Daily Papers

Using language models to scalably approximate human preferences on text quality (LLM-as-a-judge) has become a standard practice applicable to many tasks. A judgment is often extracted from the judge's textual output alone, typically with greedy decoding. However, LLM judges naturally provide distributions over judgment tokens, inviting a breadth of inference methods for extracting fine-grained preferences. We find that taking the mean of the judgment distribution consistently outperforms taking the mode (i.e. greedy decoding) in all evaluation settings (i.e. pointwise, pairwise, and listwise). We further explore novel methods of deriving preferences from judgment distributions, and find that methods incorporating risk aversion often improve performance. Lastly, we analyze LLM-as-a-judge paired with chain-of-thought (CoT) prompting, showing that CoT can collapse the spread of the judgment distribution, often harming performance. Our findings show that leveraging distributional output improves LLM-as-a-judge, as opposed to using the text interface alone.

LLMs have emerged as powerful evaluators in the LLM-as-a-Judge paradigm, offering significant efficiency and flexibility compared to human judgments. However, previous methods primarily rely on single-point evaluations, overlooking the inherent diversity and uncertainty in human evaluations. This approach leads to information loss and decreases the reliability of evaluations. To address this limitation, we propose a novel training framework that explicitly aligns the LLM-generated judgment distribution with empirical human distributions. Specifically, we propose a distributional alignment objective based on KL divergence, combined with an auxiliary cross-entropy regularization to stabilize the training process. Furthermore, considering that empirical distributions may derive from limited human annotations, we incorporate adversarial training to enhance model robustness against distribution perturbations. Extensive experiments across various LLM backbones and evaluation tasks demonstrate that our framework significantly outperforms existing closed-source LLMs and conventional single-point alignment methods, with improved alignment quality, evaluation accuracy, and robustness.

When the performance of a machine learning model varies over groups defined by sensitive attributes (e.g., gender or ethnicity), the performance disparity can be expressed in terms of the probability distributions of the input and output variables over each group. In this paper, we exploit this fact to reduce the disparate impact of a fixed classification model over a population of interest. Given a black-box classifier, we aim to eliminate the performance gap by perturbing the distribution of input variables for the disadvantaged group. We refer to the perturbed distribution as a counterfactual distribution, and characterize its properties for common fairness criteria. We introduce a descent algorithm to learn a counterfactual distribution from data. We then discuss how the estimated distribution can be used to build a data preprocessor that can reduce disparate impact without training a new model. We validate our approach through experiments on real-world datasets, showing that it can repair different forms of disparity without a significant drop in accuracy.

In this paper, we introduce the Chinese AI and Law challenge dataset (CAIL2018), the first large-scale Chinese legal dataset for judgment prediction. \dataset contains more than 2.6 million criminal cases published by the Supreme People's Court of China, which are several times larger than other datasets in existing works on judgment prediction. Moreover, the annotations of judgment results are more detailed and rich. It consists of applicable law articles, charges, and prison terms, which are expected to be inferred according to the fact descriptions of cases. For comparison, we implement several conventional text classification baselines for judgment prediction and experimental results show that it is still a challenge for current models to predict the judgment results of legal cases, especially on prison terms. To help the researchers make improvements on legal judgment prediction, both \dataset and baselines will be released after the CAIL competitionhttp://cail.cipsc.org.cn/.

The use of LLMs as automated judges ("LLM-as-a-judge") is now widespread, yet standard judges suffer from a multitude of reliability issues. To address these challenges, we introduce Verdict, an open-source library for scaling judge-time compute to enhance the accuracy, reliability, and interpretability of automated evaluators. Verdict leverages the composition of modular reasoning units (such as verification, debate, and aggregation) and increased inference-time compute to improve LLM judge quality. Across a variety of challenging tasks such as content moderation, fact-checking, and hallucination detection, Verdict judges achieves performance competitive with orders-of-magnitude larger fine-tuned judges, prompted judges, and reasoning models. Our framework establishes a foundation for scalable, interpretable, and reliable LLM-based evaluation systems for both researchers and practitioners.

Language models (LM) are capable of remarkably complex linguistic tasks; however, numerical reasoning is an area in which they frequently struggle. An important but rarely evaluated form of reasoning is understanding probability distributions. In this paper, we focus on evaluating the probabilistic reasoning capabilities of LMs using idealized and real-world statistical distributions. We perform a systematic evaluation of state-of-the-art LMs on three tasks: estimating percentiles, drawing samples, and calculating probabilities. We evaluate three ways to provide context to LMs 1) anchoring examples from within a distribution or family of distributions, 2) real-world context, 3) summary statistics on which to base a Normal approximation. Models can make inferences about distributions, and can be further aided by the incorporation of real-world context, example shots and simplified assumptions, even if these assumptions are incorrect or misspecified. To conduct this work, we developed a comprehensive benchmark distribution dataset with associated question-answer pairs that we will release publicly.

Recently, the role of LLM-as-judge in evaluating large language models has gained prominence. However, current judge models suffer from narrow specialization and limited robustness, undermining their capacity for comprehensive evaluations. In this work, we present CompassJudger-2, a novel generalist judge model that overcomes these limitations via a task-driven, multi-domain data curation strategy. Central to our approach is supervising judgment tasks with verifiable rewards, guiding intrinsic critical reasoning through rejection sampling to foster robust, generalizable judgment capabilities. We introduce a refined learning objective with margin policy gradient loss to enhance performance. Empirically, CompassJudger-2 achieves superior results across multiple judge and reward benchmarks, and our 7B model demonstrates competitive judgment accuracy with significantly larger models like DeepSeek-V3 and Qwen3-235B-A22B. Additionally, we propose JudgerBenchV2, a comprehensive benchmark evaluating cross-domain judgment accuracy and rank consistency to standardize judge model evaluation. These contributions advance robust, scalable LLM judgment and establish new performance and evaluation standards.

