pith. sign in

arxiv: 2607.00164 · v1 · pith:JQCJROG4new · submitted 2026-06-30 · 💻 cs.LG

Verifiable Rewards for Calibrated Probabilistic Forecasting

Sadanand Singh , Allam Reddy , Manan Chopra This is my paper

Pith reviewed 2026-07-02 19:43 UTC · model grok-4.3

classification 💻 cs.LG
keywords calibrated forecastingverifiable rewardsreinforcement learningBrier scoreNFL win probabilityaleatoric uncertaintygradient maskingprobabilistic forecasting
0
0 comments X

The pith

A 7B model trained solely on a verifiable empirical win rate reward matches betting market calibration for NFL forecasts without labels or supervised fine-tuning.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines whether reinforcement learning can produce calibrated probabilistic forecasts using only verifiable rewards computed from outcomes, tested on NFL in-game win probabilities with the betting market as benchmark. It shows that rewarding the single realized outcome introduces noise that degrades calibration and corrupts reasoning chains, while a state-conditioned empirical win rate drawn from past data eliminates that noise. Keeping the gradient away from reasoning steps, either by direct prediction or masking, allows training with the reward alone. The resulting model reaches market-level calibration and outperforms zero-shot frontier models on calibration metrics.

Core claim

Reinforcement learning with verifiable rewards can in principle train calibrated probabilistic forecasters because a proper scoring rule such as the Brier score is minimized in expectation by the true probability. For aleatoric forecasting where the label is one stochastic outcome, rewarding the realized per-play outcome fails due to noise and policy gradient corruption of the chain of thought. A verifiable label-free reward given by the state-conditioned empirical win rate estimated from past outcomes removes the label noise. Keeping the gradient off the reasoning by direct prediction or a gradient mask lets a 7B model trained with this reward alone reach the calibration of the betting mark

What carries the argument

State-conditioned empirical win rate estimated from past outcomes, used as verifiable reward with direct prediction or gradient mask to isolate it from reasoning.

If this is right

  • Models reach betting-market calibration on aleatoric tasks using only outcome-derived rewards without human labels.
  • Gradient masking preserves reasoning chains that ordinary chain-of-thought training corrupts.
  • The betting market's remaining edge arises from live in-game information beyond the inputs shared with the model and tabular estimator.
  • The same Brier score achieved by the frontier model, tabular estimator, and this trained model shows their shared inputs limit further improvement.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The reward construction could extend to other domains with frequent verifiable outcomes such as election or weather forecasting.
  • Separating reward computation from reasoning steps may be necessary for RL to succeed on any probabilistic output task.
  • This approach reduces dependence on supervised fine-tuning when historical outcome data is abundant.

Load-bearing premise

The state-conditioned empirical win rate estimated from past outcomes serves as an unbiased, noise-free proxy for the true conditional probability that aligns with the Brier score optimum.

What would settle it

Retraining the 7B model with the same reward on a fresh set of NFL seasons and finding its Brier score worse than the betting market on those seasons would falsify the central claim.

Figures

Figures reproduced from arXiv: 2607.00164 by Allam Reddy, Manan Chopra, Sadanand Singh.

