{"id":"ca22ef88-5926-48a5-9525-e31af3bd43d9","arxiv_id":"2507.23136","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Probabilistic classifiers inherit label randomness; this paper formalizes per-individual prediction variance as regret, gives a logistic-regression approximation, and tests it on loan and health datasets.","lead":"This paper introduces observational multiplicity: when class labels are random draws, retraining a probabilistic classifier on different label draws can change predictions for individuals, and this arbitrariness is uneven. It defines a variance based regret measure, proves an approximation for logistic regression, and tests how estimated regret can guide abstention and targeted data collection.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The gap between Definition 2.1's true-probability resampling and Theorem 3.1's model-resampling is the load-bearing weakness; the paper's model-relative reinterpretation does not close it.","rationale":"I agree with the reader's weakest assumption: the central quantity in Definition 2.1 is defined under the true conditional label probabilities, while Algorithm 1 and Theorem 3.1 operate under the fitted model's own resampling distribution. This is not a criticism of the theorem itself; Theorem 3.1 appears to be a real analytic contribution and the proof is substantially self-contained. The problem is the transfer from the proven bound to the stated notion of regret. Section 2.3 is unusually candid about the substitution, and the paper even argues that model-implied uncertainty is meaningful for deployment. That argument supports a reinterpretation of the method, but it does not establish that the method measures the quantity named in Definition 2.1. The semi-synthetic validation uses a logistic oracle, so it cannot expose the misspecification gap; the bank results already show systematic underestimation for high-true-regret points, which is the predicted failure mode. A misspecified-oracle rank test would settle whether the gap is practically severe. The reader's CONDITIONAL verdict already captures this concern, so I do not move the verdict. If anything, the paper would be strengthened by either restricting its claims to well-specified logistic settings or explicitly redefining regret as model-implied uncertainty throughout.","tokens_in":21247,"tokens_out":9554,"duration_ms":116105,"concrete_test":"Generate semi-synthetic labels from an oracle outside the logistic class, e.g., p_i = sigmoid(q(x_i)) with quadratic and interaction terms, or probabilities from a gradient-boosted model fit once to the real labels. Fit a linear logistic regression, then compute true regret by resampling labels from the oracle 300 times and refitting, and estimated regret via Algorithm 1. Compare the two on Spearman rank correlation and error by regret decile, especially the top decile. If estimated ranks do not preserve true ranks, or top-decile underestimation is severe, the estimator does not track Definition 2.1 and the central claim must be restricted to well-specified settings; if ranks are preserved, the well-specification gap may be tolerable in practice.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing gap is between Definition 2.1 and Theorem 3.1 / Algorithm 1. Definition 2.1 defines regret as the variance of f_{A(D')}(x_i) over labels resampled from the true conditional probabilities. Algorithm 1 resamples from the fitted model, and Theorem 3.1 only analyzes labels \\hat y_i ~ Bern(p_i) with p_i from an initial logistic model. Section 2.3 admits that the proxy is \"exactly the model we were attempting to learn in the first place\" and reframes the output as uncertainty \"directly implied by the model we are seeking to deploy.\" That reframing is honest but does not deliver the quantity promised in Definition 2.1. Under model misspecification, the estimated regret is a property of the deployed model, not of the true label stochasticity; the bank dataset's underestimated high-regret points and slope-greater-than-one groups are symptoms of this. Consequently, the safety applications (abstention and data collection) are conditional either on well-specification or on explicitly redefining regret as model-implied uncertainty rather than true observational multiplicity.