{"id":"0f6bca82-bc65-41f8-ab98-4e5f31848bcf","arxiv_id":"2505.03585","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"DRO-RoBAS centers MMD-based ambiguity sets on an NPL-MMD robust posterior predictive, obtains an RKHS dual formulation, and provides high-probability tolerance guarantees for containing the data-generating process.","lead":"This paper introduces DRO-RoBAS, a distributionally robust optimization method whose ambiguity sets are centered at a robust Bayesian posterior predictive that is designed to be insensitive to model misspecification. The approach matters for risk-averse decisions such as inventory and portfolio optimization, where the assumed model family may not contain the true data-generating process.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The tolerance guarantee is not actionable: Corollary 3.8 requires the unknown MMD misspecification gap to set epsilon, and no estimator for it is supplied, so the central non-conservatism claim rests on an unavailable input.","rationale":"The reader's conditional verdict identifies the same core weakness I find most load-bearing: Theorem 3.6 and Corollary 3.8 relate the required ambiguity radius to inf_theta D_k(P_theta, P_star), but the paper gives no way to determine or estimate this quantity from data. Without it, the guarantee cannot be used to choose epsilon, which is the input that determines both containment of P_star and the degree of conservatism. This is not a mathematical inconsistency in the proof; it is an operational gap in the central claim. The paper does have independent support: the dual formulation is derived carefully and the experiments include an honest acknowledgment that standard Bayesian DRO performs better in the well-specified case. However, the radius-selection issue is precisely the step between the theory and the advertised practical benefit, so the conditional verdict is appropriate. I do not see a basis for rejection, and I would not move the verdict to accept without an estimator or a demonstration that epsilon can be chosen in a data-dependent way while preserving coverage.","tokens_in":22801,"tokens_out":7637,"duration_ms":87694,"concrete_test":"Run the contaminated Gaussian Newsvendor setup (n = 20, eta = 0.1) over many repetitions. In each repetition compute the plug-in estimate g_hat = inf_{theta in Theta} D_k(P_n, P_theta) plus the analytic constant C_{n,M,alpha} from Theorem 3.6, set epsilon = C_{n,M,alpha} + g_hat, and record whether P_star lies in B_k^epsilon(hat_P_pred^NPL) and whether E_{P_star}[f_x] <= sup_{B_k^epsilon} E_P[f_x]. If empirical coverage is below the nominal 1-delta rate, or is only achieved for epsilon substantially larger than the formula, then the missing gap estimator is genuinely load-bearing. If a simple bias-corrected plug-in estimator restores coverage, the omission is minor and the paper can be made actionable with an added procedure.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing condition for the paper's central guarantee is that the decision maker can choose epsilon so that P_star is contained in the RoBAS ball with probability 1-delta. Theorem 3.6 states that epsilon >= C_{n,M,alpha} + inf_{theta in Theta} D_k(P_theta, P_star) is sufficient, and Remark 3.7 concedes that the misspecification gap 'must be reasonably approximated,' but no estimator, confidence bound, or data-dependent construction is provided. In the misspecified settings the paper targets, inf_theta D_k(P_theta, P_star) is exactly the quantity that drives the required radius: if it is underestimated, Corollary 3.8's bound on E_{P_star}[f_x] fails; if overestimated, the ambiguity set is unnecessarily large and the advertised reduction in conservatism is lost. The experiments sweep epsilon as a free parameter, which sidesteps rather than resolves this gap. The theoretical results are internally coherent and the dual derivation appears sound, but the headline probabilistic tolerance guarantee is conditional on an unknown input that the paper does not operationalize.