We study the problem of predicting numeric labels that are constrained to the integers or to a subrange of the integers. For example, the number of up-votes on social media posts, or the number of bicycles available at a public rental station. While it is possible to model these as continuous values, and to apply traditional regression, this approach changes the underlying distribution on the labels from discrete to continuous. Discrete distributions have certain benefits, which leads us to the question whether such integer labels can be modeled directly by a discrete distribution, whose parameters are predicted from the features of a given instance. Moreover, we focus on the use case of output distributions of neural networks, which adds the requirement that the parameters of the distribution be continuous so that backpropagation and gradient descent may be used to learn the weights of the network. We investigate several options for such distributions, some existing and some novel, and test them on a range of tasks, including tabular learning, sequential prediction and image generation. We find that overall the best performance comes from two distributions: Bitwise, which represents the target integer in bits and places a Bernoulli distribution on each, and a discrete analogue of the Laplace distribution, which uses a distribution with exponentially decaying tails around a continuous mean.

In a legal system, judgment consistency is regarded as one of the most important manifestations of fairness. However, due to the complexity of factual elements that impact sentencing in real-world scenarios, few works have been done on quantitatively measuring judgment consistency towards real-world data. In this paper, we propose an evaluation metric for judgment inconsistency, Legal Inconsistency Coefficient (LInCo), which aims to evaluate inconsistency between data groups divided by specific features (e.g., gender, region, race). We propose to simulate judges from different groups with legal judgment prediction (LJP) models and measure the judicial inconsistency with the disagreement of the judgment results given by LJP models trained on different groups. Experimental results on the synthetic data verify the effectiveness of LInCo. We further employ LInCo to explore the inconsistency in real cases and come to the following observations: (1) Both regional and gender inconsistency exist in the legal system, but gender inconsistency is much less than regional inconsistency; (2) The level of regional inconsistency varies little across different time periods; (3) In general, judicial inconsistency is negatively correlated with the severity of the criminal charges. Besides, we use LInCo to evaluate the performance of several de-bias methods, such as adversarial learning, and find that these mechanisms can effectively help LJP models to avoid suffering from data bias.

Legal judgment prediction is the task of automatically predicting the outcome of a court case, given a text describing the case's facts. Previous work on using neural models for this task has focused on Chinese; only feature-based models (e.g., using bags of words and topics) have been considered in English. We release a new English legal judgment prediction dataset, containing cases from the European Court of Human Rights. We evaluate a broad variety of neural models on the new dataset, establishing strong baselines that surpass previous feature-based models in three tasks: (1) binary violation classification; (2) multi-label classification; (3) case importance prediction. We also explore if models are biased towards demographic information via data anonymization. As a side-product, we propose a hierarchical version of BERT, which bypasses BERT's length limitation.

In legal decisions, split votes (SV) occur when judges cannot reach a unanimous decision, posing a difficulty for lawyers who must navigate diverse legal arguments and opinions. In high-stakes domains, understanding the alignment of perceived difficulty between humans and AI systems is crucial to build trust. However, existing NLP calibration methods focus on a classifier's awareness of predictive performance, measured against the human majority class, overlooking inherent human label variation (HLV). This paper explores split votes as naturally observable human disagreement and value pluralism. We collect judges' vote distributions from the European Court of Human Rights (ECHR), and present SV-ECHR, a case outcome classification (COC) dataset with SV information. We build a taxonomy of disagreement with SV-specific subcategories. We further assess the alignment of perceived difficulty between models and humans, as well as confidence- and human-calibration of COC models. We observe limited alignment with the judge vote distribution. To our knowledge, this is the first systematic exploration of calibration to human judgements in legal NLP. Our study underscores the necessity for further research on measuring and enhancing model calibration considering HLV in legal decision tasks.

Large Reasoning Models have demonstrated remarkable performance with the advancement of test-time scaling techniques, which enhances prediction accuracy by generating multiple candidate responses and selecting the most reliable answer. While prior work has analyzed that internal model signals like confidence scores can partly indicate response correctness and exhibit a distributional correlation with accuracy, such distributional information has not been fully utilized to guide answer selection. Motivated by this, we propose DistriVoting, which incorporates distributional priors as another signal alongside confidence during voting. Specifically, our method (1) first decomposes the mixed confidence distribution into positive and negative components using Gaussian Mixture Models, (2) then applies a reject filter based on positive/negative samples from them to mitigate overlap between the two distributions. Besides, to further alleviate the overlap from the perspective of distribution itself, we propose SelfStepConf, which uses step-level confidence to dynamically adjust inference process, increasing the separation between the two distributions to improve the reliability of confidences in voting. Experiments across 16 models and 5 benchmarks demonstrate that our method significantly outperforms state-of-the-art approaches.

Popular approaches for quantifying predictive uncertainty in deep neural networks often involve distributions over weights or multiple models, for instance via Markov Chain sampling, ensembling, or Monte Carlo dropout. These techniques usually incur overhead by having to train multiple model instances or do not produce very diverse predictions. This comprehensive and extensive survey aims to familiarize the reader with an alternative class of models based on the concept of Evidential Deep Learning: For unfamiliar data, they aim to admit "what they don't know", and fall back onto a prior belief. Furthermore, they allow uncertainty estimation in a single model and forward pass by parameterizing distributions over distributions. This survey recapitulates existing works, focusing on the implementation in a classification setting, before surveying the application of the same paradigm to regression. We also reflect on the strengths and weaknesses compared to other existing methods and provide the most fundamental derivations using a unified notation to aid future research.

In this paper we argue for the fundamental importance of the value distribution: the distribution of the random return received by a reinforcement learning agent. This is in contrast to the common approach to reinforcement learning which models the expectation of this return, or value. Although there is an established body of literature studying the value distribution, thus far it has always been used for a specific purpose such as implementing risk-aware behaviour. We begin with theoretical results in both the policy evaluation and control settings, exposing a significant distributional instability in the latter. We then use the distributional perspective to design a new algorithm which applies Bellman's equation to the learning of approximate value distributions. We evaluate our algorithm using the suite of games from the Arcade Learning Environment. We obtain both state-of-the-art results and anecdotal evidence demonstrating the importance of the value distribution in approximate reinforcement learning. Finally, we combine theoretical and empirical evidence to highlight the ways in which the value distribution impacts learning in the approximate setting.