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"Across probabilistic classification, the paper argues that a single observed label vector is one draw from a stochastic process, so different draws produce different plausible training sets and different models. It defines observational multiplicity as the resulting arbitrariness and formalizes a per-example regret (Definition 2.1) as the variance of the prediction f_A(D')(x_i) when D' is created by resampling labels from the true conditional probabilities. Algorithm 1 estimates this variance by resampling labels from a fitted model, and Theorem 3.1 gives a finite-sample bound for logistic regression: if epsilon < 1, the variance of the resampled prediction is within a factor (1 +/- epsilon) of Q_i = p_i^2(1-p_i)^2 x_i^T H^{-1} x_i. The paper validates the estimator on three semi-synthetic datasets, applies it to 13 million Fannie Mae loans with a GBDT, and demonstrates abstention and active-learning uses.","tokens_in":21515,"tokens_out":9507,"duration_ms":117562,"significance":"The paper tackles a timely and important question: when multiple equally performant models disagree on an individual, how much of that disagreement is irreducible randomness in the observation process? The proof of Theorem 3.1 is a genuine technical contribution, applying Bach's self-concordance bounds in a nontrivial way and producing an explicit, actionable formula. The semi-synthetic study is careful, and the Fannie Mae experiment shows the framework scales. The significance is conditional, however, on the identification of the model-resampled variance with the true-probability variance of Definition 2.1; the paper itself acknowledges the proxy, and the experimental evidence does not close the gap under misspecification. With a redefinition or a transfer theorem, the work would be a solid contribution to model multiplicity and uncertainty quantification.","major_comments":[{"comment":"Definition 2.1 defines regret as variance under resampling from the true conditional probabilities of the labels, but Algorithm 1 resamples from the fitted model's own probabilities and Theorem 3.1 analyzes exactly that model-resampled process (labels ~ Bern(p_i) from an initial logistic fit). Section 2.3 concedes that the proxy is 'exactly the model we were attempting to learn in the first place.' These are different stochastic mechanisms unless the fitted model coincides with the true conditional probability function; under misspecification, Algorithm 1 estimates a property of the deployed model rather than the quantity in Definition 2.1. The underestimation of high-regret points in the bank data is consistent with this divergence. The paper should either prove a bound that transfers from model-resampled variance to true regret under well-specification, or reframe regret and the safety applications as explicitly model-implied rather than as estimates of the true observational multiplicity.","section":"2.3, Algorithm 1, Definition 2.1"},{"comment":"The central theorem applies only when epsilon < 1, but the paper never reports whether this condition holds in any of its experiments, and the condition depends on d, Xmax/Xmin, ||theta*||_2, and the smallest Hessian eigenvalue lambda_min, so it is most likely to fail in exactly the ill-conditioned or high-dimensional settings where regret matters. In the loan experiment the model is a GBDT, which is outside the theorem's scope entirely. Please state how a practitioner can verify epsilon < 1 and provide at least one experiment, synthetic or real, where the condition and the bound are actually checked; otherwise the empirical support for the theorem's regime is missing.","section":"Theorem 3.1 and Section 4"},{"comment":"The semi-synthetic validation in Section 4.1 generates ground-truth labels from a logistic model whose parameters are themselves fit to the original data, so the experiments are confined to the well-specified logistic family. The Fannie Mae experiment is fully real and therefore lacks any ground-truth regret to compare against, and the GBDT estimator has no supporting theory. There is no experiment that quantifies the gap between true regret (Definition 2.1) and Algorithm 1's output under a deliberately misspecified model (e.g., nonlinear label-generating function with a logistic estimator). Such an experiment would directly address the main weakness identified in Major Comment 1 and would determine whether the reported bank underestimation is an isolated finite-sample artifact or a systematic feature.","section":"4.1 and 4.2"}],"minor_comments":[{"comment":"The abstract contains several grammatical errors, including 'can can undermine', 'as a result of different training labels change', and 'estimating regret promote safety'; these should be corrected.","section":"Abstract"},{"comment":"Lemma 5.5 states Pr(B_c) <= 2d exp(...), but the proof bounds the complement B_c^c; the event/complement notation is also inconsistent in the lower-bound part of Section 5.4, which uses 'B_c' for both the event and its complement.","section":"5.3"},{"comment":"Algorithm 1 does not list k (the number of resamples) as an input; the experiments use k = 300, but the pseudocode should state k explicitly.","section":"Algorithm 1"},{"comment":"The claim that the tail of the regret distribution 'appears to follow a power law' is supported only by an eyeballed line y = 3.5e-10 x^{-3}; a fitted exponent and goodness-of-fit measure would make the claim precise.","section":"4.2"},{"comment":"The caption 'The five points at the top and five points at the bottom all have an 80% probability of lying in the positive and