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes DRO-RoBAS, a distributionally robust optimization method whose ambiguity set is an MMD ball centered at the NPL-MMD posterior predictive, with the goal of mitigating model misspecification in Bayesian DRO. The authors derive a dual formulation in the RKHS using the kernel DRO duality theorem of Zhu et al. (2021), prove a finite-sample tolerance guarantee bounding the MMD between the data-generating process and the approximate posterior predictive by the model's irreducible MMD misspecification gap plus a concentration term, and provide an additional guarantee under Huber contamination. The method is evaluated on Newsvendor and Portfolio problems under various misspecification regimes, with comparisons against DRO-BAS, BDRO, and empirical MMD baselines.","tokens_in":23035,"tokens_out":7021,"duration_ms":74889,"significance":"If the claims hold, the paper makes a useful contribution by extending Bayesian DRO to settings where the parametric model is misspecified, a setting where standard Bayesian ambiguity sets can become overly conservative. The derivation is not circular: the tolerance guarantee is stated relative to the external misspecification gap inf_theta D_k(P_theta, P_star), and the dual formulation is a correct application of an existing kernel DRO duality theorem. The concentration analysis is a standard McDiarmid argument building on existing MMD posterior bootstrap results, and the experiments cover several misspecification regimes against relevant baselines. The paper also clearly identifies the computational costs and limitations of the method. The main weakness is that the central radius-selection guarantee is conditional on an unknown quantity that the paper does not operationalize, which undermines the practical actionability of the headline non-conservatism claim.","major_comments":[{"comment":"The tolerance guarantee is not directly actionable: Corollary 3.8 requires choosing epsilon at least C_{n,M,alpha} + inf_{theta in Theta} D_k(P_theta, P_star), but inf_theta D_k(P_theta, P_star) is the unknown misspecification gap that the method is designed to be robust against. Remark 3.7 states that the gap must be 'reasonably approximated', yet no estimator, confidence bound, or data-dependent construction is provided. If the gap is underestimated, the containment guarantee and hence the upper bound on the true objective can fail; if it is overestimated, the ambiguity set is unnecessarily large and the advertised reduction in conservatism is lost. The experiments in Section 4 sweep epsilon as a free parameter, which sidesteps rather than resolves this issue. The paper should either provide an estimator for the gap with its own finite-sample guarantee, or reframe the contribution so that the radius selection problem is explicitly acknowledged as an open component.","section":"Theorem 3.6, Remark 3.7, Corollary 3.8"},{"comment":"The verification that ri(K_{C*}) is nonempty is incomplete. The proof states that 'it suffices to prove that K_{C*} is non-empty and convex', but this implication is not generally true in infinite-dimensional spaces, where a nonempty convex set can have empty relative interior. Since the dual formulation relies on Theorem 3.1 of Zhu et al. (2021), the paper should either prove directly that the MMD ball has nonempty relative interior in the relevant topology, or cite/verify a version of the duality theorem with weaker assumptions that are satisfied here.","section":"Appendix A.2, Lemma A.2"}],"minor_comments":[{"comment":"The text refers to the 'NLP-MMD posterior' in the paragraph before Assumption 3.4; this should be 'NPL-MMD posterior'.","section":"Section 3.2"},{"comment":"The DGP specification for the contaminated Exponential case contains a duplicated and incomplete line: 'P⋆ train := (1−η)Exp(0.05) +ηN (100, 0.5) P⋆ train := (1−η)Exp(θ⋆) +ηN (µ,σ )'. This should be cleaned up.","section":"Appendix B"},{"comment":"The caption reads 'Average (standard derivation)solve time'; 'derivation' should be 'deviation'.","section":"Appendix B, Table 1 caption"},{"comment":"The legend entry 'Posterior' is ambiguous; it should be labelled 'standard posterior' to distinguish it from the NPL-MMD posterior shown in the same plot.","section":"Figure 2"},{"comment":"The concentration parameter of the DP prior is denoted alpha in the main text but c in Lemmas A.3 and A.4; the notation should be unified to avoid confusion.","section":"Appendix A.3"},{"comment":"The approximate DP in (17) uses a truncation level tau, but the concentration bound and the constant C_{n,M,alpha} do not explicitly depend on tau. The