Proper Scoring Rules for Agentic Uncertainty Quantification
Suresh Raghu, Satwik Pandey, Shashwat Pandey
Abstract
Language-model agents increasingly emit uncertainty signals throughout a trajectory, but existing agentic UQ evaluations often conflate ranking usefulness with probabilistic truthfulness. AUROC, AUPRC, risk-coverage, Trajectory ECE, and scalarized trajectory scores evaluate discrimination, binwise calibration, or collapsed summaries, but do not strictly elicit the full prefix-conditioned success-probability trace . Building on prequential proper scoring, we introduce the Trajectory Proper Score (TPS), a predictor-agnostic family of strictly proper trajectory-level scoring rules for any per-step uncertainty signal calibrated into a probability of eventual success. We prove that TPS strictly elicits the success-probability process under complete observation, within the chosen score family and weight schedule. We extend the construction to administratively censored trajectories by projecting the complete-data score onto the observable stopped prefix, yielding an exact -weighted reduced score and a tractable approximation when is unestimated. We further show that common trajectory evaluators target weaker objects than the full prefix-conditioned probability process: Trajectory ECE is resolution-blind, while scalarized Trajectory Brier elicits only the collapsed scalar, not the full trace. Experiments on StrategyQA, Tau2-Bench, HotpotQA, and WebShop show that these theoretical distinctions are operationally visible: probability recalibration can substantially change TPS while leaving rank metrics nearly unchanged, and the tractable censored approximation can change the verdict relative to complete-only evaluation.
AI Impact Assessments
(1 models)Scientific Impact Assessment: Proper Scoring Rules for Agentic Uncertainty Quantification
Core Contribution
This paper identifies a precise gap in the evaluation toolbox for agentic uncertainty quantification: existing metrics (AUROC, AUPRC, Trajectory ECE, scalarized Trajectory Brier) do not strictly elicit the full prefix-conditioned success-probability process . The authors introduce the Trajectory Proper Score (TPS), a weighted sum of strictly proper binary scores applied at each trajectory prefix, and prove it strictly elicits this process under complete observation. They extend TPS to administratively censored trajectories via conditional projection, yielding an exact -weighted reduced score and a tractable pessimistic approximation.
The conceptual contribution is clarifying what *object* different evaluators target. The paper formalizes that Trajectory ECE is resolution-blind (Theorem A), and scalarized proper scores under common aggregators () elicit only a collapsed scalar, not the full trace (Theorem B). These are not surprising results to scoring rule specialists, but their explicit formalization for the agentic UQ community is valuable.
Methodological Rigor
The theoretical framework is sound. Theorem 4.1 follows straightforwardly from per-step conditional strict propriety summed with positive weights—the proof is essentially a direct application of known proper scoring rule theory lifted to a sequential setting. The authors are transparent about this, which is appropriate. The beta-family parameterization via the Schervish-Buja threshold-mixture construction is well-established, and its application here is clean.
The censored extension (Theorems 4.2, 4.3) is more technically interesting. The conditional projection onto the observable stopped prefix, yielding an exact -weighted form under non-informative censoring assumptions, connects the agentic setting to survival analysis methodology. The assumptions (non-informative censoring, administrative stop) are clearly stated, and the paper is disciplined about when they hold—the exclusion of 192 parse-error trajectories in WebShop as informative censoring demonstrates careful assumption enforcement.
The Monte Carlo continuation audit on HotpotQA (Appendix H.2) validates the exact reduced form to numerical precision (), confirming implementation correctness. The artificial censoring validation on three datasets with closed-form decomposition matching is thorough.
However, the experiments are somewhat limited: one model (Gemma 4 31B), one harness (ReAct), and the predictor streams are relatively basic (verbal confidence, token probabilities, entropy). The paper acknowledges this limitation explicitly, arguing evaluator-side claims don't require predictor-side diversity—a reasonable but not fully satisfying position.
Potential Impact
Immediate utility: TPS provides a principled evaluation metric for the growing agentic UQ community. As agents are increasingly deployed with deferral, human handoff, and reflection mechanisms that consume probabilities (not just rankings), having an evaluator that strictly elicits calibrated probabilities is practically important.