LLM-as-a-Judge leverages the generative and reasoning capabilities of large language models (LLMs) to evaluate LLM responses across diverse scenarios, providing accurate preference signals. This approach plays a vital role in aligning LLMs with human values, ensuring ethical and reliable AI outputs that align with societal norms. Recent studies have raised many methods to train LLM as generative judges, but most of them are data consuming or lack accuracy, and only focus on LLM's judge ability. In this work, we regard judge ability as a general ability of LLM and implement a two-stage training approach, comprising supervised fine-tuning (SFT) warm-up and direct preference optimization (DPO) enhancement, to achieve judge style adaptation and improve judgment accuracy. Additionally, we introduce an efficient data synthesis method to generate judgmental content. Experimental results demonstrate that our approach, utilizing only about 2% to 40% of the data required by other methods, achieves SOTA performance on RewardBench. Furthermore, our training method enhances the general capabilities of the model by constructing complicated judge task, and the judge signals provided by our model have significantly enhanced the downstream DPO training performance of our internal models in our test to optimize policy model with Judge Model. We also open-source our model weights and training data to facilitate further research.

The rapid deployment of AI systems in high-stakes domains, including those classified as high-risk under the The EU AI Act (Regulation (EU) 2024/1689), has intensified the need for reliable compliance auditing. For binary classifiers, regulatory risk assessment often relies on global fairness metrics such as the Disparate Impact ratio, widely used to evaluate potential discrimination. In typical auditing settings, the auditee provides a subset of its dataset to an auditor, while a supervisory authority may verify whether this subset is representative of the full underlying distribution. In this work, we investigate to what extent a malicious auditee can construct a fairness-compliant yet representative-looking sample from a non-compliant original distribution, thereby creating an illusion of fairness. We formalize this problem as a constrained distributional projection task and introduce mathematically grounded manipulation strategies based on entropic and optimal transport projections. These constructions characterize the minimal distributional shift required to satisfy fairness constraints. To counter such attacks, we formalize representativeness through distributional distance based statistical tests and systematically evaluate their ability to detect manipulated samples. Our analysis highlights the conditions under which fairness manipulation can remain statistically undetected and provides practical guidelines for strengthening supervisory verification. We validate our theoretical findings through experiments on standard tabular datasets for bias detection. Code is publicly available at https://github.com/ValentinLafargue/Inspection.

Quantile regression is a fundamental problem in statistical learning motivated by a need to quantify uncertainty in predictions, or to model a diverse population without being overly reductive. For instance, epidemiological forecasts, cost estimates, and revenue predictions all benefit from being able to quantify the range of possible values accurately. As such, many models have been developed for this problem over many years of research in statistics, machine learning, and related fields. Rather than proposing yet another (new) algorithm for quantile regression we adopt a meta viewpoint: we investigate methods for aggregating any number of conditional quantile models, in order to improve accuracy and robustness. We consider weighted ensembles where weights may vary over not only individual models, but also over quantile levels, and feature values. All of the models we consider in this paper can be fit using modern deep learning toolkits, and hence are widely accessible (from an implementation point of view) and scalable. To improve the accuracy of the predicted quantiles (or equivalently, prediction intervals), we develop tools for ensuring that quantiles remain monotonically ordered, and apply conformal calibration methods. These can be used without any modification of the original library of base models. We also review some basic theory surrounding quantile aggregation and related scoring rules, and contribute a few new results to this literature (for example, the fact that post sorting or post isotonic regression can only improve the weighted interval score). Finally, we provide an extensive suite of empirical comparisons across 34 data sets from two different benchmark repositories.

The progress of AI is bottlenecked by the quality of evaluation, and powerful LLM-as-a-Judge models have proved to be a core solution. Improved judgment ability is enabled by stronger chain-of-thought reasoning, motivating the need to find the best recipes for training such models to think. In this work we introduce J1, a reinforcement learning approach to training such models. Our method converts both verifiable and non-verifiable prompts to judgment tasks with verifiable rewards that incentivize thinking and mitigate judgment bias. In particular, our approach outperforms all other existing 8B or 70B models when trained at those sizes, including models distilled from DeepSeek-R1. J1 also outperforms o1-mini, and even R1 on some benchmarks, despite training a smaller model. We provide analysis and ablations comparing Pairwise-J1 vs Pointwise-J1 models, offline vs online training recipes, reward strategies, seed prompts, and variations in thought length and content. We find that our models make better judgments by learning to outline evaluation criteria, comparing against self-generated reference answers, and re-evaluating the correctness of model responses.

This article introduces Regression Discontinuity Design (RDD) with Distribution-Valued Outcomes (R3D), extending the standard RDD framework to settings where the outcome is a distribution rather than a scalar. Such settings arise when treatment is assigned at a higher level of aggregation than the outcome-for example, when a subsidy is allocated based on a firm-level revenue cutoff while the outcome of interest is the distribution of employee wages within the firm. Since standard RDD methods cannot accommodate such two-level randomness, I propose a novel approach based on random distributions. The target estimand is a "local average quantile treatment effect", which averages across random quantiles. To estimate this target, I introduce two related approaches: one that extends local polynomial regression to random quantiles and another based on local Fr\'echet regression, a form of functional regression. For both estimators, I establish asymptotic normality and develop uniform, debiased confidence bands together with a data-driven bandwidth selection procedure. Simulations validate these theoretical properties and show existing methods to be biased and inconsistent in this setting. I then apply the proposed methods to study the effects of gubernatorial party control on within-state income distributions in the US, using a close-election design. The results suggest a classic equality-efficiency tradeoff under Democratic governorship, driven by reductions in income at the top of the distribution.