paper should state whether the bound is uniform in tau or clarify how tau is absorbed into the constant.","section":"Equation (17) and Theorem 3.6"},{"comment":"The abstract claims the method 'outperforms other Bayesian and empirical DRO approaches', but the experiments show that empirical MMD outperforms DRO-RoBAS in the multivariate bimodal case, and DRO-BAS formulations outperform DRO-RoBAS in the well-specified contamination case. The abstract should qualify the claim to match the paper's own conclusions.","section":"Abstract and Section 4"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this if you care about Bayesian DRO under misspecification. The paper constructs a new object—an MMD ambiguity set centered at an NPL-MMD posterior predictive—derives its support function, and proves a high-probability containment result. The math is sound and the empirical section is more honest than most: it openly reports that DRO-BAS beats RoBAS when the model is well specified.\n\nWhat is genuinely new is the center of the ambiguity set. Using the NPL-MMD posterior predictive in place of the standard posterior is natural and does buy robustness in the contaminated examples, as the toy experiment shows. The dual formulation is a direct application of Zhu et al. (2021), but Proposition 3.2 is their own support-function computation, and Corollary 3.8 follows correctly from the concentration lemma plus McDiarmid. The Huber contamination corollary is a nice extra. I found no circularity: the posterior and the ambiguity set both use MMD, but the bound is against the model's own irreducible MMD error, which is external.\n\nThe real soft spot is the tolerance guarantee. Corollary 3.8 requires the decision maker to choose epsilon larger than C + inf_theta D_k(P_theta, P_star). That gap is exactly what the method is supposed to be robust against, and Remark 3.7 just says it must be 'reasonably approximated.' No estimator or confidence interval is provided. If you under-estimate the gap, the containment claim fails; if you over-estimate it, you are back to the large, conservative ambiguity sets the method was designed to avoid. The experiments sweep epsilon as a free parameter, which side-steps the issue. This is not a fatal flaw in the mathematics—the theorem is stated conditionally—but it does mean the headline guarantee is not directly actionable in the misspecified settings the paper targets.\n\nTwo smaller issues. The abstract claims RoBAS outperforms empirical DRO, but in the bimodal Gaussian experiments empirical MMD is comparable or better, especially in the multivariate case; the paper's own text admits this. And no code or data is released, which hurts reproducibility given the complex optimization.\n\nOverall: the core construction is a real contribution and the proofs look correct. With a data-dependent way to estimate or bound the gap, the paper would be strong. As is, it is a good methods paper with a known limitation, and it deserves serious refereeing rather than a desk reject.","headline":"Sound new construction for Bayesian DRO under misspecification; the main caveat is a tolerance guarantee that depends on the unknown misspecification gap.","tokens_in":23569,"tokens_out":2522,"would_cite":true,"duration_ms":23206,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F15","90C15","90C47"],"pacs":[],"model":"deepseek-v4-flash","headline":"An MMD ball around a robust nonparametric posterior predictive, instead of a KL ball around the standard posterior, keeps DRO decisions from being overly conservative under model misspecification, with a high-probability tolerance…","keywords":["distributionally robust optimization","model misspecification","maximum mean discrepancy","Bayesian nonparametric learning","NPL posterior predictive","ambiguity set","newsvendor problem","portfolio optimization"],"falsifier":"Run repeated trials of a misspecified problem (say, a unimodal Gaussian model with a bimodal Gaussian DGP) and compare the two sides of Corollary 3.8: set $\\epsilon = C_{n,M,\\alpha} + \\inf_\\theta D_k(P_\\theta, P_\\star)$ and check whether the realized out-of-sample cost $E_{\\xi\\sim P_\\star}[f_{x^\\star}(\\xi)]$ at the RoBAS decision exceeds $\\sup_{P \\in B_k^\\epsilon(\\hat{P}^{\\mathrm{NPL}}_{n})} E_{\\xi\\sim P}[f_{x^\\star}(\\xi)]$ more often than the claimed rate $\\delta$. A sharper test of Theorem 3.6 is to measure $D_k(P_\\star, \\hat{P}^{\\mathrm{NPL}}_{n}) - \\inf_\\theta D_k(P_\\theta, P_\\star)$ as $n$ grows: if it does not decay at the claimed $1/\\sqrt{n}$ rate, or if it levels off at a positive floor under bounded kernels and light-tailed DGPs, the concentration result that anchors the whole method is wrong.","tokens_in":22607,"feed_emoji":"🛡️","tokens_out":21455,"duration_ms":158073,"temperature":0.7,"pith_summary":"Distributionally robust optimisation (DRO) lets a risk-averse decision maker minimize worst-case cost over an ambiguity set of distributions, and Bayesian versions center that set on the posterior. When the model is misspecified, the standard posterior drifts away from the data-generating process (DGP), so the set must be stretched to contain the truth and decisions become overly conservative. This paper claims that the problem is fixed by changing both ingredients at once: center the ambiguity set on the NPL-MMD posterior predictive, a robust posterior that puts a Dirichlet process prior directly on the DGP, and measure distance with the Maximum Mean Discrepancy (MMD), a kernel-based metric on distributions, instead of KL divergence. It proves that a radius of $C_{n,M,\\alpha} + \\inf_{\\theta} D_k(P_\\theta, P_\\star)$ is enough for the set to contain the DGP with probability at least $1-\\delta$, and that the true expected cost is then bounded by the worst-case cost over the set (Corollary 3.8). The resulting problem has a computable dual in a reproducing kernel Hilbert space, and Newsvendor and Portfolio experiments show better out-of-sample performance than KL-based Bayesian DRO and empirical MMD under misspecification and contamination.","feed_headline":"Stop model misspecification from inflating worst-case decision sets","feed_subtitle":"Robust Bayesian ambiguity sets keep out-of-sample cost bounds tight when the model is wrong.","key_machinery":"The load-bearing object is the pair formed by the NPL-MMD posterior predictive and the MMD ball. The predictive is $P^{\\mathrm{NPL}}_{n} := E_{Q\\sim \\mathrm{DP}(\\alpha', F')}[P_{\\theta_k(Q)}]$, where $\\theta_k(Q) := \\arg\\min_{\\theta} D_k(Q, P_\\theta)$ selects the model closest to a drawn DGP $Q$; the Dirichlet process prior states uncertainty directly about the DGP, and the push-forward posterior is sampled by the posterior bootstrap. The ambiguity set is $B_k^\\epsilon(P) := \\{P' : D_k(P', P) \\le \\epsilon\\}$, where the MMD is a kernel-based metric on probability measures (zero only when the measures agree, for a characteristic kernel), so unlike KL balls it needs no absolute continuity. The argument runs through the mean embedding: with $C_\\star := \\{\\mu \\in \\mathcal{H}_k : \\|\\mu - \\mu_{P^{\\mathrm{NPL}}_{n}}\\|_k \\le \\epsilon\\}$, the support function is $\\delta^\\star_{C_\\star}(g) = E_{Q\\sim\\mathrm{DP}(\\alpha',F')}\\left[E_{\\xi\\sim P_{\\theta_k(Q)}}[g(\\xi)]\\right] + \\epsilon\\|g\\|_k$, and applying the kernel-DRO duality theorem turns the minimax problem into the minimization of $g_0 + \\delta^\\star_{C_\\star}(g)$ over functions dominating $f_x$. The probabilistic tolerance guarantee is carried by a McDiarmid concentration bound on $E_{Q\\sim\\mathrm{DP}}[D_k(P_\\star, Q)]$, which yields the radius constant $C_{n,M,\\alpha}$ with rate $1/\\sqrt{n}$.","core_discovery":"The central claim is that the NPL-MMD posterior predictive, $\\hat{P}^{\\mathrm{NPL}}_{n}$, inherits robustness from both its construction and its metric, so the MMD ball around it — the RoBAS ambiguity set — contains the DGP at a radius that scales with the minimal misspecification gap rather than with the posterior's sensitivity to outliers. Theorem 3.6 states that with probability at least $1-\\delta$, $D_k(P_\\star, \\hat{P}^{\\mathrm{NPL}}_{n}) \\le \\inf_{\\theta\\in\\Theta} D_k(P_\\theta, P_\\star) + C_{n,M,\\alpha}$, where $C_{n,M,\\alpha}$ decays as $1/\\sqrt{n}$ and is fully known given the kernel bound $M$. Corollary 3.8 then gives the tolerance guarantee: for $\\epsilon \\ge C_{n,M,\\alpha} + \\inf_{\\theta} D_k(P_\\theta, P_\\star)$, the true expected cost is bounded above by the worst-case expectation over the set, $E_{\\xi\\sim P_\\star}[f_x(\\xi)] \\le \\sup_{P \\in B_k^\\epsilon(\\hat{P}^{\\mathrm{NPL}}_{n})} E_{\\xi\\sim P}[f_x(\\xi)]$, with probability at