bm{x}|\theta)$ as the relevant reference likelihood.
Recall that the mixture likelihood places greater weight on likelihood
1 when $K_{1}<K_{2}$. In the extreme case where$K_{1}\to0$ ---
that is, Alice has no misspecification concerns about likelihood 1
--- we obtain $\alpha\to1$ and $\lambda\to\infty$, so the optimal
decision reduces to the minimax-optimal decision under likelihood
1 alone, irrespective of the degree of misspecification concern about
likelihood 2. This holds even if likelihood 2 is more efficient than
likelihood 1 under correct specification. Intuitively, when likelihood
2 is globally misspecified but likelihood 1 is not, the likelihoods
are far apart in terms of misspecification risk, and it is always
optimal to place all weight on the correctly specified model. To generate
more meaningful tradeoffs between efficiency and misspecification,
one would need to bring the misspecification risks closer together
by taking $|K_{1}-K_{2}|=O(1/n)$. We leave the analysis of such a
regime of closely competing models to future research.
\section{Conclusion}
In this article, we have introduced a framework for evaluating statistical
decisions under both prior ambiguity and likelihood misspecification.
Misspecification manifests as an exponential tilting of the loss function,
while ambiguity corresponds to a search for the least favorable prior.
We also develop a theory of local asymptotics under global misspecification,
achieved by localizing the priors around a reference parameter, and
use this theory to characterize optimal estimation and treatment-assignment
decisions. Remarkably, in both cases, optimal decisions coincide with
those under correct likelihood specification.
The proposed framework opens several avenues for further research.
While we discuss some examples of asymmetric loss functions in Appendix
C, a general theory for characterizing optimal decisions under such
losses remains to be developed. As noted there, optimal decisions
under asymmetric loss may depend on the degree of misspecification.
On model selection, while Section \ref{subsec:Model-selection} provides
an initial treatment, a richer characterization would require taking
the misspecification risks of competing likelihoods to converge to
each other, so that meaningful tradeoffs between efficiency and misspecification
robustness may emerge. A further extension would be to separate ambiguity
concerns over the prior from those over candidate likelihoods, employing
smooth ambiguity aversion for the latter, as discussed in Section
\ref{subsec:Relaxing-caution}. This would bring the framework closer
to the literature on Bayesian model averaging and model selection.
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