Large language model (LLM)-based judges are widely adopted for automated evaluation and reward modeling, yet their judgments are often affected by judgment biases. Accurately evaluating these biases is essential for ensuring the reliability of LLM-based judges. However, existing studies typically investigate limited biases under a single judge formulation, either generative or discriminative, lacking a comprehensive evaluation. To bridge this gap, we propose JudgeBiasBench, a benchmark for systematically quantifying biases in LLM-based judges. JudgeBiasBench defines a taxonomy of judgment biases across 4 dimensions, and constructs bias-augmented evaluation instances through a controlled bias injection pipeline, covering 12 representative bias types. We conduct extensive experiments across both generative and discriminative judges, revealing that current judges exhibit significant and diverse bias patterns that often compromise the reliability of automated evaluation. To mitigate judgment bias, we propose bias-aware training that explicitly incorporates bias-related attributes into the training process, encouraging judges to disentangle task-relevant quality from bias-correlated cues. By adopting reinforcement learning for generative judges and contrastive learning for discriminative judges, our methods effectively reduce judgment biases while largely preserving general evaluation capability.

Customer Lifetime Value (CLTV) prediction is a critical task in business applications. Accurately predicting CLTV is challenging in real-world business scenarios, as the distribution of CLTV is complex and mutable. Firstly, there is a large number of users without any consumption consisting of a long-tailed part that is too complex to fit. Secondly, the small set of high-value users spent orders of magnitude more than a typical user leading to a wide range of the CLTV distribution which is hard to capture in a single distribution. Existing approaches for CLTV estimation either assume a prior probability distribution and fit a single group of distribution-related parameters for all samples, or directly learn from the posterior distribution with manually predefined buckets in a heuristic manner. However, all these methods fail to handle complex and mutable distributions. In this paper, we propose a novel optimal distribution selection model OptDist for CLTV prediction, which utilizes an adaptive optimal sub-distribution selection mechanism to improve the accuracy of complex distribution modeling. Specifically, OptDist trains several candidate sub-distribution networks in the distribution learning module (DLM) for modeling the probability distribution of CLTV. Then, a distribution selection module (DSM) is proposed to select the sub-distribution for each sample, thus making the selection automatically and adaptively. Besides, we design an alignment mechanism that connects both modules, which effectively guides the optimization. We conduct extensive experiments on both two public and one private dataset to verify that OptDist outperforms state-of-the-art baselines. Furthermore, OptDist has been deployed on a large-scale financial platform for customer acquisition marketing campaigns and the online experiments also demonstrate the effectiveness of OptDist.

Large Language Model (LLM)-based judgments leverage powerful LLMs to efficiently evaluate candidate content and provide judgment scores. However, the inherent biases and vulnerabilities of LLM-generated judgments raise concerns, underscoring the urgent need for distinguishing them in sensitive scenarios like academic peer reviewing. In this work, we propose and formalize the task of judgment detection and systematically investigate the detectability of LLM-generated judgments. Unlike LLM-generated text detection, judgment detection relies solely on judgment scores and candidates, reflecting real-world scenarios where textual feedback is often unavailable in the detection process. Our preliminary analysis shows that existing LLM-generated text detection methods perform poorly given their incapability to capture the interaction between judgment scores and candidate content -- an aspect crucial for effective judgment detection. Inspired by this, we introduce J-Detector, a lightweight and transparent neural detector augmented with explicitly extracted linguistic and LLM-enhanced features to link LLM judges' biases with candidates' properties for accurate detection. Experiments across diverse datasets demonstrate the effectiveness of J-Detector and show how its interpretability enables quantifying biases in LLM judges. Finally, we analyze key factors affecting the detectability of LLM-generated judgments and validate the practical utility of judgment detection in real-world scenarios.

We consider the problem of performing Bayesian inference in probabilistic models where observations are accompanied by uncertainty, referred to as "uncertain evidence." We explore how to interpret uncertain evidence, and by extension the importance of proper interpretation as it pertains to inference about latent variables. We consider a recently-proposed method "distributional evidence" as well as revisit two older methods: Jeffrey's rule and virtual evidence. We devise guidelines on how to account for uncertain evidence and we provide new insights, particularly regarding consistency. To showcase the impact of different interpretations of the same uncertain evidence, we carry out experiments in which one interpretation is defined as "correct." We then compare inference results from each different interpretation illustrating the importance of careful consideration of uncertain evidence.

In the past couple of years, various approaches to representing and quantifying different types of predictive uncertainty in machine learning, notably in the setting of classification, have been proposed on the basis of second-order probability distributions, i.e., predictions in the form of distributions on probability distributions. A completely conclusive solution has not yet been found, however, as shown by recent criticisms of commonly used uncertainty measures associated with second-order distributions, identifying undesirable theoretical properties of these measures. In light of these criticisms, we propose a set of formal criteria that meaningful uncertainty measures for predictive uncertainty based on second-order distributions should obey. Moreover, we provide a general framework for developing uncertainty measures to account for these criteria, and offer an instantiation based on the Wasserstein distance, for which we prove that all criteria are satisfied.

The rise of Large Language Models (LLMs) as evaluators offers a scalable alternative to human annotation, yet existing Supervised Fine-Tuning (SFT) for judges approaches often fall short in domains requiring complex reasoning. In this work, we investigate whether LLM judges truly benefit from enhanced reasoning capabilities. Through a detailed analysis of reasoning requirements across evaluation tasks, we reveal a negative correlation between SFT performance gains and the proportion of reasoning-demanding samples - highlighting the limitations of SFT in such scenarios. To address this, we introduce JudgeLRM, a family of judgment-oriented LLMs trained using reinforcement learning (RL) with judge-wise, outcome-driven rewards. JudgeLRM models consistently outperform both SFT-tuned and state-of-the-art reasoning models. Notably, JudgeLRM-3B surpasses GPT-4, and JudgeLRM-7B outperforms DeepSeek-R1 by 2.79% in F1 score, particularly excelling in judge tasks requiring deep reasoning.

