{"id":"b6c7a19c-4a9d-4cad-b515-65c6800a3405","arxiv_id":"2607.16975","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A nonparametric calibration method enforces a known marginal for control parameters by augmenting the model output with those parameters and using optimal transport when paired data are unavailable.","lead":"This paper adds a way to calibrate computer models when some inputs have known distributions (control parameters) and others unknown (calibration parameters), producing an input distribution that matches field data while preserving the known control-parameter distribution. The core idea is to fold the known inputs into the model output and, when paired measurements are missing, to reconstruct a joint distribution by optimal transport.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unpaired case hinges on an arbitrary OT surrogate for the unobserved augmented pushforward; absent sensitivity analysis, the calibration estimate is not shown to be scientifically meaningful.","rationale":"The reader's weakest assumption is essentially the same as mine: the unobserved joint pushforward is replaced by an OT surrogate, and the final calibration depends on it. After reviewing the manuscript, I find no internal contradiction in Theorem 1 or Algorithm 2; the proof has a small typo (D\\times U should be D\\times B in the second chain) but the argument is valid. The true soft spot is that Algorithm 3 does not satisfy the hypothesis of Theorem 1 — it matches an estimated \\hat\\Pi, not \\tilde Q P^t. Since any coupling with the given marginals is observationally equivalent, the method cannot, without additional assumptions, identify a unique or scientifically preferred \\hatP_\\Lambda. The examples do not exercise this weakness because their TGDs are product measures, so the unpaired and paired problems are closer than they would be under dependence. The paper's own remark that the SK solution 'certainly differs from the true distribution' concedes the point, and no sensitivity analysis is provided. Thus the central practical claim is conditional on the surrogate being a plausible stand-in. Because the reader already returned CONDITIONAL for these reasons, I recommend UNCHANGED. If the sensitivity test shows instability, the appropriate verdict would be REJECT or a stronger CONDITIONAL requiring revision; if it shows stability, ACCEPT becomes defensible.","tokens_in":18347,"tokens_out":6842,"duration_ms":68245,"concrete_test":"Construct a synthetic TGD with the same marginals as Example 4.1 but with strong dependence between \\lambda and U, e.g., U = ((\\lambda_1+\\lambda_2)^2)/40 + \\varepsilon with small noise. Draw paired data from this TGD. Run Algorithm 3 on the unpaired q-marginal and U-marginal, and Algorithm 2 on the true paired data. Compare the estimated \\Lambda-marginals and stacked \\Lambda\\times U densities. If they diverge materially, the OT surrogate is load-bearing. As a second check, rerun Algorithm 3 with c=0 (the independent coupling) as a null cost; a material change in \\hatP_\\Lambda confirms sensitivity to the arbitrary choice in Eq. (4).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The formal result Theorem 1 is correct but definitional: it states that matching the augmented pushforward \\tilde Q P^t suffices for the constrained SCP. Algorithm 3, however, never has \\tilde Q P^t in the unpaired case. It constructs \\hat\\Pi by solving (3) with cost (4), then feeds samples from \\hat\\Pi into Algorithm 2. The output \\hatP therefore satisfies Q\\hatP \\approx P_D and \\pi_U\\hatP \\approx P_U by construction, but this is true for every coupling in F(P_D, P_U). The joint dependence between q and u is unidentifiable from the two marginals, and the particular choice (4) is not derived from any physical or statistical principle beyond averaging Q against the analyst's prior \\rho_\\Lambda. No theorem shows the induced \\hatP_\\Lambda is invariant to this choice or close to the paired-data solution. The examples use TGDs with \\lambda and U independent (Example 4.1 explicitly; Example 4.2 as an independent product of shifted betas), so the surrogate is not stress-tested against dependence. The paper acknowledges the SK joint 'certainly differs from the true distribution' and displays only marginal agreement; Section 5 flags surrogate-model regularity but not the OT-coupling choice. Thus the practical validity of the unpaired pipeline rests on an unvalidated modeling assumption. This is the load-bearing concern: the central claim of delivering a scientifically plausible estimate is not supported for dependent TGDs.