Censoring awareness: The censored extension addresses a real and underappreciated problem. Many benchmarks impose step budgets, and complete-only evaluation silently discards the hardest trajectories. The WebShop result—where 47% of trajectories are administratively censored and censored-aware scoring shifts the verdict by 0.159 nats—demonstrates this is not merely theoretical.
Calibration loss: The dual use of TPS as both evaluator and calibration loss (Appendix J.4) for training confidence heads or post-hoc calibrators under fixed policies could influence future predictor-side methods.
Limitations on breadth: The restriction to binary terminal outcomes is significant. Many real agent tasks have graded success, partial credit, or multiple objectives. The paper acknowledges this but doesn't sketch extensions. The non-informative censoring assumption excludes adaptive stopping, which is increasingly common in production agent systems with monitors.
Timeliness & Relevance
This paper arrives at an opportune moment. Agentic UQ is a rapidly growing subfield (SAUP, UProp, AUQ, STeCa are all 2024-2026), and the evaluation side has not kept pace with predictor-side innovation. The paper correctly diagnoses that the community is evaluating trajectory uncertainty with tools designed for static classification or ranking, not for sequential probability elicitation. The gap between rank metrics and probabilistic truthfulness is especially relevant as agents are deployed in high-stakes settings requiring reliable probability-based decision-making.
Strengths
1. Clean problem formulation: The paper precisely identifies what existing evaluators target versus what they should target, with formal theorems backing each claim.
2. Assumption discipline: The treatment of censoring is notably careful—excluding informative failures, validating assumptions, and clearly distinguishing exact from approximate scores.
3. Operational visibility: The Tau2-Bench calibration experiment ( for TPS vs. for AUROC) compellingly demonstrates the theoretical distinction is empirically large.
4. Comprehensive appendices: The extensive robustness checks across score families, weight schedules, censoring rates, and predictor streams substantially strengthen the empirical claims.
Limitations
1. Limited novelty in core theory: The main theorem is a direct application of known proper scoring rule theory to a sequential setting. The intellectual contribution is more in *framing* than in *technique*.
2. Single model/harness: All experiments use Gemma 4 31B with ReAct, limiting generalizability claims.
3. Binary outcomes only: Real agentic tasks frequently involve partial success or continuous rewards.
4. Weight schedule sensitivity: The choice of weight schedule (linear-front, uniform, etc.) is evaluator-chosen and influences results, but guidance on principled selection is limited.
5. Simple censored approximation: The operational approximation elicits a different target than the original , which somewhat undermines the strict propriety motivation when exact continuation weights are unavailable.
Overall Assessment
This is a well-executed paper that brings needed rigor to agentic UQ evaluation. Its primary value is conceptual clarity and formalization rather than deep technical novelty. The censored extension and the empirical demonstration that evaluator choice is consequential are the strongest contributions. It should influence how the agentic UQ community reports results, though adoption depends on whether the community values probabilistic truthfulness over simpler rank-based diagnostics.
Generated May 26, 2026
Comparison History (19)
Paper 2 is likely to have higher scientific impact because it introduces a principled, broadly applicable evaluation framework (strictly proper trajectory-level scoring rules) with formal elicitation guarantees, addressing a foundational measurement problem in agentic uncertainty. Its methodological rigor (theorems, censored-trajectory extension) and clarity about what existing metrics do/do not elicit make it useful across many agent settings and benchmarks, improving comparability and scientific validity of UQ research. Paper 1 is practically impactful, but its contribution is more system/engineering-centric and may date faster as agent-building tools evolve.
Paper 2 introduces a principled, theoretically grounded scoring framework (TPS) for evaluating uncertainty quantification in agentic AI systems—a rapidly growing area. Its contribution is foundational: it provides strictly proper scoring rules with formal proofs, addresses censored trajectories, and demonstrates that existing metrics are theoretically deficient. This has broad applicability across all agentic LLM systems. Paper 1, while addressing an interesting niche (creative physical reasoning in LMMs), is more application-specific with a narrower benchmark contribution and incremental alignment technique. Paper 2's methodological rigor and generalizability give it higher potential impact.