Figure 1
Figure 1. Figure 1: The same game state, two ways to train a 7B model against outcomes. Rewarding the single realized outcome and updating the whole completion drives the chain of thought to extreme, overconfident numbers. Rewarding a state￾conditioned empirical win rate pˆ(x) and confining the gradient to the answer span leaves the reasoning intact and calibrates the forecaster to the betting market, using no human labels an… view at source ↗
Figure 2
Figure 2. Figure 2: Held-out calibration during training (fixed n = 128). Applied to the full completion, the reward of eq. (1) drives a chain-of-thought policy away from calibration on both Brier (a) and ECE (b). Removing the reasoning from the gradient, by direct prediction or by masking the gradient to the answer span, recovers calibration to the teacher’s level. The full-completion and masked runs share the same reward, p… view at source ↗
Figure 3
Figure 3. Figure 3: Rate at which the stated probability does not follow from the model’s own chain of thought, on a blinded-judge sample of held-out plays (95% bootstrap intervals). Full-completion chain-of-thought reinforcement learning leaves the base rate unchanged; masked training reduces it sharply, beyond what the masked prompt format alone accounts for. Example 1. Q4, 2:21 left. WAS trailing by 12, 2nd & 6 at the NYG … view at source ↗
Figure 4
Figure 4. Figure 4: Reasoning before and after gradient-masked CoT training, on identical game states (blinded-judge sample, 2023). In the base completions the stated probability does not follow from the text: an addition that sums to 108% reports one of its branches, and a leading favorite is rated below an even chance. The masked-CoT model reads the live state and states a probability consistent with it, close to the empiri… view at source ↗
Figure 5
Figure 5. Figure 5: Reliability on held-out 2024; points on the diagonal are perfectly calibrated and their area is proportional to the plays in the bin. The trained forecasters follow the diagonal as closely as the market, while the untrained chain-of-thought model is overconfident. market. On the held-out 2024 season the direct model reaches an ECE of 0.029, against the market’s 0.027 and the untrained model’s 0.057. Its ma… view at source ↗
Figure 6
Figure 6. Figure 6: Held-out 2024 Brier with 95% bootstrap intervals. A label-free 7B model trained by direct prediction, a frontier model prompted zero-shot, and the empirical-rate teacher converge near 0.143 and trail the market by the same significant margin, placing the residual gap in the available information rather than in model capacity. Static-feature forecaster Brier ECE Coarse pˆ (our reward) 0.1432 0.0437 nflverse… view at source ↗
read the original abstract

Reinforcement learning with verifiable rewards can in principle train calibrated probabilistic forecasters, since a proper scoring rule such as the Brier score is computed from outcomes alone and is minimized in expectation by the true probability. In practice it degrades calibration, and existing remedies address epistemic uncertainty, where a model's confidence accompanies a verifiably correct or incorrect answer. We study aleatoric forecasting, where the forecast itself is the output and the label is one stochastic outcome, taking NFL in-game win probability as a testbed with the betting market as a reference. Rewarding the realized per-play outcome fails, because the single outcome is a noisy target and the policy gradient corrupts the chain of thought. We introduce a verifiable, label-free reward, a state-conditioned empirical win rate estimated from past outcomes, that removes the label noise, and we keep the gradient off the reasoning, by direct prediction or a gradient mask, so it cannot be corrupted. Trained with this reward alone, without human labels or supervised fine-tuning, a 7B model reaches the calibration of the betting market by direct prediction and is better calibrated than a zero-shot frontier model. That frontier model and a tabular estimator reach the same Brier score as this model, identifying the market's small remaining edge as live in-game information beyond their shared inputs. Masking the gradient, rather than dropping the chain of thought, preserves reasoning from which the forecast follows, which ordinary chain-of-thought training corrupts.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces a verifiable reward for RL-based training of calibrated probabilistic forecasters in aleatoric settings, using a state-conditioned empirical win rate r(s) estimated from historical NFL outcomes instead of single noisy play results. It claims that a 7B model trained solely with this reward (via direct prediction or gradient masking) achieves calibration matching the betting market and exceeding zero-shot frontier models, while identifying the market's residual edge as arising from live in-game information beyond shared inputs; gradient masking is shown to preserve reasoning chains that ordinary CoT training corrupts.

Significance. If the central empirical claims hold after addressing verification gaps, the work demonstrates a label-free route to calibrated probabilistic forecasting via proper scoring rules on historical aggregates, with direct applicability to other aleatoric domains. The explicit separation of direct prediction versus masked-gradient training, and the identification that tabular estimators and frontier models match the trained model's Brier score, provides a concrete baseline for when additional live information matters.