negative class' is ambiguous about which points are positive and which are negative; please clarify.","section":"Figure 1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is honest about the proxy in Section 2.3, and the authors should be credited for that, but the title and abstract overstate the connection to true observational multiplicity. The most viable path to acceptance is to either (a) add a bridging result or a misspecified-model experiment that shows when Algorithm 1's output is a conservative or tight estimate of Definition 2.1, or (b) explicitly redefine regret as model-implied uncertainty and adjust the contributions and safety framing. I also note the paper cites relevant prior work on predictive multiplicity and dataset multiplicity; the novelty relative to [19] should be sharpened."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper gives the probabilistic classification community something real: a per-point regret measure (variance of the predicted probability under label resampling) and, for logistic regression, a finite-sample variance approximation with a rigorous proof. Theorem 3.1 is the main event. The proof follows Bach's self-concordance analysis, and the bound |Var(p̂_i) − Q_i| ≤ εQ_i with Q_i = p_i²(1−p_i)² x_iᵀH⁻¹x_i is nontrivial and clearly stated. That alone is worth a referee's time.\n\nThe paper also does several things well. The semi-synthetic validation is thoughtfully set up: ground-truth logistic parameters are fit to real data, so the estimator can be compared to true regret, and the figures show close agreement in well-specified settings. The use cases (abstention and data collection) are simple but demonstrate where regret could help. The Fannie Mae experiment is a concrete observation: 13 million loans, a power-law tail in regret, and high regret concentrated among delinquent borrowers.\n\nThe soft spot is the one the stress-test note flags, and I agree it is real. Definition 2.1 defines regret as variance under resampling from the true conditional probabilities. Algorithm 1 resamples from the fitted model, and Theorem 3.1 only analyzes that model-resampling case. The paper is honest about this in Section 2.3—it says the proxy is 'exactly the model we were attempting to learn' and reframes the output as uncertainty 'directly implied by the model we are seeking to deploy.' That honesty is to its credit, but it does not close the gap. Under misspecification, the estimator measures a property of the deployed model, not the aleatoric uncertainty of the label process. The underestimated high-regret points in the bank dataset are a symptom. So the framing outsells the estimator slightly: the safety applications are conditional on trusting the model, not on the Definition 2.1 quantity.\n\nMinor issues: no code or data released, which matters for a method paper; the semi-synthetic ground truth is fit to the same data, so the well-specified case is partly self-confirming; and the ε<1 condition restricts the theorem's practical range, though the paper notes the constant is loose.\n\nWho should read this? Anyone working on predictive multiplicity, uncertainty quantification, or selective prediction for probabilistic classifiers. It sits close to Watson-Daniels et al. and Nagaraj et al., but the variance bound is a new card on the table. I would send it to peer review: a serious referee can push on the definition/estimator gap and the missing artifacts, but the theorem deserves a venue.\n\nRecommendation: accept for review, with the expectation of substantial revision on framing and reproducibility.","headline":"Introduces a useful regret measure for probabilistic classification with a rigorous logistic-regression variance bound, but the estimator answers model-implied uncertainty rather than the true-label regret of Definition 2.1 unless the model is well-specified.","tokens_in":21992,"tokens_out":6024,"would_cite":true,"duration_ms":57764,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A closed-form estimate shows which probability predictions are arbitrary, because the observed labels themselves are a random draw.","keywords":["observational multiplicity","predictive multiplicity","regret","probabilistic classification","model multiplicity","uncertainty quantification","logistic regression","safe machine learning"],"falsifier":"Take a dataset whose labels are generated by a known nonlinear probability model, fit a logistic regression to it, and compute both true regret (resampling labels from the known generator, retraining, and taking variances) and estimated regret from Algorithm 1. If the two diverge systematically, and in particular if estimated regret underestimates true regret for points whose fitted probabilities are extreme, then the well-specification premise fails and Algorithm 1 measures the model's internal plausibility, not the label randomness the definition targets.","tokens_in":21043,"feed_emoji":"🎲","tokens_out":6771,"duration_ms":69783,"temperature":0.7,"pith_summary":"This