Paper 1 introduces a rigorous mathematical framework (Trajectory Proper Score) addressing a fundamental gap in evaluating uncertainty quantification for language model agents—a rapidly growing area. Its contributions are broadly applicable across agentic AI systems, providing theoretical guarantees (strict properness proofs) and practical tools (censored trajectory handling). Paper 2 offers valuable empirical insights into refusal mechanisms in reasoning models, but its scope is narrower (specific to activation steering of a single model family) and more incremental. Paper 1's methodological contribution has broader potential to shape evaluation standards across the field.
Paper 2 (Hera) likely has higher impact due to strong real-world applicability and timeliness: step-level device–cloud routing directly targets deployment cost/latency constraints for long-horizon LLM agents and shows sizable practical gains on established embodied/web agent benchmarks. Its two-stage IL→cost-aware RL framework is methodologically substantial and broadly relevant to systems + RL + agent communities. Paper 1 is theoretically novel and rigorous for agentic uncertainty evaluation, but its impact may be narrower (metrics/assessment) and more indirect on deployments than a coordination method that materially changes performance–cost tradeoffs.
Paper 1 addresses a fundamental and highly timely challenge in AI—uncertainty quantification for language model agents. By introducing theoretically grounded, strictly proper scoring rules with mathematical proofs and evaluating them on broad benchmarks, it offers rigorous methodology with wide-ranging implications for AI safety and reliability. Paper 2, while offering a useful practical pipeline for Business Process Management, focuses on a much narrower application domain and lacks the broad, cross-disciplinary foundational impact of Paper 1.
Paper 1 offers a more fundamental, broadly applicable methodological contribution: a family of strictly proper trajectory-level scoring rules that elicit the full prefix-conditioned success-probability process, with theory for censored trajectories and clear demonstrations that common metrics target weaker objects. This is novel and likely to influence evaluation practice across agentic LMs, RL, and sequential decision-making. Paper 2 is timely and important for mental-health safety, but relies heavily on LLM simulation/judging (potential validity/reproducibility concerns) and is narrower in scope (primarily SI scenarios), making its cross-field methodological impact likely smaller.
Paper 2 provides foundational theoretical contributions to the rapidly expanding field of LLM agents, specifically addressing the critical bottleneck of uncertainty quantification. While Paper 1 offers a valuable applied solution for sustainable EV charging, Paper 2's introduction of the predictor-agnostic Trajectory Proper Score provides rigorous methodological advancements (including proofs) that will likely be widely adopted by AI researchers, granting it broader cross-disciplinary impact and higher citation potential.
Paper 1 has higher potential impact due to a more general, theoretically grounded contribution: a family of strictly proper trajectory-level scoring rules that provably elicit the full prefix-conditioned success-probability process, including a principled treatment of censored trajectories. This is broadly applicable across agent evaluation, calibration, and decision-making in many domains beyond any single benchmark. Paper 2 is timely and practically valuable for human-AI coordination, but its contribution is more domain-tied (Overcooked-style collaboration) and may generalize less broadly than a foundational evaluation/uncertainty framework.
Paper 1 introduces a highly innovative paradigm shift in reward modeling by replacing weight optimization with agentic context evolution. Its massive improvement in data efficiency (using only 0.05% of preference data) directly addresses a critical bottleneck in AI alignment. While Paper 2 offers rigorous theoretical advancements in uncertainty evaluation, Paper 1 provides a highly practical framework with immediate, transformative applicability to multimodal model alignment and RLHF fine-tuning.
Paper 2 addresses a fundamental challenge in LLM agents—uncertainty quantification—with a mathematically rigorous, domain-agnostic theoretical framework based on proper scoring rules. Its foundational nature gives it much broader applicability across AI safety and reliability. In contrast, Paper 1 presents a highly specific, application-focused architecture for crypto portfolio management, limiting its overall scientific reach compared to Paper 2.