major comments (2)
  1. [§4] §4 (empirical results) and the definition of r(s): the central claim that the 7B model reaches betting-market calibration rests on r(s) serving as an unbiased proxy for E[outcome|s]; however, the manuscript provides no quantitative details on state binning (clock, score differential, down, yard line), historical sample sizes per bin, variance of r(s) in sparse states, or handling of non-stationarity across rule changes and roster turnover. Without these, it is impossible to verify that gradient steps on the Brier objective converge to the true conditional probability rather than finite-sample historical averages.
  2. [Abstract, §5] Abstract and §5 (comparisons): the claim that the trained model 'reaches the calibration of the betting market' and is 'better calibrated than a zero-shot frontier model' is stated without reported Brier scores, ECE values, sample sizes, statistical significance tests, or calibration plots with confidence intervals. This leaves the strongest empirical assertion without verifiable support and prevents assessment of whether the result is load-bearing or sensitive to implementation choices in the empirical estimator.
minor comments (2)
  1. [§3] Notation for the reward r(s) and the gradient mask should be introduced with an explicit equation early in §3 to avoid ambiguity when comparing direct-prediction versus masked variants.
  2. [Appendix] The manuscript should include a table or appendix listing the exact state variables and binning thresholds used to construct the empirical win rates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments. We respond point-by-point to the major comments below and indicate revisions that will be incorporated to strengthen the verifiability of the empirical results.

read point-by-point responses

  1. Referee: [§4] §4 (empirical results) and the definition of r(s): the central claim that the 7B model reaches betting-market calibration rests on r(s) serving as an unbiased proxy for E[outcome|s]; however, the manuscript provides no quantitative details on state binning (clock, score differential, down, yard line), historical sample sizes per bin, variance of r(s) in sparse states, or handling of non-stationarity across rule changes and roster turnover. Without these, it is impossible to verify that gradient steps on the Brier objective converge to the true conditional probability rather than finite-sample historical averages.

    Authors: We agree that the manuscript omits these quantitative details on r(s) construction. In revision we will add a new subsection to §4 specifying the exact binning rules for clock, score differential, down, and yard line; reporting minimum and average historical sample sizes per bin; providing variance estimates for sparse bins; and describing our handling of non-stationarity via season-restricted data windows. These additions will allow readers to assess how closely r(s) approximates the conditional expectation. We note that the Brier objective remains proper even with finite-sample r(s), and the empirical results show the trained model matches the market's calibration, but the requested details will make this convergence verifiable. revision: yes

  2. Referee: [Abstract, §5] Abstract and §5 (comparisons): the claim that the trained model 'reaches the calibration of the betting market' and is 'better calibrated than a zero-shot frontier model' is stated without reported Brier scores, ECE values, sample sizes, statistical significance tests, or calibration plots with confidence intervals. This leaves the strongest empirical assertion without verifiable support and prevents assessment of whether the result is load-bearing or sensitive to implementation choices in the empirical estimator.

    Authors: The referee is correct that specific numerical values, statistical tests, and plots with intervals are not reported in the abstract or §5. We will revise both sections to include the Brier scores and ECE for the 7B model, betting market, tabular estimator, and zero-shot frontier models; the evaluation sample size; results of paired significance tests; and calibration plots with bootstrap confidence intervals. These additions will directly support the calibration claims and allow assessment of robustness. The underlying argument that the verifiable reward enables label-free calibration to market levels remains unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation uses an external historical dataset to compute state-conditioned empirical win rates as the verifiable reward target, then applies a proper scoring rule (Brier) to train the model against that fixed target. This is independent of the model's own outputs or any fitted parameters from the current run. The final calibration claim is benchmarked against an external betting market and a tabular estimator on the same data, with no load-bearing self-citation, no renaming of known results as new derivations, and no step where a prediction is definitionally equivalent to its input. The approach is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that historical empirical rates from conditioned states approximate the true probability distribution and that gradient isolation preserves reasoning quality; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)
  • standard math A proper scoring rule such as the Brier score is minimized in expectation by the true conditional probability.
    Invoked in the first sentence of the abstract as the theoretical basis for using verifiable rewards.

pith-pipeline@v0.9.1-grok · 5789 in / 1225 out tokens · 21965 ms · 2026-07-02T19:43:25.768441+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

39 extracted references · 22 canonical work pages · 9 internal anchors

  1. [1]

    Zhihong Shao, Peiyi Wang, Qihao Zhu, Runxin Xu, Junxiao Song, Xiao Bi, Haowei Zhang, Mingchuan Zhang, Y. K. Li, Y. Wu, and Daya Guo. DeepSeek- Math: Pushing the limits of mathematical rea- soning in open language models.arXiv preprint arXiv:2402.03300, 2024

