Discrepancy-Based Ambiguity Sets

• Discrepancy-based ambiguity sets are constructs that bound plausible probability distributions using metrics like KL divergence, Wasserstein, and MMD.
• They enable robust decision-making by addressing model misspecification and integrating decision dependence in Bayesian and DRO frameworks.
• Applications span robust inference, dynamic control, and optimization, balancing conservatism with efficiency across diverse statistical models.

Discrepancy-based ambiguity sets are sets of plausible probability distributions, posterior laws, or joint statistical models defined by placing a bound on a distance or discrepancy from a reference object. Across recent work, the reference object may be an empirical distribution, a posterior predictive distribution, a fixed nominal law, or an entire structured class such as $\mathcal Q=\{\pi\otimes p_\theta\}$; the discrepancy may be a Kullback–Leibler divergence, a Wasserstein or Sinkhorn discrepancy, a maximum mean discrepancy in an RKHS, a Kolmogorov–Smirnov distance, or a discrepancy functional built from test sets such as anchored or periodic boxes (Adusumilli, 7 Apr 2026, Nemmour et al., 2022, Luo et al., 2018, Cescon et al., 26 Mar 2025, Dick et al., 2014).

1. Reference classes, discrepancy functionals, and set geometry

A common template is

$\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$

where $P_0$ is a nominal distribution and $d$ is a discrepancy. In decision-dependent DRO, the same structure is made endogenous: $$\mathcal P(x)=\{P:\text{dist}(P,P_0)\le \text{radius}(x)\},$$ with the radius or related ambiguity parameters depending on the decision $x$ (Luo et al., 2018). In robust Bayesian and semi-parametric settings, the “reference” may instead be a structured class or a posterior predictive law rather than a single $P_0$ (Adusumilli, 7 Apr 2026, Dellaporta et al., 6 May 2025).

Several representative constructions recur.

Discrepancy	Reference object	Representative set
KL on joint models	$$\mathcal Q=\{\pi(\theta)\otimes p_\theta(\mathbf x)\}$$	$\mathcal M=\{m:R_{\mathcal Q}(m)\le K\}$ (Adusumilli, 7 Apr 2026)
MMD in an RKHS	$\hat P_N$ or $\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$0	$\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$1, $\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$2 (Nemmour et al., 2022, Dellaporta et al., 6 May 2025)
Sinkhorn discrepancy	empirical $\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$3 plus reference $\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$4	$\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$5 (Cescon et al., 26 Mar 2025, Cescon et al., 5 May 2026)
Wasserstein/CDF discrepancy	empirical input laws or propagated empirical CDFs	pointwise Wasserstein balls and CDF envelope bands (Boso et al., 2020)
Weighted star discrepancy	empirical measure on a point set	$\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$6 (Dick et al., 2014)

This taxonomy already shows that discrepancy-based ambiguity is not restricted to balls around empirical measures. In the KL construction of misspecification and ambiguity, the ball is taken around a set of joint models; in robust Bayesian DRO, the ball is centered at a robust posterior predictive; in hyperbolic PDEs, the ambiguity set can be a time-dependent band of CDFs; and in decision-dependent DRO, the geometry itself changes with the control variable (Adusumilli, 7 Apr 2026, Dellaporta et al., 6 May 2025, Boso et al., 2020, Luo et al., 2018).

2. KL discrepancy on structured statistical models

A particularly explicit decision-theoretic construction starts from the joint distribution over the payoff-relevant state $\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$7,

$\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$8

and defines the ambiguity set over Bayesian models as

$\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$9

Here ambiguity is over priors only, while the likelihood $P_0$0 is held fixed as a reference specification (Adusumilli, 7 Apr 2026).

Likelihood misspecification is then introduced by surrounding $P_0$1 with a KL ball in the space of joint distributions: $P_0$2 Writing $P_0$3, the KL projection onto $P_0$4 yields

$P_0$5

The resulting ambiguity set is therefore

$P_0$6

so prior ambiguity is unrestricted while misspecification enters through an integrated KL constraint on the conditional laws (Adusumilli, 7 Apr 2026).

For a decision rule $P_0$7 with loss $P_0$8, the robust criterion is

$P_0$9

Using Donsker–Varadhan duality,

$d$0

the problem becomes the minimax program

$d$1

This yields the paper’s separation principle: ambiguity appears as maximization over priors, while misspecification appears as exponential tilting of the loss (Adusumilli, 7 Apr 2026).