The research field of Legal Natural Language Processing (NLP) has been very active recently, with Legal Judgment Prediction (LJP) becoming one of the most extensively studied tasks. To date, most publicly released LJP datasets originate from countries with civil law. In this work, we release, for the first time, a challenging LJP dataset focused on class action cases in the US. It is the first dataset in the common law system that focuses on the harder and more realistic task involving the complaints as input instead of the often used facts summary written by the court. Additionally, we study the difficulty of the task by collecting expert human predictions, showing that even human experts can only reach 53% accuracy on this dataset. Our Longformer model clearly outperforms the human baseline (63%), despite only considering the first 2,048 tokens. Furthermore, we perform a detailed error analysis and find that the Longformer model is significantly better calibrated than the human experts. Finally, we publicly release the dataset and the code used for the experiments.

Common law courts need to refer to similar precedents' judgments to inform their current decisions. Generating high-quality summaries of court judgment documents can facilitate legal practitioners to efficiently review previous cases and assist the general public in accessing how the courts operate and how the law is applied. Previous court judgment summarization research focuses on civil law or a particular jurisdiction's judgments. However, judges can refer to the judgments from all common law jurisdictions. Current summarization datasets are insufficient to satisfy the demands of summarizing precedents across multiple jurisdictions, especially when labeled data are scarce for many jurisdictions. To address the lack of datasets, we present CLSum, the first dataset for summarizing multi-jurisdictional common law court judgment documents. Besides, this is the first court judgment summarization work adopting large language models (LLMs) in data augmentation, summary generation, and evaluation. Specifically, we design an LLM-based data augmentation method incorporating legal knowledge. We also propose a legal knowledge enhanced evaluation metric based on LLM to assess the quality of generated judgment summaries. Our experimental results verify that the LLM-based summarization methods can perform well in the few-shot and zero-shot settings. Our LLM-based data augmentation method can mitigate the impact of low data resources. Furthermore, we carry out comprehensive comparative experiments to find essential model components and settings that are capable of enhancing summarization performance.

Recent advances in LegalAI have primarily focused on individual case judgment analysis, often overlooking the critical appellate process within the judicial system. Appeals serve as a core mechanism for error correction and ensuring fair trials, making them highly significant both in practice and in research. To address this gap, we present the AppealCase dataset, consisting of 10,000 pairs of real-world, matched first-instance and second-instance documents across 91 categories of civil cases. The dataset also includes detailed annotations along five dimensions central to appellate review: judgment reversals, reversal reasons, cited legal provisions, claim-level decisions, and whether there is new information in the second instance. Based on these annotations, we propose five novel LegalAI tasks and conduct a comprehensive evaluation across 20 mainstream models. Experimental results reveal that all current models achieve less than 50% F1 scores on the judgment reversal prediction task, highlighting the complexity and challenge of the appeal scenario. We hope that the AppealCase dataset will spur further research in LegalAI for appellate case analysis and contribute to improving consistency in judicial decision-making.

Evaluating large language models (LLMs) on open-ended tasks without ground-truth labels is increasingly done via the LLM-as-a-judge paradigm. A critical but under-modeled issue is that judge LLMs differ substantially in reliability; treating all judges equally can yield biased leaderboards and misleading uncertainty estimates. More data can make evaluation more confidently wrong under misspecified aggregation. We propose a judge-aware ranking framework that extends the Bradley-Terry-Luce model by introducing judge-specific discrimination parameters, jointly estimating latent model quality and judge reliability from pairwise comparisons without reference labels. We establish identifiability up to natural normalizations and prove consistency and asymptotic normality of the maximum likelihood estimator, enabling confidence intervals for score differences and rank comparisons. Across multiple public benchmarks and a newly collected dataset, our method improves agreement with human preferences, achieves higher data efficiency than unweighted baselines, and produces calibrated uncertainty quantification for LLM rankings.

Legal judgment prediction (LJP) aims to function as a judge by making final rulings based on case claims and facts, which plays a vital role in the judicial domain for supporting court decision-making and improving judicial efficiency. However, existing methods often struggle with logical errors when conducting complex legal reasoning. We propose LegalReasoner, which enhances LJP reliability through step-wise verification and correction of the reasoning process. Specifically, it first identifies dispute points to decompose complex cases, and then conducts step-wise reasoning while employing a process verifier to validate each step's logic from correctness, progressiveness, and potential perspectives. When errors are detected, expert-designed attribution and resolution strategies are applied for correction. To fine-tune LegalReasoner, we release the LegalHK dataset, containing 58,130 Hong Kong court cases with detailed annotations of dispute points, step-by-step reasoning chains, and process verification labels. Experiments demonstrate that LegalReasoner significantly improves concordance with court decisions from 72.37 to 80.27 on LLAMA-3.1-70B. The data is available at https://huggingface.co/datasets/weijiezz/LegalHK.

LLM-as-a-Judge refers to the automatic modeling of preferences for responses generated by Large Language Models (LLMs), which is of significant importance for both LLM evaluation and reward modeling. Although generative LLMs have made substantial progress in various tasks, their performance as LLM-Judge still falls short of expectations. In this work, we propose Think-J, which improves generative LLM-as-a-Judge by learning how to think. We first utilized a small amount of curated data to develop the model with initial judgment thinking capabilities. Subsequently, we optimize the judgment thinking traces based on reinforcement learning (RL). We propose two methods for judgment thinking optimization, based on offline and online RL, respectively. The offline RL requires training a critic model to construct positive and negative examples for learning. The online method defines rule-based reward as feedback for optimization. Experimental results showed that our approach can significantly enhance the evaluation capability of generative LLM-Judge, surpassing both generative and classifier-based LLM-Judge without requiring extra human annotations.

This research introduces the Judge's Verdict Benchmark, a novel two-step methodology to evaluate Large Language Models (LLMs) as judges for response accuracy evaluation tasks. We assess how well 54 LLMs can replicate human judgment when scoring responses from RAG (Retrieval-Augmented Generation) or Agentic pipelines against ground truth answers. Our methodology progresses from traditional correlation analysis to comprehensive Cohen's Kappa analysis that measures actual agreement patterns. The two-step approach includes: (1) a correlation test that filters judges with strong alignment, followed by (2) a human-likeness test using z-scores to identify two distinct judgment patterns: human-like judgment (|z| < 1) that mimics natural human variation, and super-consistent judgment (z > 1) that exceeds typical human-to-human agreement levels. This methodology reveals that 27 out of 54 tested LLMs achieve Tier 1 performance: 23 models exhibit human-like patterns that preserve the nuances of human judgment, while 4 models demonstrate super-consistent behavior, a pattern that could indicate either enhanced reliability or oversimplification of complex judgments. Testing 43 open-source models (1B-405B parameters) and 11 closed models (GPT, Gemini, Claude variants), we demonstrate that judge excellence is not solely dependent on model size but on specific training strategies. Our key contributions include: (1) establishing that correlation alone is insufficient for judge evaluation, (2) introducing a "Turing Test for judges" based on agreement patterns, and (3) providing a standardized benchmark for classifying LLM judges into distinct performance tiers for different evaluation needs.