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  2. [2]

    DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning

    Daya Guo, Dejian Yang, Haowei Zhang, Junxiao Song, et al. DeepSeek-R1 incentivizes reasoning in LLMs through reinforcement learning.Nature, 645 (8081):633–638, 2025. arXiv:2501.12948

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  3. [3]

    Glenn W. Brier. Verification of forecasts expressed in terms of probability.Monthly Weather Review, 78(1):1–3, 1950

    1950

  4. [4]

    Tilmann Gneiting and Adrian E. Raftery. Strictly proper scoring rules, prediction, and estimation. Journal of the American Statistical Association, 102 (477):359–378, 2007

    2007

  5. [5]

    Taming overconfidence in LLMs: Reward calibration in RLHF.arXiv preprint arXiv:2410.09724, 2024

    Jixuan Leng, Chengsong Huang, Banghua Zhu, and Jiaxin Huang. Taming overconfidence in LLMs: Reward calibration in RLHF.arXiv preprint arXiv:2410.09724, 2024

    work page arXiv 2024

  6. [6]

    Decoupling Reasoning and Confidence: Resurrecting Calibration in Reinforcement Learning from Verifiable Rewards

    Zhengzhao Ma, Xueru Wen, Boxi Cao, Yaojie Lu, Hongyu Lin, Jinglin Yang, Min He, Xianpei Han, and Le Sun. Decoupling reasoning and confidence: Resurrecting calibration in reinforcement learning from verifiable rewards. InProceedings of the 43rd International Conference on Machine Learning (ICML), 2026. arXiv:2603.09117

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  7. [7]

    Uncal- ibrated reasoning: GRPO induces overconfi- dence for stochastic outcomes.arXiv preprint arXiv:2508.11800, 2025

    Michael Bereket and Jure Leskovec. Uncal- ibrated reasoning: GRPO induces overconfi- dence for stochastic outcomes.arXiv preprint arXiv:2508.11800, 2025

    work page arXiv 2025

  8. [8]

    Beyond Binary Rewards: Training LMs to Reason About Their Uncertainty

    Mehul Damani, Isha Puri, Stewart Slocum, Idan Shenfeld, Leshem Choshen, Yoon Kim, and Jacob Andreas. Beyond binary rewards: Training LMs to reason about their uncertainty.arXiv preprint arXiv:2507.16806, 2025

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  9. [9]

    arXiv preprint arXiv:2503.02623 , year=

    David Bani-Harouni, Chantal Pellegrini, Paul Stan- gel, Ege Özsoy, Kamilia Zaripova, Nassir Navab, and Matthias Keicher. Rewarding doubt: A re- inforcement learning approach to calibrated confi- dence expression of large language models.arXiv preprint arXiv:2503.02623, 2025

    work page arXiv 2025

  10. [10]

    Linguistic calibration of long-form gen- erations

    Neil Band, Xuechen Li, Tengyu Ma, and Tatsunori Hashimoto. Linguistic calibration of long-form gen- erations. InProceedings of the 41st International Conference on Machine Learning (ICML), 2024. arXiv:2404.00474

    work page arXiv 2024

  11. [11]

    Approaching human-level fore- casting with language models

    Danny Halawi, Fred Zhang, Chen Yueh-Han, and Jacob Steinhardt. Approaching human-level fore- casting with language models. InAdvances in Neural Information Processing Systems (NeurIPS),

  12. [12]

    Can language models use forecasting strategies?arXiv preprint arXiv:2406.04446, 2024

    Sarah Pratt, Seth Blumberg, Pietro Kreitlon Car- olino, and Meredith Ringel Morris. Can language models use forecasting strategies?arXiv preprint arXiv:2406.04446, 2024

    work page arXiv 2024

  13. [13]

    Outcome- based reinforcement learning to predict the future

    Benjamin Turtel, Danny Franklin, Kris Skotheim, Luke Hewitt, and Philipp Schoenegger. Outcome- based reinforcement learning to predict the future. arXiv preprint arXiv:2505.17989, 2025. 9

    work page arXiv 2025

  14. [14]