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends non-parametric, measure-disintegration-based computer model calibration to the setting where the input space contains both calibration parameters with unknown distribution and control parameters with a known marginal distribution. The constrained SCP (Definition 3.1) requires the solution to match both the data pushforward and the known control marginal. The paper's formal device is to augment the computer model to \\tilde Q=(Q,\\pi_U) (Eq. (1)) and prove (Theorem 1) that matching the augmented pushforward of the trial-generating distribution is sufficient for both constraints. For paired data, the existing SCP algorithm is applied directly with \\tilde Q. For unpaired data, the paper constructs a surrogate joint distribution \\hat\\Pi on D\\times U via entropic optimal transport with cost c(q,u)=∫|q-Q(λ,u)|^2ρ_Λ(λ)dλ (Eq. (4)), samples synthetic pairs from \\hat\\Pi, and then runs the paired algorithm. Two synthetic experiments (a quadratic model and a heat-equation model with a surrogate) are used to argue that the unpaired method performs nearly as well as the paired method.","tokens_in":18742,"tokens_out":8625,"duration_ms":88389,"significance":"If the unpaired-data extension were valid, the paper would contribute a practical method for calibrating computer models with stochastic control parameters whose marginal is known but whose joint dependence with outputs is unobserved. The manuscript has several genuine strengths: it ships reproducible code, Theorem 1 is a clean and correct marginalization argument, Theorem 2 is a standard existence result, and the regularity conditions are discussed carefully in an appendix. However, the central empirical claim — that the unpaired method is nearly as good as paired calibration — is not supported by the evidence presented. The unpaired pipeline relies on an arbitrary optimal-transport surrogate for an unidentifiable joint distribution, and the paper offers no sensitivity analysis, no dependent-TGD experiments, and no quantitative comparison. These issues are load-bearing because the unpaired case is the main methodological novelty; the formal paired case is a straightforward application of existing methods.","major_comments":[{"comment":"In the unpaired case, Algorithm 3 replaces the unavailable augmented pushforward \\tilde Q P^t_{\\Lambda\\times U} with the entropic-OT coupling \\hat\\Pi computed by Eq. (3) under cost (4). Since only the marginals P_D and P^t_U are identified from unpaired data, every coupling in F(P_D,P^t_U) satisfies the two constraints in Definition 3.1 by construction; the final \\hat P_{\\Lambda\\times U} depends on which coupling is chosen. The paper does not show that \\hat P_\\Lambda is insensitive to this choice or close to the paired-data solution, and the authors themselves note that the Sinkhorn joint 'certainly differs from the true distribution.' This is load-bearing: the advertised scientific plausibility of the unpaired estimate is not established. I request (i) a sensitivity analysis with respect to the cost function and the regularization \\epsilon, and (ii) experiments where the TGD has non-tri","section":"§3.3, Eq. (4)"},{"comment":"The central empirical claim that the unpaired method 'yields similar estimates as paired data' rests on two single-run synthetic examples with visual comparison only. No quantitative distance or discrepancy measure between the paired and unpaired \\hat P_\\Lambda is reported, and there are no repeated simulations with different random seeds or Sinkhorn draws. Since this claim is the main motivation for the unpaired algorithm, the evidence is insufficient as it stands. Please add quantitative comparisons (e.g., L1 or Wasserstein distance between estimated marginal densities of \\Lambda), repeated-run variability, and ideally a check of whether the unpaired solution remains close to paired when the surrogate coupling is changed.","section":"§4, Figures 2, 5, 6"},{"comment":"Theorem 1 is correct, but it only reduces the constrained problem to exact matching of \\tilde Q P^t_{\\Lambda\\times U}. In the unpaired pipeline this target is replaced by the surrogate \\hat\\Pi, so Theorem 1 does not validate Algorithm 3. The U-marginal agreement of the final solution is imposed by construction in the OT step and then reproduced by Algorithm 2; it is not a data-informed property. The paper should state this limitation explicitly and separate the formal contribution (Theorem 1 and the paired Algorithm 2) from the heuristic unpaired extension, which currently requires an unvalidated modeling assumption.","section":"Theorem 1, Definition 3.1"}],"minor_comments":[{"comment":"The text says 'We return to the heat transfer problem of Example 4.2' but the running example is Example 1.1.","section":"§4.2"},{"comment":"The statement that without the KL term '\\hat\\Pi would overfit to the graph of Q and not admit a density in Lebesgue measure' is imprecise for the discrete implementation described in Algorithm 3. Clarify whether this refers to the continuous population problem or to the discrete Sinkhorn approximation.","section":"§3.3"},{"comment":"The histogram estimator has undefined 0/0 ratios in bins that contain no prior samples. The paper should state how zero-count bins are handled in the implementation.","section":"Algorithms 1, 2"},{"comment":"The heat-equation example uses an XGBoost surrogate for the PDE solver, and Section 5 acknowledges that this introduces additional statistical error and may violate regularity conditions. A diagnostic of surrogate accuracy (e.g., held-out error) would help the reader assess the example.","section":"§4.2"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a competent, honest extension of [9] to the constrained setting where some stochastic inputs have known marginals. The formal reduction in Theorem 1 is correct but elementary—matching the augmented pushforward is exactly matching the two marginals plus an arbitrary coupling, so the paired-data algorithm is just [9] applied to a modified map. That is worth stating cleanly, and the paper does it cleanly. The genuinely new piece is the unpaired-data pipeline with entropic OT and a model-aware cost, and that is where I would focus referee attention.