Given the rapid progress of generative AI, there is a pressing need to systematically compare and choose between the numerous models and configurations available. The scale and versatility of such evaluations make the use of LLM-based judges a compelling solution for this challenge. Crucially, this approach requires first to validate the quality of the LLM judge itself. Previous work has focused on instance-based assessment of LLM judges, where a judge is evaluated over a set of responses, or response pairs, while being agnostic to their source systems. We argue that this setting overlooks critical factors affecting system-level ranking, such as a judge's positive or negative bias towards certain systems. To address this gap, we conduct the first large-scale study of LLM judges as system rankers. System scores are generated by aggregating judgment scores over multiple system outputs, and the judge's quality is assessed by comparing the resulting system ranking to a human-based ranking. Beyond overall judge assessment, our analysis provides a fine-grained characterization of judge behavior, including their decisiveness and bias.

Scaling test-time computation, or affording a generator large language model (LLM) extra compute during inference, typically employs the help of external non-generative evaluators (i.e., reward models). Concurrently, LLM-judges, models trained to generate evaluations and critiques (explanations) in natural language, are becoming increasingly popular in automatic evaluation. Despite judge empirical successes, their effectiveness as evaluators in test-time scaling settings is largely unknown. In this paper, we introduce the Judge Evaluation for Test-Time Scaling (JETTS) benchmark, which evaluates judge performance in three domains (math reasoning, code generation, and instruction following) under three task settings: response reranking, step-level beam search, and critique-based response refinement. We evaluate 10 different judge models (7B-70B parameters) for 8 different base generator models (6.7B-72B parameters). Our benchmark shows that while judges are competitive with outcome reward models in reranking, they are consistently worse than process reward models in beam search procedures. Furthermore, though unique to LLM-judges, their natural language critiques are currently ineffective in guiding the generator towards better responses.

In this work, we build on recent advances in distributional reinforcement learning to give a generally applicable, flexible, and state-of-the-art distributional variant of DQN. We achieve this by using quantile regression to approximate the full quantile function for the state-action return distribution. By reparameterizing a distribution over the sample space, this yields an implicitly defined return distribution and gives rise to a large class of risk-sensitive policies. We demonstrate improved performance on the 57 Atari 2600 games in the ALE, and use our algorithm's implicitly defined distributions to study the effects of risk-sensitive policies in Atari games.

LLM-as-judge evaluation has become the de facto standard for scaling model assessment, but the practice is statistically unsound: uncalibrated scores can invert preferences, naive confidence intervals on uncalibrated scores achieve near-0% coverage, and importance-weighted estimators collapse under limited overlap despite high effective sample size (ESS). We introduce Causal Judge Evaluation (CJE), a framework that fixes all three failures. On n=4,961 Chatbot Arena prompts (after filtering from 5k), CJE achieves 99% pairwise ranking accuracy at full sample size (94% averaged across configurations), matching oracle quality, at 14x lower cost (for ranking 5 policies) by calibrating a 16x cheaper judge on just 5% oracle labels (~250 labels). CJE combines three components: (i) AutoCal-R, reward calibration via mean-preserving isotonic regression; (ii) SIMCal-W, weight stabilization via stacking of S-monotone candidates; and (iii) Oracle-Uncertainty Aware (OUA) inference that propagates calibration uncertainty into confidence intervals. We formalize the Coverage-Limited Efficiency (CLE) diagnostic, which explains why IPS-style estimators fail even when ESS exceeds 90%: the logger rarely visits regions where target policies concentrate. Key findings: SNIPS inverts rankings even with reward calibration (38% pairwise, negative Kendall's tau) due to weight instability; calibrated IPS remains near-random (47%) despite weight stabilization, consistent with CLE; OUA improves coverage from near-0% to ~86% (Direct) and ~96% (stacked-DR), where naive intervals severely under-cover.

LLM-as-a-Judge has emerged as a promising alternative to human evaluators across various tasks, yet inherent biases - particularly position bias, the tendency to favor solutions based on their position within the prompt - compromise its reliability. This exploratory study evaluates position bias in LLM judges across pairwise and list-wise comparison settings, introducing three metrics: repetition stability, position consistency, and preference fairness. Our experiments, involving 15 LLM judges across MTBench and DevBench with 22 tasks and approximately 40 solution-generating models, result in over 150,000 evaluation instances. We identify Judge-Level, Candidate-Level, and Task-Level factors contributing to bias. The findings confirm that position bias is not due to random chance and varies significantly across judges and tasks. While position bias is weakly influenced by the length of prompt components, it is strongly affected by the quality gap between solutions. Our agreement and disagreement analysis among judges further provides insights into the distribution of judging difficulty across the dataset, and highlights the potential for dataset modifications.

Legal judgment prediction is essential for enhancing judicial efficiency. In this work, we identify that existing large language models (LLMs) underperform in this domain due to challenges in understanding case complexities and distinguishing between similar charges. To adapt LLMs for effective legal judgment prediction, we introduce the Ask-Discriminate-Predict (ADAPT) reasoning framework inspired by human judicial reasoning. ADAPT involves decomposing case facts, discriminating among potential charges, and predicting the final judgment. We further enhance LLMs through fine-tuning with multi-task synthetic trajectories to improve legal judgment prediction accuracy and efficiency under our ADAPT framework. Extensive experiments conducted on two widely-used datasets demonstrate the superior performance of our framework in legal judgment prediction, particularly when dealing with complex and confusing charges.