    LLMs can teach themselves to better predict the future.arXiv preprint arXiv:2502.05253, 2025

    Benjamin Turtel, Danny Franklin, and Philipp Schoenegger. LLMs can teach themselves to better predict the future.arXiv preprint arXiv:2502.05253, 2025

    work page arXiv 2025

  15. [15]

    Pitfalls in evaluating language model forecasters.arXiv preprint arXiv:2506.00723, 2025

    Daniel Paleka, Shashwat Goel, Jonas Geiping, and Florian Tramèr. Pitfalls in evaluating language model forecasters.arXiv preprint arXiv:2506.00723, 2025

    work page arXiv 2025

  16. [16]

    ForecastBench: Adynamicbench- mark of AI forecasting capabilities.arXiv preprint arXiv:2409.19839, 2024

    Ezra Karger, Houtan Bastani, Chen Yueh-Han, Zachary Jacobs, Danny Halawi, Fred Zhang, and PhilipE.Tetlock. ForecastBench: Adynamicbench- mark of AI forecasting capabilities.arXiv preprint arXiv:2409.19839, 2024

    work page arXiv 2024

  17. [17]

    DAPO: An Open-Source LLM Reinforcement Learning System at Scale

    Qiying Yu, Zheng Zhang, Ruofei Zhu, Yufeng Yuan, Xiaochen Zuo, Yu Yue, et al. DAPO: An open- source LLM reinforcement learning system at scale. arXiv preprint arXiv:2503.14476, 2025

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  18. [18]

    GPG: A simple and strong reinforcement learning baseline for model reasoning

    Xiangxiang Chu, Hailang Huang, Xiao Zhang, Fei Wei, and Yong Wang. GPG: A simple and strong reinforcement learning baseline for model reasoning. InInternational Conference on Learning Represen- tations (ICLR), 2026. arXiv:2504.02546

    work page arXiv 2026

  19. [19]

    On the design of KL-regularized policy gradient algo- rithms for LLM reasoning

    YifanZhang, YifengLiu, HuizhuoYuan, YangYuan, Quanquan Gu, and Andrew Chi-Chih Yao. On the design of KL-regularized policy gradient algo- rithms for LLM reasoning. InInternational Con- ference on Learning Representations (ICLR), 2026. arXiv:2505.17508

    work page arXiv 2026

  20. [20]

    Beyond the 80/20 Rule: High-Entropy Minority Tokens Drive Effective Reinforcement Learning for LLM Reasoning

    Shenzhi Wang, Le Yu, Chang Gao, Chujie Zheng, Shixuan Liu, Rui Lu, Kai Dang, Xionghui Chen, JianxinYang, ZhenruZhang, YuqiongLiu, AnYang, Andrew Zhao, Yang Yue, Shiji Song, Bowen Yu, Gao Huang, and Junyang Lin. Beyond the 80/20 rule: High-entropy minority tokens drive effective reinforcement learning for LLM reasoning. InAd- vances in Neural Information P...

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  21. [21]

    Stabilizing Knowledge, Promoting Reasoning: Dual-Token Constraints for RLVR

    Jiakang Wang, Runze Liu, Fuzheng Zhang, Xiu Li, Guorui Zhou, and Ling Pan. Stabilizing knowledge, promoting reasoning: Dual-token constraints for RLVR.arXiv preprint arXiv:2507.15778, 2025

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  22. [22]

    arXiv preprint arXiv:2508.04349 , year=

    Hongze Tan, Zihan Wang, Jianfei Pan, Jinghao Lin, Hao Wang, Yifan Wu, Tao Chen, Zhihang Zheng, Zhihao Tang, and Haihua Yang. GTPO and GRPO- S: Token and sequence-level reward shaping with policy entropy.arXiv preprint arXiv:2508.04349, 2025

    work page arXiv 2025

  23. [23]

    Using random forests to estimate win probability before each play of an NFL game.Journal of Quantitative Analysis in Sports, 10(2):197–205, 2014

    Dennis Lock and Dan Nettleton. Using random forests to estimate win probability before each play of an NFL game.Journal of Quantitative Analysis in Sports, 10(2):197–205, 2014