\n\nThe paper does several things well. It defines the constrained SCP precisely, shows why the naive unconstrained solution fails (Example 3.1), and gives a concrete algorithm with code. Theorem 2 and the measurability result in Appendix B are fine. The authors are also unusually candid: they say the Sinkhorn joint \"certainly differs from the true distribution,\" they flag surrogate-model regularity in Section 5, and they do not oversell the paired case.\n\nThe soft spot is the unpaired case, and it is load-bearing. From the two marginals alone, the joint dependence between q and u is unidentifiable. The cost function (4) is a heuristic—averaging squared model error over the prior—and there is no argument that the resulting calibration estimate is invariant to that choice, or close to the paired-data answer, for any realistic dependence. The examples both use TGDs with lambda and U independent, so they do not stress-test the surrogate at all. The numerical support is also single-run and visual: no error bars, no sensitivity analysis for epsilon (despite the claim of robustness), no repeated-seed variation. The marginal-preservation property is enforced by construction rather than demonstrated to be statistically meaningful.\n\nNone of this is fatal. The paired case stands on its own, and the unpaired algorithm is a reasonable proposal that needs stronger validation, not a fundamentally wrong idea. But as written, the central empirical claim—that unpaired data incur minimal cost—is supported only by two favorable examples.\n\nWho is this for? People working in nonparametric computer-model calibration and inverse UQ. It is a within-subfield contribution, not a field-reorganizing one. I would send it to peer review, with the expectation of major revisions: add a dependent-TGD example, report variability across runs and a sweep over epsilon, and ideally give some theoretical or at least empirical argument that the OT coupling choice matters less than the marginals. The authors should get credit for a clean, honest framework, but the unpaired pipeline needs more evidence before I would trust it in practice.","headline":"A correct but mostly definitional extension of the disintegration-based calibration program to known control-parameter marginals; the real question is whether the unpaired-data OT surrogate can be trusted, and the paper's evidence on that is thin.","tokens_in":19170,"tokens_out":1748,"would_cite":true,"duration_ms":20798,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F15","62G05","49Q22"],"pacs":[],"model":"deepseek-v4-flash","headline":"Matching output–control pairs solves constrained calibration.","keywords":["statistical calibration","computer model calibration","disintegration of measure","optimal transport","entropic regularization","uncertainty quantification","Bayesian inverse problems","nonparametric estimation"],"falsifier":"Construct a synthetic problem where the true coupling has strong negative dependence between Q and u (e.g., u = 1 − Q), run both the paired and unpaired algorithms, and compare the estimated marginal distributions on Λ. If the unpaired estimate substantially differs from the paired estimate despite both matching the desired marginals, then the optimal transport surrogate fails to recover scientifically plausible dependence.","tokens_in":18213,"feed_emoji":"⚙️","tokens_out":4373,"duration_ms":44517,"temperature":0.7,"pith_summary":"This paper tackles the problem of calibrating a computer model when some stochastic inputs have unknown distributions (calibration parameters) while others have known distributions (control parameters). The authors show that the constrained calibration problem—fitting field data while preserving the known marginal of the control parameters—can be reduced to the standard unconstrained calibration problem by augmenting the model output with the control parameter itself. The key theoretical result is that matching the pushforward of the augmented map (Q(λ,u), u) automatically satisfies both constraints: matching the observed data distribution and matching the known control marginal. For the practically important case where control parameter values are not recorded alongside field observations, the paper builds a surrogate joint distribution using entropic optimal transport with a model-aware cost function, then applies the standard disintegration-based estimator. The result is a distributional estimate over the input space that respects both constraints, demonstrated on a quadratic model and a heat-equation example.","feed_headline":"Matching output–control pairs solves constrained calibration","feed_subtitle":"Estimates input distributions consistent with field data while preserving known control-parameter marginals.","key_machinery":"The central object is the augmented computer model Q̃(λ,u)=(Q(λ,u),u), which embeds the control parameter into the output space. Any probability measure that matches the trial-generating measure's pushforward under Q̃ automatically satisfies both the data-matching condition