Building on developments in machine learning and prior work in the science of judicial prediction, we construct a model designed to predict the behavior of the Supreme Court of the United States in a generalized, out-of-sample context. To do so, we develop a time evolving random forest classifier which leverages some unique feature engineering to predict more than 240,000 justice votes and 28,000 cases outcomes over nearly two centuries (1816-2015). Using only data available prior to decision, our model outperforms null (baseline) models at both the justice and case level under both parametric and non-parametric tests. Over nearly two centuries, we achieve 70.2% accuracy at the case outcome level and 71.9% at the justice vote level. More recently, over the past century, we outperform an in-sample optimized null model by nearly 5%. Our performance is consistent with, and improves on the general level of prediction demonstrated by prior work; however, our model is distinctive because it can be applied out-of-sample to the entire past and future of the Court, not a single term. Our results represent an important advance for the science of quantitative legal prediction and portend a range of other potential applications.

Large Language Models (LLMs) as automatic evaluators, commonly referred to as LLM-as-a-Judge, have also attracted growing attention. This approach plays a vital role in aligning LLMs with human judgments, providing accurate and reliable assessments. However, LLM-based judgment models often exhibit judgment preference bias during the evaluation phase, tending to favor responses generated by themselves, undermining the reliability of their judgments. This paper introduces the Group-Based Polling Optimization (Genii), an unsupervised multi-agent collaborative optimization framework that mitigates the inherent judgment preference bias of judgment models. Specifically, Genii integrates various LLM-based judgment models into a multi-agent system and simulates the interactive client-server polling mechanism to optimize each client agent unsupervisedly. Our experiments demonstrate that Genii outperforms supervised models trained on annotated judgment data, while requiring no human-labeled annotations. Genii consistently improves performance across different client agents during the polling, even when weaker models act as server agents. Further analysis reveals that Genii effectively mitigates judgment preference bias of LLM-based judgment models, demonstrating its effectiveness. All codes are available at https://github.com/NEUIR/Genii.

Common evaluation paradigms for language models focus on scoring single responses through accuracy metrics or proper scoring rules, failing to capture the full richness of a model's belief state. Recent work illustrates that language models hallucinate in-part because they are optimised to be good test-takers under binary scoring schemes that reward any answer over abstention. While this insight naturally leads to penalty-based approaches, they ignore crucial distinctions in how models distribute uncertainty, for example between hedging toward incorrect answers versus hedging toward "I don't know" responses. A novel evaluation metric, the Distributional Correctness Score (DCS), is introduced to solve this problem, i.e., of not considering a model's entire probability distribution over answer choices. DCS naturally distinguishes between harmful overconfidence in wrong answers and uncertainty expressed through abstention, providing scores in an interpretable default range. Through theoretical analysis and illustrative examples, DCS is demonstrated to offer a more nuanced and aligned evaluation paradigm that incentivises models to express genuine uncertainty rather than guessing. Adapting 12 existing evaluation benchmarks to DCS's variants and measuring performance on six language models reveals that for half of the tested benchmarks scores are negative across all tested models, indicating significant tendencies towards hallucination.

We develop an econometric framework integrating heavy-tailed Student's t distributions with behavioral probability weighting while preserving infinite divisibility. Using 432{,}752 observations across 86 assets (2004--2024), we demonstrate Student's t specifications outperform Gaussian models in 88.4\% of cases. Bounded probability-weighting transformations preserve mathematical properties required for dynamic pricing. Gaussian models underestimate 99\% Value-at-Risk by 19.7\% versus 3.2\% for our specification. Joint estimation procedures identify tail and behavioral parameters with established asymptotic properties. Results provide robust inference for asset-pricing applications where heavy tails and behavioral distortions coexist.

LLM-as-a-Judge has been widely adopted as an evaluation method and served as supervised rewards in model training. However, existing benchmarks for LLM-as-a-Judge are mainly relying on human-annotated ground truth, which introduces human bias that undermines the assessment of reliability and imposes scalability constraints. To overcome these limitations, we introduce Sage, a novel evaluation suite that assesses the quality of LLM judges without necessitating any human annotation. Inspired by axioms of rational choice theory, Sage introduces two new lenses for measuring LLM-as-a-Judge: local self-consistency (pair-wise preference stability) and global logical consistency (transitivity across a full set of preferences). We curate a dataset of 650 questions by combining structured benchmark problems with real-world user queries. Our experiments demonstrate both the stability of our metrics and their high correlation with supervised benchmarks like LLMBar and RewardBench2, confirming Sage's reliability as an evaluation suite for the robustness and accuracy of LLM-as-a-Judge. Based on Sage, we reveal that current state-of-the-art LLMs exhibit significant reliability problems when acting as judges in both scoring and pairwise settings; even the top-performing models, Gemini-2.5-Pro and GPT-5, fail to maintain consistent preferences in nearly a quarter of difficult cases. We attribute this to a new phenomenon called situational preference, which explains why explicit rubrics or criteria can help the model judge consistently across answer pairs. Our further analysis shows that finetuned LLM-as-a-Judge is a feasible method to boost performance, and the panel-based judge as well as deep reasoning can enhance the judging consistency. We also find substantial inconsistency in human judgments, which indicates that human annotation may not be a reliable gold standard.

LLM-as-Judge frameworks are increasingly popular for AI evaluation, yet research findings on the relationship between models' generation and judgment abilities remain inconsistent. We investigate this relationship through systematic dataset- and instance-level analyses across 11 models and 21 diverse tasks. Despite both capabilities relying on the same underlying knowledge, our analyses reveal they are only weakly correlated, primarily due to LLMs' sensitivity to the responses being judged. To address this, we propose a self-reference-guided evaluation strategy that leverages a model's own answers as references. This approach significantly strengthens the correlation between generation and judgment abilities, offering a practical path to align these skills and providing a reliable proxy for model selection in evaluation tasks.