    2014

  24. [24]

    nflWAR: A Reproducible Method for Offensive Player Evaluation in Football

    Ronald Yurko, Samuel Ventura, and Maksim Horowitz. nflWAR: A reproducible method for offensive player evaluation in football.Journal of Quantitative Analysis in Sports, 15(3), 2019. arXiv:1802.00998

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  25. [25]

    Sebastian Carl and Ben Baldwin.nflfastR: Func- tions to Efficiently Access NFL Play by Play Data,

  26. [26]

    R pack- age

    URL https://www.nflfastr.com/. R pack- age

  27. [27]

    Brill, Ronald Yurko, and Abraham J

    Ryan S. Brill, Ronald Yurko, and Abraham J. Wyner. Exploring the difficulty of estimating win probability: A simulation study.Journal of Quan- titative Analysis in Sports, 2025. arXiv:2406.16171

    work page arXiv 2025

  28. [28]

    Polson and Hal S

    Nicholas G. Polson and Hal S. Stern. The implied volatility of a sports game.Journal of Quantitative Analysis in Sports, 11(3):145–153, 2015

    2015

  29. [29]

    Boulier and H

    Bryan L. Boulier and H. O. Stekler. Predicting the outcomes of national football league games.In- ternational Journal of Forecasting, 19(2):257–270, 2003

    2003

  30. [30]

    Steven D. Levitt. Why are gambling markets or- ganised so differently from financial markets?The Economic Journal, 114(495):223–246, 2004

    2004

  31. [31]

    Prediction accuracy of different market structures: Bookmakers versus a betting exchange.Interna- tional Journal of Forecasting, 26(3):448–459, 2010

    Egon Franck, Erwin Verbeek, and Stephan Nüesch. Prediction accuracy of different market structures: Bookmakers versus a betting exchange.Interna- tional Journal of Forecasting, 26(3):448–459, 2010

    2010

  32. [32]

    Schwartz, Bonnie F

    Justin Cox, Adam L. Schwartz, Bonnie F. Van Ness, and Robert A. Van Ness. The predictive power of college football spreads: Regular season versus bowl games.Journal of Sports Economics, 22(3):251–273, 2021

    2021

  33. [33]

    On determining probability fore- casts from betting odds.International Journal of Forecasting, 30(4):934–943, 2014

    Erik Štrumbelj. On determining probability fore- casts from betting odds.International Journal of Forecasting, 30(4):934–943, 2014

    2014

  34. [34]

    Allan H. Murphy. A new vector partition of the probability score.Journal of Applied Meteorology, 12(4):595–600, 1973

    1973

  35. [35]

    On Calibration of Modern Neural Networks

    Chuan Guo, Geoff Pleiss, Yu Sun, and Kilian Q. Weinberger. On calibration of modern neural net- works. InProceedings of the 34th International Conference on Machine Learning (ICML), 2017. arXiv:1706.04599

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  36. [36]

    Jochen Bröcker and Leonard A. Smith. Increasing the reliability of reliability diagrams.Weather and Forecasting, 22(3):651–661, 2007. 10

    2007

  37. [37]

    Timo Dimitriadis, Tilmann Gneiting, and Alexan- der I. Jordan. Stable reliability diagrams for prob- abilistic classifiers.Proceedings of the National Academy of Sciences, 118(8):e2016191118, 2021

    2021

  38. [38]

    Bootstrap methods: Another look at the jackknife.The Annals of Statistics, 7(1):1–26, 1979

    Bradley Efron. Bootstrap methods: Another look at the jackknife.The Annals of Statistics, 7(1):1–26, 1979

    1979

  39. [39]

    Data analysis using Stein’s estimator and its generalizations.Journal of the American Statistical Association, 70(350): 311–319, 1975

    Bradley Efron and Carl Morris. Data analysis using Stein’s estimator and its generalizations.Journal of the American Statistical Association, 70(350): 311–319, 1975. 11 A The empirical-rate target The reward targetˆp(x)is a state-conditioned empirical win rate estimated from training-season outcomes. Each play is placed in a bucket defined by three game-l...

    1975