least $1-\\delta$. In the Huber contamination model the required gap is only $4\\eta + 2C_{n,M,\\alpha}$ (Corollary 3.9). Computationally, the set is a ball in the kernel mean embedding space, and its support function evaluates to $E_{Q\\sim\\mathrm{DP}(\\alpha',F')}\\left[E_{\\xi\\sim P_{\\theta_k(Q)}}[g(\\xi)]\\right] + \\epsilon\\|g\\|_k$, which converts the minimax problem into a finite-dimensional kernel optimization via the kernel-DRO duality theorem and the representer theorem.","pith_inferences":["The missing estimator for $\\inf_\\theta D_k(P_\\theta, P_\\star)$ could be supplied in a follow-up: for instance, a nested bootstrap on the posterior bootstrap samples (or a resampling-based upper confidence bound on the MMD gap) plugged into Remark 3.7's 'reasonable approximation' would make the radius fully data-driven, at the price of extra concentration terms; the paper stops short of that.","A testable corollary of Theorem 3.6 is that RoBAS's advantage over KL-based Bayesian DRO should widen as the gap between model and DGP widens: the baselines pay a radius that tracks posterior sensitivity, while RoBAS pays only the minimal MMD gap plus $C_{n,M,\\alpha}$; re-running the Newsvendor experiments while sweeping a misspecification parameter would expose this scaling directly.","Since $C_{n,M,\\alpha}$ depends on the kernel bound $M$ and the gap depends on the kernel's resolving power, kernel choice is part of the trade-off, not a free hyperparameter: a wider-bandwidth kernel shrinks the gap but inflates the constant; tuning the bandwidth against both terms is a natural extension the paper leaves open.","The appendix's identity $B_k^\\epsilon(P^{\\mathrm{NPL}}_{n}) \\equiv A_k^{\\epsilon^2 + v(\\mathrm{DP})}$ suggests the expected-squared-MMD set carries an extra posterior-variance term $v(\\mathrm{DP})$, so the RoBAS ball should be systematically less conservative than the expected-MMD set for the same nominal — an empirical comparison the paper does not report."],"forward_implications":["Setting $\\epsilon = C_{n,M,\\alpha} + \\inf_\\theta D_k(P_\\theta, P_\\star)$ yields a decision whose true expected cost is, with probability at least $1-\\delta$, no larger than the worst-case cost over the RoBAS set — so misspecification protection does not require an arbitrarily inflated ambiguity set (Corollary 3.8).","Because the dual problem is a kernel optimization solvable by sampling plus the representer theorem, RoBAS applies to any model family, including likelihood-free models, where DRO-BAS-PE (exponential families only) and KL-based methods needing closed-form posteriors do not apply.","Under Huber contamination the radius requirement grows only linearly in the contamination level ($4\\eta + 2C_{n,M,\\alpha}$), so mildly contaminated settings do not push decisions into extreme conservatism (Corollary 3.9).","In the Newsvendor and Portfolio experiments, RoBAS delivers better out-of-sample mean-variance trade-offs than DRO-BAS and BDRO under moderate-to-high misspecification, while KL-based Bayesian DRO remains competitive in the well-specified regime.","The duality and tolerance results are stated for a general posterior, so the RoBAS construction can be re-instantiated with other robust posteriors (generalized Bayesian inference, MMD-Bayes) without re-deriving the machinery."],"supporting_citations":[{"why":"Supplies Theorem 3.1, the kernel-DRO duality theorem that converts the RoBAS primal into the RKHS dual, and the SAA/discretization scheme plus representer lemma used for computation.","marker":"Zhu et al. (2021)"},{"why":"Defines the NPL-MMD posterior and MMD posterior bootstrap; its Lemmas 6 and 11 give the concentration bounds that Theorem 3.6 rests on.","marker":"Dellaporta et al. (2022)"},{"why":"Introduces the NPL posterior, the DGP-level nonparametric prior that makes the RoBAS center robust to misspecification.","marker":"Lyddon et al. (2018)"},{"why":"Provides the posterior bootstrap algorithm used to sample from the Dirichlet process posterior in practice.","marker":"Fong et al. (2019)"},{"why":"The DRO-BAS framework (BASPE and BASPP) that RoBAS extends, and the main Bayesian baseline in the experiments.","marker":"Dellaporta et al. (2024)"},{"why":"Introduces MMD-ball ambiguity sets; serves as the empirical-MMD baseline.","marker":"Staib & Jegelka (2019)"},{"why":"Bayesian