The same paper shows that this separation is especially convenient for local asymptotics. The reference likelihood in the minimax criterion remains the classical $d$2, so standard LAN/Le Cam arguments apply, but with nonlinear loss $d$3. In the local Gaussian experiment, the optimal estimation rule is the scaled MLE and the optimal treatment rule is the threshold rule $d$4, both independent of $d$5. The lower-bound and achievability results imply that the same efficient rules that are minimax under correct specification are also minimax optimal for every KL radius $d$6; the paper extends this conclusion to semi-parametric models and draws explicit procedural implications for maximum likelihood versus SMM and efficient versus diagonally weighted GMM (Adusumilli, 7 Apr 2026).

3. Kernel and MMD ambiguity sets

In MMD-based DRO, the discrepancy is the RKHS integral probability metric

$d$7

where $d$8 is the kernel mean embedding. The corresponding ambiguity set is the MMD ball around the empirical distribution

$d$9

Under $$\mathcal P(x)=\{P:\text{dist}(P,P_0)\le \text{radius}(x)\},$$0, the paper gives the finite-sample bound

$$\mathcal P(x)=\{P:\text{dist}(P,P_0)\le \text{radius}(x)\},$$1

with probability at least $$\mathcal P(x)=\{P:\text{dist}(P,P_0)\le \text{radius}(x)\},$$2; the rate is $$\mathcal P(x)=\{P:\text{dist}(P,P_0)\le \text{radius}(x)\},$$3 and dimension-independent (Nemmour et al., 2022).

This geometry is used to formulate distributionally robust chance-constrained programs with general nonlinear constraints: $$\mathcal P(x)=\{P:\text{dist}(P,P_0)\le \text{radius}(x)\},$$4 Using Zhu’s MMD-DRO duality, the exact robust feasibility condition becomes the existence of $$\mathcal P(x)=\{P:\text{dist}(P,P_0)\le \text{radius}(x)\},$$5 and $$\mathcal P(x)=\{P:\text{dist}(P,P_0)\le \text{radius}(x)\},$$6 such that

$$\mathcal P(x)=\{P:\text{dist}(P,P_0)\le \text{radius}(x)\},$$7

A CVaR-based relaxation replaces the indicator by $$\mathcal P(x)=\{P:\text{dist}(P,P_0)\le \text{radius}(x)\},$$8, and a robust representer theorem yields a tractable finite-dimensional program in the Gram matrix variables. The paper proves a finite-sample constraint-satisfaction guarantee for the approximate algorithm with $$\mathcal P(x)=\{P:\text{dist}(P,P_0)\le \text{radius}(x)\},$$9, again dimension-independent, and proposes a bootstrap calibration of $x$0 that avoids cross-validation (Nemmour et al., 2022).

A second line of work centers the MMD ball at a robust posterior predictive rather than $x$1. In DRO-RoBAS, a Dirichlet-process posterior on the data-generating process is pushed through the minimum-MMD target

$x$2

and the robust posterior predictive is

$x$3

The ambiguity set is then

$x$4

Its RKHS dual takes the form

$x$5

subject to $x$6 for all $x$7. The paper also proves that, with high probability,

$x$8

where $x$9, thereby linking the ambiguity radius to both finite-sample error and model approximation error under misspecification (Dellaporta et al., 6 May 2025).

4. Optimal transport, Sinkhorn regularization, and decision-dependent sets

Sinkhorn ambiguity sets regularize optimal transport by adding an entropic penalty on couplings: $P_0$0 and define

$P_0$1

In control applications, the center $P_0$2 is typically empirical, the transportation cost is quadratic, and $P_0$3 is a Gaussian reference distribution encoding prior information. The regularization parameter $P_0$4 interpolates between Wasserstein robustness and control under the reference law: as $P_0$5, Sinkhorn DRO reduces to Wasserstein DRO; as $P_0$6, and under the stated feasibility condition, the ambiguity set collapses to $P_0$7 (Cescon et al., 26 Mar 2025).