paper argues that a familiar source of arbitrariness in probabilistic classification is the randomness of the observed labels: a dataset is one draw of the labels, and a different draw would produce an equally valid but different model. The authors call this observational multiplicity and propose to measure it per individual as regret, defined as the variance of a point's predicted probability when the training labels are resampled from the true conditional probabilities. For logistic regression they prove that this variance is, up to a relative error, the closed-form quantity $p_i^2(1-p_i)^2 x_i^\\top H^{-1} x_i$, where $H$ is the Hessian of the logistic loss. If the claim holds, a practitioner can estimate which individuals receive arbitrary predictions without collecting new labels, and can abstain on those points or collect their labels first. The stakes are safety and fairness: the burden of arbitrariness is not shared equally, and it concentrates on identifiable subgroups.","feed_headline":"Which probability predictions are hostage to label luck?","feed_subtitle":"A new regret formula flags individuals whose predicted probabilities flip when observed labels are redrawn.","key_machinery":"The load-bearing object is the regret identity $Q_i = p_i^2(1-p_i)^2 x_i^\\top H^{-1} x_i$, with $H = \\sum_j p_j(1-p_j)x_j x_j^\\top$ the Hessian of the logistic loss at the fitted parameters. It expresses per-point instability as the product of label noise, $p_i^2(1-p_i)^2$, and leverage in the inverse-Hessian metric, $x_i^\\top H^{-1} x_i$, so regret is large near $p_i=1/2$ and in directions where the data barely constrain the model. The proof machinery is a quantitative self-concordance bound on one Newton step for logistic regression, which controls the difference between the retrained parameters and a single linearized step, plus Bernstein concentration on the resampled-label gradient. Algorithm 1 supplies the estimator: resample labels from the fitted model, retrain, record the point's predicted probability, repeat, and take the variance.","core_discovery":"The central discovery is that arbitrariness in probabilistic classification can be defined as a per-point variance and, for logistic regression, estimated without retraining on fresh data. Regret is the variance of $f_{A(D')}(x_i)$ when $D'$ is formed by resampling each label from the true conditional probability $p_i^*$; since those probabilities are unknown, Algorithm 1 resamples from the fitted model and averages the resulting predictions. Theorem 3.1 gives that when the resampling model is the initial logistic model, the resulting variance obeys $|\\operatorname{Var}(\\hat p_i) - Q_i| \\le \\epsilon Q_i$ with $Q_i = p_i^2(1-p_i)^2 x_i^\\top H^{-1} x_i$ and $\\epsilon$ a small constant depending on dimension, feature norms, and the Hessian's smallest eigenvalue, provided $\\epsilon<1$. The paper shows this matches true regret in semi-synthetic experiments and uses it to flag high-regret points for abstention and targeted data collection, including on a large mortgage-loan dataset where estimated regret follows a heavy-tailed distribution.","pith_inferences":["Inference: the same $Q_i$ formula suggests a cheap diagnostic for feature engineering: adding features that raise the minimum eigenvalue of $H$ in a cluster's subspace should shrink regret there, which is testable.","Inference: regret as defined is a property of the pairing of a model class and a dataset, so comparing regret across model classes on the same data would separate label noise from model flexibility; the paper's gradient-boosting experiment on loan data is a step in that direction.","Inference: because the estimator resamples from the deployed model, a model that is confidently wrong will report low regret; calibration audits should run alongside regret estimates before using abstention in high-stakes decisions.","Inference: the power-law tail found on the loan dataset, if it persists, implies that fairness audits should report regret distributions per group rather than aggregate accuracy, since group-level means can hide a small set of arbitrarily predicted individuals."],"forward_implications":["For logistic regression, per-point regret can be computed or bounded from a single dataset using $Q_i$, so no fresh label collection is needed to know which predictions are arbitrary.","Regret concentrates: points with predicted probability near $1/2$ and points lying in low-curvature directions of the Hessian carry most of the instability.","Estimated regret supports selective abstention: withholding predictions for high-regret points reduces average KL divergence to the true probabilities, close to the best possible in the bank dataset.","Regret-guided data collection outperforms uniform sampling in two of the three semi-synthetic datasets, reaching full-dataset accuracy after a few additional samples.","On a real mortgage-loan dataset, 98% of