DRO (BDRO), the expected-worst-case baseline RoBAS is compared against.","marker":"Shapiro et al. (2023)"},{"why":"Gives the closed-form MMD in terms of kernels and the kernel mean embedding characterization used throughout the dual derivation.","marker":"Gretton et al. (2012)"},{"why":"Its Lemma 3.3 on MMD robustness under contamination is used to derive the Huber-contamination bound (Corollary 3.9).","marker":"Chérief-Abdellatif & Alquier (2022)"},{"why":"The McDiarmid-based proof technique behind the concentration lemma (Lemma A.4) for the approximate DP posterior.","marker":"Briol et al. (2019)"}],"fun_headline_variants":["RoBAS: Bayesian DRO that resists model misspecification","Robust Bayesian ambiguity sets tighten DRO under model errors","Keep worst-case risk low even when the model is wrong","New DRO with robust Bayesian sets beats misspecification","DRO-RoBAS: decision making that shrugs off wrong models"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The radius guarantee only gives a high-probability bound if the decision maker knows or can closely approximate the misspecification gap $\\inf_{\\theta\\in\\Theta} D_k(P_\\theta, P_\\star)$ — precisely the unknown quantity the method is designed to be robust against — and the paper does not provide an estimator or data-dependent construction for it (Remark 3.7 only asks that it be 'reasonably approximated').","fun_headline_variants_meta":{"raw":{"variants":["RoBAS: Bayesian DRO that resists model misspecification","Robust Bayesian ambiguity sets tighten DRO under model errors","Keep worst-case risk low even when the model is wrong","New DRO with robust Bayesian sets beats misspecification","DRO-RoBAS: decision making that shrugs off wrong models"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000316,"raw_usage":{"total_tokens":1880,"prompt_tokens":1124,"completion_tokens":756,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":740,"completion_tokens_details":{"reasoning_tokens":670}},"tokens_in":740,"tokens_out":756,"duration_ms":7124,"temperature":1.0,"reasoning_tokens":670,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:47:52.908660+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run repeated trials of a misspecified problem (say, a unimodal Gaussian model with a bimodal Gaussian DGP) and compare the two sides of Corollary 3.8: set $\\epsilon = C_{n,M,\\alpha} + \\inf_\\theta D_k(P_\\theta, P_\\star)$ and check whether the realized out-of-sample cost $E_{\\xi\\sim P_\\star}[f_{x^\\star}(\\xi)]$ at the RoBAS decision exceeds $\\sup_{P \\in B_k^\\epsilon(\\hat{P}^{\\mathrm{NPL}}_{n})} E_{\\xi\\sim P}[f_{x^\\star}(\\xi)]$ more often than the claimed rate $\\delta$. A sharper test of Theorem 3.6 is to measure $D_k(P_\\star, \\hat{P}^{\\mathrm{NPL}}_{n}) - \\inf_\\theta D_k(P_\\theta, P_\\star)$ as $n$ grows: if it does not decay at the claimed $1/\\sqrt{n}$ rate, or if it levels off at a positive floor under bounded kernels and light-tailed DGPs, the concentration result that anchors the whole method is wrong.","supporting_citations":[{"cited_title":"Kernel distributionally robust optimization: G eneralized duality theorem and stochastic approximation","cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 3.1, the kernel-DRO duality theorem that converts the RoBAS primal into the RKHS dual, and the SAA/discretization scheme plus representer lemma used for computation."},{"cited_title":"Robust B ayesian inference for simulator-based models via the MMD posterior bootstrap","cited_arxiv_id":null,"evidence_quote":"Defines the NPL-MMD posterior and MMD posterior bootstrap; its Lemmas 6 and 11 give the concentration bounds that Theorem 3.6 rests on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the NPL posterior, the DGP-level nonparametric prior that makes the RoBAS center robust to misspecification."},{"cited_title":"Decision Making under the Exponential Family: Distributionally Robust Optimisation with Bayesian Ambiguity Sets","cited_arxiv_id":"2411.16829","evidence_quote":"The DRO-BAS framework (BASPE and BASPP) that RoBAS extends, and the main Bayesian baseline in the experiments."},{"cited_title":"and Alquier, P","cited_arxiv_id":null,"evidence_quote":"Its Lemma 3.3 on MMD robustness under contamination is used to derive the Huber-contamination bound (Corollary 3.9)."}],"review_version":1}