Two structural facts are central. First,

$P_0$8

so Sinkhorn balls are contained in the corresponding OT balls, and they shrink monotonically as $P_0$9 increases. Second, because feasible couplings satisfy $$\mathcal Q=\{\pi(\theta)\otimes p_\theta(\mathbf x)\}$$0, every feasible $$\mathcal Q=\{\pi(\theta)\otimes p_\theta(\mathbf x)\}$$1 satisfies $$\mathcal Q=\{\pi(\theta)\otimes p_\theta(\mathbf x)\}$$2; when $$\mathcal Q=\{\pi(\theta)\otimes p_\theta(\mathbf x)\}$$3 is continuous, worst-case Sinkhorn distributions are continuous even if the empirical center is discrete. This directly addresses a limitation of Wasserstein DRO with empirical centers, where worst-case distributions are discrete and supported on at most $$\mathcal Q=\{\pi(\theta)\otimes p_\theta(\mathbf x)\}$$4 points (Cescon et al., 5 May 2026).

The same paper establishes convexity and weak compactness of $$\mathcal Q=\{\pi(\theta)\otimes p_\theta(\mathbf x)\}$$5 under standard assumptions on the cost function. Convexity follows from the convexity of $$\mathcal Q=\{\pi(\theta)\otimes p_\theta(\mathbf x)\}$$6, and weak compactness follows from lower semicontinuity together with containment in a weakly compact OT/Wasserstein ball. These properties underpin minimax interchanges and strong duality in distributionally robust control (Cescon et al., 5 May 2026).

For finite-horizon linear-quadratic control with linear policies, the robust cost

$$\mathcal Q=\{\pi(\theta)\otimes p_\theta(\mathbf x)\}$$7

and robust CVaR safety constraints

$$\mathcal Q=\{\pi(\theta)\otimes p_\theta(\mathbf x)\}$$8

admit exact convex reformulations involving LMIs and log-det terms. The tractable program remains convex even with DR safety constraints, and the empirical study shows lower conservatism than Wasserstein DR control when only few noise samples are available (Cescon et al., 5 May 2026).

Decision dependence can also be imposed directly on discrepancy radii. In finite-support DD-DRO, the paper considers decision-dependent Wasserstein, $$\mathcal Q=\{\pi(\theta)\otimes p_\theta(\mathbf x)\}$$9-divergence, and Kolmogorov–Smirnov sets,

$\mathcal M=\{m:R_{\mathcal Q}(m)\le K\}$0

and analogous KS sets. Linear, conic, and Lagrangian duality yield finite reformulations in the support probabilities, but the overall optimization is typically nonconvex in $\mathcal M=\{m:R_{\mathcal Q}(m)\le K\}$1, even when the inner ambiguity problem is convex (Luo et al., 2018).

5. Dynamic propagation, state dependence, and control under ambiguity

A major development is the transition from static ambiguity sets to propagated ambiguity tubes. For nonlinear data-driven dynamics represented by Koopman operators and conditional mean embeddings, kernel ambiguity sets are MMD balls in the RKHS: $\mathcal M=\{m:R_{\mathcal Q}(m)\le K\}$2 Because the learned dynamics act linearly on embeddings, the paper derives exact multi-step propagation formulas. If $\mathcal M=\{m:R_{\mathcal Q}(m)\le K\}$3 is the empirical embedded push-forward operator, $\mathcal M=\{m:R_{\mathcal Q}(m)\le K\}$4, and $\mathcal M=\{m:R_{\mathcal Q}(m)\le K\}$5, then Algorithm 1 updates the center and radius by

$\mathcal M=\{m:R_{\mathcal Q}(m)\le K\}$6

This yields an ambiguity tube $\mathcal M=\{m:R_{\mathcal Q}(m)\le K\}$7 in $\mathcal M=\{m:R_{\mathcal Q}(m)\le K\}$8, and bootstrap procedures estimate the operator error $\mathcal M=\{m:R_{\mathcal Q}(m)\le K\}$9 from kernel matrices (Zhu, 2023).

A different dynamic construction arises for hyperbolic conservation laws with uncertain inputs. There the relevant object is the single-point CDF $\hat P_N$0, which satisfies a linear hyperbolic PDE in the augmented $\hat P_N$1-space: $\hat P_N$2 Input ambiguity is first built from Wasserstein concentration bounds in parameter space and then pushed forward to CDFs. For general nonlinear hyperbolic equations with smooth solutions, upper and lower envelopes of pointwise ambiguity bands are propagated through the CDF equation, while for linear dynamics the 1-Wasserstein radius itself satisfies a transport-reaction equation. In both cases the propagated ambiguity sets retain the prescribed confidence level and contain the true distribution throughout the space-time domain (Boso et al., 2020).