points have negligible regret while a small tail follows an approximate power law, so arbitrariness is a minority phenomenon that is identifiable."],"supporting_citations":[{"why":"It supplies the self-concordance bound on one Newton step for logistic regression that proves Theorem 3.1.","marker":"[1]"},{"why":"It motivates the term regret for irreducible error under label noise; the paper extends that idea from hard labels to probability predictions.","marker":"[13]"},{"why":"It introduces predictive multiplicity, the phenomenon for which observational multiplicity is presented as a new mechanism.","marker":"[10]"},{"why":"It defines predictive multiplicity for probabilistic classifiers, the closest prior notion the paper distinguishes from observational multiplicity.","marker":"[19]"},{"why":"It frames the broader dataset multiplicity problem from which observational multiplicity is isolated as a distinct source.","marker":"[11]"},{"why":"It provides the bank telemarketing data used for semi-synthetic validation and the group-level regret analysis.","marker":"[12]"},{"why":"It provides the sleep apnea data used in the semi-synthetic validation experiments.","marker":"[15]"},{"why":"It provides the Wisconsin breast cancer data used in the semi-synthetic validation experiments.","marker":"[20]"},{"why":"It provides the large mortgage-loan dataset on which the paper computes estimated regret for a real-world classifier.","marker":"[6]"}],"fun_headline_variants":["How to spot predictions that flip when labels shift","A formula for arbitrariness in probability models","When reshuffling labels changes a model's verdict","Flagging predictions that depend on label luck"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the fitted model's probabilities are the true conditional label probabilities, so that resampling labels from the model mimics resampling labels from nature; when the model is misspecified, the estimated regret describes arbitrariness implied by the model, not by the data-generating process, and the paper reports exactly this kind of underestimation for high-regret points in the bank dataset.","fun_headline_variants_meta":{"raw":{"variants":["How to spot predictions that flip when labels shift","A formula for arbitrariness in probability models","When reshuffling labels changes a model's verdict","Flagging predictions that depend on label luck"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000462,"raw_usage":{"total_tokens":2316,"prompt_tokens":954,"completion_tokens":1362,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":570,"completion_tokens_details":{"reasoning_tokens":1303}},"tokens_in":570,"tokens_out":1362,"duration_ms":11038,"temperature":1.0,"reasoning_tokens":1303,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T11:00:46.908631+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a dataset whose labels are generated by a known nonlinear probability model, fit a logistic regression to it, and compute both true regret (resampling labels from the known generator, retraining, and taking variances) and estimated regret from Algorithm 1. If the two diverge systematically, and in particular if estimated regret underestimates true regret for points whose fitted probabilities are extreme, then the well-specification premise fails and Algorithm 1 measures the model's internal plausibility, not the label randomness the definition targets.","supporting_citations":[{"cited_title":"Self-concordant analysis for logistic regression.Electronic Journal of Statistics, 4: 384–414, 2010","cited_arxiv_id":null,"evidence_quote":"It supplies the self-concordance bound on one Newton step for logistic regression that proves Theorem 3.1."},{"cited_title":"Regretful decisions under label noise","cited_arxiv_id":null,"evidence_quote":"It motivates the term regret for irreducible error under label noise; the paper extends that idea from hard labels to probability predictions."},{"cited_title":"Predictive multiplicity in classification","cited_arxiv_id":null,"evidence_quote":"It introduces predictive multiplicity, the phenomenon for which observational multiplicity is presented as a new mechanism."},{"cited_title":"Parkes, and Berk Ustun","cited_arxiv_id":null,"evidence_quote":"It defines predictive multiplicity for probabilistic classifiers, the closest prior notion the paper distinguishes from observational multiplicity."},{"cited_title":"Brandon Westover, Cynthia Rudin, and Matt T","cited_arxiv_id":null,"evidence_quote":"It provides the sleep apnea data used in the semi-synthetic validation experiments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the Wisconsin breast cancer data used in the semi-synthetic validation experiments."},{"cited_title":"Fannie Mae single-family loan performance data, 2025","cited_arxiv_id":null,"evidence_quote":"It provides the large mortgage-loan dataset on which the paper computes estimated regret for a real-world classifier."}],"review_version":1}