# Quantitative LLM Judges

arXiv: https://arxiv.org/abs/2506.02945 · HTML: https://mlsanity.com/paper/2506.02945/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 54 references that are in the map.
- First posted on arXiv 3 Jun 2025.

## Builds on (most cited first, 30 of 41)

- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [On the Opportunities and Risks of Foundation Models](https://mlsanity.com/paper/2108.07258/) (2021)
- [Learning to summarize from human feedback](https://mlsanity.com/paper/2009.01325/) (2020)
- [A Survey on LLM-as-a-Judge](https://mlsanity.com/paper/2411.15594/) (2024)
- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [Large Language Models are not Fair Evaluators](https://mlsanity.com/paper/2305.17926/) (2023)
- [Can LLMs Express Their Uncertainty? An Empirical Evaluation of Confidence Elicitation in LLMs](https://mlsanity.com/paper/2306.13063/) (2023)
- [Probing Classifiers: Promises, Shortcomings, and Advances](https://mlsanity.com/paper/2102.12452/) (2021)
- [Just Ask for Calibration: Strategies for Eliciting Calibrated Confidence Scores from Language Models Fine-Tuned with Human Feedback](https://mlsanity.com/paper/2305.14975/) (2023)
- [Can Large Language Models Be an Alternative to Human Evaluations?](https://mlsanity.com/paper/2305.01937/) (2023)
- [Self-Rewarding Language Models](https://mlsanity.com/paper/2401.10020/) (2024)
- [LMSYS-Chat-1M: A Large-Scale Real-World LLM Conversation Dataset](https://mlsanity.com/paper/2309.11998/) (2023)
- [Prometheus 2: An Open Source Language Model Specialized in Evaluating Other Language Models](https://mlsanity.com/paper/2405.01535/) (2024)
- [JudgeLM: Fine-tuned Large Language Models are Scalable Judges](https://mlsanity.com/paper/2310.17631/) (2023)
- [JudgeBench: A Benchmark for Evaluating LLM-based Judges](https://mlsanity.com/paper/2410.12784/) (2024)
- [PandaLM: An Automatic Evaluation Benchmark for LLM Instruction Tuning Optimization](https://mlsanity.com/paper/2306.05087/) (2023)
- [Evaluating Large Language Models at Evaluating Instruction Following](https://mlsanity.com/paper/2310.07641/) (2023)
- [LLMs instead of Human Judges? A Large Scale Empirical Study across 20 NLP Evaluation Tasks](https://mlsanity.com/paper/2406.18403/) (2024)
- [ToxicChat: Unveiling Hidden Challenges of Toxicity Detection in Real-World User-AI Conversation](https://mlsanity.com/paper/2310.17389/) (2023)
- [SALAD-Bench: A Hierarchical and Comprehensive Safety Benchmark for Large Language Models](https://mlsanity.com/paper/2402.05044/) (2024)
- [Judging the Judges: Evaluating Alignment and Vulnerabilities in LLMs-as-Judges](https://mlsanity.com/paper/2406.12624/) (2024)
- [Repairing the Cracked Foundation: A Survey of Obstacles in Evaluation Practices for Generated Text](https://mlsanity.com/paper/2202.06935/) (2022)
- [HelpSteer2-Preference: Complementing Ratings with Preferences](https://mlsanity.com/paper/2410.01257/) (2024)
- [Preference Leakage: A Contamination Problem in LLM-as-a-judge](https://mlsanity.com/paper/2502.01534/) (2025)
- [Taming Overconfidence in LLMs: Reward Calibration in RLHF](https://mlsanity.com/paper/2410.09724/) (2024)
- [An Empirical Study of LLM-as-a-Judge for LLM Evaluation: Fine-tuned Judge Model is not a General Substitute for GPT-4](https://mlsanity.com/paper/2403.02839/) (2024)
- [OffsetBias: Leveraging Debiased Data for Tuning Evaluators](https://mlsanity.com/paper/2407.06551/) (2024)
- [The BiGGen Bench: A Principled Benchmark for Fine-grained Evaluation of Language Models with Language Models](https://mlsanity.com/paper/2406.05761/) (2024)
- [Systematic Evaluation of LLM-as-a-Judge in LLM Alignment Tasks: Explainable Metrics and Diverse Prompt Templates](https://mlsanity.com/paper/2408.13006/) (2024)

## Built on by (most cited first, 6 of 6)

- [Hidden Measurement Error in LLM Pipelines Distorts Annotation, Evaluation, and Benchmarking](https://mlsanity.com/paper/2604.11581/) (2026)
- [Efficient Inference for Noisy LLM-as-a-Judge Evaluation](https://mlsanity.com/paper/2601.05420/) (2026)
- [Calibrate, Don't Curate: Label-Efficient Estimation from Noisy LLM Judges](https://mlsanity.com/paper/2605.09702/) (2026)
- [EmeraldMind: A Knowledge Graph-Augmented Framework for Greenwashing Detection](https://mlsanity.com/paper/2512.11506/) (2025)
- [Learning to Reason in LLMs by Expectation Maximization](https://mlsanity.com/paper/2512.20169/) (2025)
- [Share the Judge, Learn the Deferral: Where Specialization Helps LLM Evaluation](https://mlsanity.com/paper/2607.27984/) (2026)

## Abstract (arXiv)

LLM-as-a-judge is a framework where a large language model (LLM) evaluates the output of another LLM. While LLMs excel at producing qualitative textual evaluations, they often struggle to predict human preferences and numeric scores. We propose quantitative LLM judges, which align evaluation scores of existing LLM judges to humans in a given domain using regression models. The models are trained to improve the score of the original judge using its rationale and score. We present four quantitative judges for different types of absolute and relative feedback, which showcases the generality and versatility of our framework. Our framework is more computationally efficient than supervised fine-tuning and can be more statistically efficient when human feedback is limited, which is expected in practice. We validate these claims empirically on four datasets using two base judges. Our experiments show that quantitative judges can improve the predictive power of existing judges through post-hoc modeling.