Continuous-time robust Bayesian portfolio optimization introduces yet another state-dependent formulation. The posterior mean drift estimate $\hat P_N$3 is surrounded by a discrepancy-based posterior ambiguity set,

$\hat P_N$4

with Wasserstein, $\hat P_N$5, sample-path, or multiple discrepancy constraints. The naive global ambiguity set is time inconsistent, so the paper introduces the feedback-type local set

$\hat P_N$6

This leads to a modified HJBI equation and, for exponential utility, to the reduced PDE

$\hat P_N$7

The optimal feedback portfolio then combines a Merton-type term under the worst-case drift with a hedging term involving $\hat P_N$8 (Liang et al., 16 Jun 2026).

6. Tractability, deterministic discrepancy constructions, and unresolved issues

Discrepancy-based ambiguity sets are not limited to stochastic metrics. In quasi-Monte Carlo, weighted star discrepancy induces a set-valued uncertainty model

$\hat P_N$9

where the discrepancy is the supremum of weighted deviations on anchored boxes. Korobov’s $\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$00-sets provide explicit deterministic scenario sets that are independent of the weights, and for product weights the paper proves strong polynomial tractability when $\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$01, and polynomial tractability when $\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$02 for some $\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$03 (Dick et al., 2014).

Related asymptotic results for extreme and periodic $\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$04 discrepancy show that, for fixed dimension $\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$05, the minimal discrepancy order is $\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$06. The focused synthesis further states that, after normalization by $\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$07, the corresponding best-case ambiguity radius behaves like

$\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$08

which quantifies the smallest discrepancy radii achievable by optimal deterministic designs (Kritzinger et al., 2021).

Several comparative lessons recur across the literature. One is that the reference object matters as much as the discrepancy: KL may be taken around an entire structured class $\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$09, MMD around an empirical or robust posterior predictive law, Sinkhorn around an empirical law together with a reference $\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$10, and posterior ambiguity around a time-varying Bayesian filter (Adusumilli, 7 Apr 2026, Dellaporta et al., 6 May 2025, Cescon et al., 26 Mar 2025, Liang et al., 16 Jun 2026). Another is that stronger discrepancy notions are not automatically more useful. In the KL-based variational framework, many common $\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$11-divergences, including total variation, Hellinger, and Pearson $\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$12, fail to accommodate unbounded losses when priors are rich, leading to trivially infinite robust values; KL and Neyman $\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$13 satisfy the required growth condition, with KL producing the larger misspecification set (Adusumilli, 7 Apr 2026).

A further point concerns conservatism and model misspecification. Under clean and well-specified models, KL-based Bayesian DRO can outperform more elaborate robust Bayesian MMD constructions, whereas under contamination or misspecification the robust posterior predictive center in DRO-RoBAS produces better out-of-sample performance than standard Bayesian and empirical DRO baselines (Dellaporta et al., 6 May 2025). Sinkhorn sets exhibit an analogous bias-variance trade-off: increasing $\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$14 shrinks the ambiguity set and interpolates between Wasserstein robustness and optimization under the reference distribution, which is particularly useful when data are scarce (Cescon et al., 26 Mar 2025).

Open issues remain distribution-specific. Periodic $\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$15 discrepancy is characterized sharply for $\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$16, but the same upper bounds for $\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$17 remain open; explicit optimal periodic constructions for arbitrary $\mathcal P=\{Q:\, d(Q,P_0)\le \varepsilon\},$18 are also open (Kritzinger et al., 2021). In DD-DRO, dualization survives decision dependence, but global nonconvexity remains the main computational obstacle (Luo et al., 2018). In dynamic kernel models, the propagation theory is open-loop and translating RKHS ambiguity tubes back to state-space control constraints requires further work (Zhu, 2023). These limitations notwithstanding, the current literature shows that discrepancy-based ambiguity sets now form a broad family of constructions linking robust statistics, variational decision theory, kernel methods, optimal transport, PDE dynamics, and control.