Paper 2 likely has higher impact: it introduces a general, theoretically grounded evaluation framework (trajectory-level strictly proper scoring rules) that cleanly targets a well-defined object (prefix-conditioned success probability), with proofs and extensions to censored trajectories. This is broadly applicable across agent benchmarks and model classes, timely for agentic LM evaluation, and can become a standard tool. Paper 1 is novel in linking child cognition and LLM agents via program-induction formulations, but its impact is more specialized (cognitive modeling + LLM behavior) and depends on task-specific generalization and empirical validity.
Paper 1 establishes fundamental, computable limits on transformer reasoning depth and generalizes impossibility results across multiple AI subfields. Its broad implications for AI scaling, architecture design, and safety give it significantly higher potential scientific impact than Paper 2, which offers a narrower, albeit rigorous, methodological improvement for uncertainty quantification.
Paper 2 likely has higher scientific impact: it targets a timely, high-stakes problem (trustworthy self-improvement in agentic search) and proposes a broadly applicable, tool- and model-agnostic mechanism (evidence-verifiable training signal via marginal utility of evidence) with clear real-world relevance (auditable, grounded agents). If validated empirically, it can influence training paradigms across IR, RL/self-training, and agent design. Paper 1 is methodologically rigorous and novel in evaluation theory, but its impact is narrower (metrics/UQ evaluation) and more incremental to practice.
Paper 1 introduces a principled, broadly applicable evaluation framework (trajectory-level strictly proper scoring rules) with formal guarantees about what is elicited, plus extensions for censored trajectories. This is methodological infrastructure likely to impact many agent/LLM evaluation settings beyond a single domain, with clear novelty and rigor. Paper 2 targets an important, timely practical issue (look-ahead bias in LLM financial backtests) and proposes a clever inference-time mitigation, but its impact is narrower (finance/backtesting), with domain-specific assumptions and more limited theoretical generality.
Density estimation is a fundamental mathematical primitive across machine learning and statistics. A unified framework bridging continuous and discrete domains provides a significant theoretical advancement with broad applicability in generative AI, statistical physics, and probabilistic modeling. Paper 1, while highly relevant to current LLM research, focuses on the narrower domain of evaluation metrics for agentic uncertainty, giving it a more limited scope of impact compared to the foundational statistical contribution of Paper 2.
Paper 2 introduces a theoretically rigorous, mathematically grounded contribution (strictly proper scoring rules for trajectory-level uncertainty quantification in LLM agents) that addresses a fundamental gap in evaluating agentic AI systems. Its formal proofs of strict properness, novel handling of censored trajectories, and demonstration that existing metrics target weaker objects represent deeper methodological innovation. Paper 1, while useful, primarily assembles existing psychometric concepts into an evaluation framework without comparable theoretical novelty. Paper 2's contribution is more foundational and likely to influence the rapidly growing agentic AI evaluation field broadly.
Paper 1 offers a more methodologically novel and generalizable contribution: a theoretically grounded family of strictly proper trajectory-level scoring rules that elicit the full prefix-conditioned success-probability process, including treatment of censored trajectories. This is likely to influence broad swaths of ML (UQ, evaluation, RL/agentic systems) and provides reusable tools with clear rigor and proofs. Paper 2 is timely with strong real-world relevance and an important benchmark, but is primarily an applied evaluation framework whose impact may be narrower and less theoretically foundational.
Paper 2 offers a technically novel, rigorously justified metric family (TPS) with proofs of strict propriety/elicitation for the full prefix-conditioned success-probability process, plus a principled extension to censored trajectories and empirical validation across multiple benchmarks. It directly addresses a timely, high-leverage problem in LM agents—reliable uncertainty quantification and evaluation—enabling clearer comparisons and better calibration methods with immediate real-world applicability. Paper 1 is conceptually interesting and potentially broad, but is largely programmatic/philosophical with less methodological concreteness and near-term measurable impact.
Paper 1 addresses a fundamental theoretical gap in evaluating uncertainty quantification for language model agents by introducing a strictly proper scoring rule (TPS) backed by mathematical proofs. This foundational contribution to safety, reliability, and evaluation methodology has deeper long-term scientific implications than Paper 2, which offers a practical systems-level optimization (parallel context compaction) that may become obsolete as native context windows and architectures evolve.