and the known-marginal condition. To handle unpaired data, the paper constructs a surrogate for the joint distribution of (Q,u) by solving the entropic Kantorovich problem with the model-aware cost c(q,u)=∫_Λ |q−Q(λ,u)|² ρ_Λ(λ)dλ, regularized by KL divergence to the independent coupling; the Sinkhorn–Knopp algorithm computes the coupling. The disintegration-based estimator is then applied on the augmented space, which req","core_discovery":"The paper's central claim is Theorem 1: if a candidate joint distribution on Λ×U has the same pushforward as the trial-generating distribution under the augmented map Q̃(λ,u)=(Q(λ,u),u), then that candidate automatically solves the constrained statistical calibration problem—its pushforward under Q matches the observed field data, and its marginal on U matches the known control distribution. This reduction turns the constrained problem into an ordinary unconstrained SCP on the augmented output space D×U, for which a disintegration-based estimator already exists. When samples are unpaired, the paper supplies the missing joint pushforward by solving an entropically regularized optimal transpor","pith_inferences":["The surrogate coupling for unpaired data is fundamentally unidentifiable from the two marginals alone; any coupling with the given marginals yields a valid constrained SCP solution, but only the true coupling captures the actual physical dependence, so the calibrated distribution on Λ can be misleading even when both constraints are satisfied.","Different choices of the transport cost (e.g., copula-based or domain-informed costs) would produce different—still constraint-satisfying—calibrations, meaning the optimal transport step is not just computational but encodes a modeling assumption about the joint structure.","The known continuity results for the unconstrained SCP suggest a limiting argument: as the control distribution concentrates to a point, the constrained solution should degenerate into a conditional calibration at that fixed control value, bridging to classical fixed-control calibration.","A testable extension is to run the unpaired algorithm on synthetic data where the true coupling is deliberately anti-correlated with the model-aware cost, then compare the resulting Λ-marginal against the paired-data benchmark to reveal potential bias from the surrogate coupling."],"forward_implications":["When paired data exist, the constrained calibration problem is solved by applying the known unconstrained algorithm directly to the augmented model, requiring no new estimation scheme.","When only unpaired data exist, the model-aware optimal transport coupling provides a surrogate for the missing joint, and the resulting calibration preserves both the data distribution and the known control marginal by construction.","The augmented approach imposes a dimension condition (m ≤ n) but relaxes the Jacobian rank condition to only the λ-derivatives of Q, which may ease verification of regularity conditions.","In the examples, the calibration solution is stable across a broad range of the entropy regularization parameter ε, even though the transport coupling itself varies substantially, suggesting an unexpected robustness worth theoretical investigation.","The method extends the scope of non-parametric calibration to problems with partially specified stochastic inputs, including physical settings where control parameters follow well-established distributions."],"fun_headline_variants":["One pushforward equality solves constrained calibration","Same augmented pushforward ⇒ calibration constraints met","Match Q~, not Q, to satisfy control marginals","Theorem: unconstrained SCP on augmented output suffices","Disentangling calibration: match augmented output pairs"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"For unpaired data, the surrogate coupling obtained by optimal transport stands in for the unobserved joint distribution of outputs and control parameters, and the final calibration inherits this assumption without any way to verify it from the data alone.","fun_headline_variants_meta":{"raw":{"variants":["One pushforward equality solves constrained calibration","Same augmented pushforward ⇒ calibration constraints met","Match Q~, not Q, to satisfy control marginals","Theorem: unconstrained SCP on augmented output suffices","Disentangling calibration: match augmented output pairs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000479,"raw_usage":{"total_tokens":2141,"prompt_tokens":609,"completion_tokens":1532,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":353,"completion_tokens_details":{"reasoning_tokens":1473}},"tokens_in":353,"tokens_out":1532,"duration_ms":12926,"temperature":1.0,"reasoning_tokens":1473,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T19:22:29.254847+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a synthetic problem where the true coupling has strong negative dependence between Q and u (e.g., u = 1 − Q), run both the paired and unpaired algorithms, and compare the estimated marginal distributions on Λ. If the unpaired estimate substantially differs from the paired estimate despite both matching the desired marginals, then the optimal transport surrogate fails to recover scientifically plausible dependence.","supporting_citations":[],"review_version":1}