Human reasoning often involves working over limited information to arrive at probabilistic conclusions. In its simplest form, this involves making an inference that is not strictly entailed by a premise, but rather only likely given the premise. While reasoning LLMs have demonstrated strong performance on logical and mathematical tasks, their behavior on such open-ended, non-deterministic inferences remains largely unexplored. We introduce ProbCOPA, a dataset of 210 handcrafted probabilistic inferences in English, each annotated for inference likelihood by 25--30 human participants. We find that human responses are graded and varied, revealing probabilistic judgments of the inferences in our dataset. Comparing these judgments with responses from eight state-of-the-art reasoning LLMs, we show that models consistently fail to produce human-like distributions. Finally, analyzing LLM reasoning chains, we find evidence of a common reasoning pattern used to evaluate such inferences. Our findings reveal persistent differences between humans and LLMs, and underscore the need to evaluate reasoning beyond deterministic settings.

Large language models (LLMs) are increasingly deployed in high-stakes domains, where rare but severe failures can result in irreversible harm. However, prevailing evaluation benchmarks often reduce complex social risk to mean-centered scalar scores, thereby obscuring distributional structure, cross-dimensional interactions, and worst-case behavior. This paper introduces Social Harm Analysis via Risk Profiles (SHARP), a framework for multidimensional, distribution-aware evaluation of social harm. SHARP models harm as a multivariate random variable and integrates explicit decomposition into bias, fairness, ethics, and epistemic reliability with a union-of-failures aggregation reparameterized as additive cumulative log-risk. The framework further employs risk-sensitive distributional statistics, with Conditional Value at Risk (CVaR95) as a primary metric, to characterize worst-case model behavior. Application of SHARP to eleven frontier LLMs, evaluated on a fixed corpus of n=901 socially sensitive prompts, reveals that models with similar average risk can exhibit more than twofold differences in tail exposure and volatility. Across models, dimension-wise marginal tail behavior varies systematically across harm dimensions, with bias exhibiting the strongest tail severities, epistemic and fairness risks occupying intermediate regimes, and ethical misalignment consistently lower; together, these patterns reveal heterogeneous, model-dependent failure structures that scalar benchmarks conflate. These findings indicate that responsible evaluation and governance of LLMs require moving beyond scalar averages toward multidimensional, tail-sensitive risk profiling.

Real-world datasets follow an imbalanced distribution, which poses significant challenges in rare-category object detection. Recent studies tackle this problem by developing re-weighting and re-sampling methods, that utilise the class frequencies of the dataset. However, these techniques focus solely on the frequency statistics and ignore the distribution of the classes in image space, missing important information. In contrast to them, we propose FRActal CALibration (FRACAL): a novel post-calibration method for long-tailed object detection. FRACAL devises a logit adjustment method that utilises the fractal dimension to estimate how uniformly classes are distributed in image space. During inference, it uses the fractal dimension to inversely downweight the probabilities of uniformly spaced class predictions achieving balance in two axes: between frequent and rare categories, and between uniformly spaced and sparsely spaced classes. FRACAL is a post-processing method and it does not require any training, also it can be combined with many off-the-shelf models such as one-stage sigmoid detectors and two-stage instance segmentation models. FRACAL boosts the rare class performance by up to 8.6% and surpasses all previous methods on LVIS dataset, while showing good generalisation to other datasets such as COCO, V3Det and OpenImages. We provide the code at https://github.com/kostas1515/FRACAL.

Post-hoc out-of-distribution (OOD) detection has garnered intensive attention in reliable machine learning. Many efforts have been dedicated to deriving score functions based on logits, distances, or rigorous data distribution assumptions to identify low-scoring OOD samples. Nevertheless, these estimate scores may fail to accurately reflect the true data density or impose impractical constraints. To provide a unified perspective on density-based score design, we propose a novel theoretical framework grounded in Bregman divergence, which extends distribution considerations to encompass an exponential family of distributions. Leveraging the conjugation constraint revealed in our theorem, we introduce a ConjNorm method, reframing density function design as a search for the optimal norm coefficient p against the given dataset. In light of the computational challenges of normalization, we devise an unbiased and analytically tractable estimator of the partition function using the Monte Carlo-based importance sampling technique. Extensive experiments across OOD detection benchmarks empirically demonstrate that our proposed ConjNorm has established a new state-of-the-art in a variety of OOD detection setups, outperforming the current best method by up to 13.25% and 28.19% (FPR95) on CIFAR-100 and ImageNet-1K, respectively.

LLM-as-a-judge has emerged as a cornerstone technique for evaluating large language models by leveraging LLM reasoning to score prompt-response pairs. Since LLM judgments are stochastic, practitioners commonly query each pair multiple times to estimate mean scores accurately. This raises a critical challenge: given a fixed computational budget B, how to optimally allocate queries across K prompt-response pairs to minimize estimation error? % We present a principled variance-adaptive approach leveraging multi-armed bandit theory and concentration inequalities. Our method dynamically allocates queries based on estimated score variances, concentrating resources where uncertainty is highest. Further, our algorithm is shown to achieve a worst-case score-estimation error of Oleft(frac{sum_{i=1^K σ_i^2}{B}}right), σ_i^2 being the unknown score variance for pair i in [K] with near-optimal budget allocation. % Experiments on Summarize-From-Feedback and HelpSteer2 demonstrate that our method significantly outperforms uniform allocation, reducing worst-case estimation error while maintaining identical budgets. Our work establishes a theoretical foundation for efficient LLM evaluation with practical implications for AI safety, model alignment, and automated assessment at scale.

In reinforcement learning an agent interacts with the environment by taking actions and observing the next state and reward. When sampled probabilistically, these state transitions, rewards, and actions can all induce randomness in the observed long-term return. Traditionally, reinforcement learning algorithms average over this randomness to estimate the value function. In this paper, we build on recent work advocating a distributional approach to reinforcement learning in which the distribution over returns is modeled explicitly instead of only estimating the mean. That is, we examine methods of learning the value distribution instead of the value function. We give results that close a number of gaps between the theoretical and algorithmic results given by Bellemare, Dabney, and Munos (2017). First, we extend existing results to the approximate distribution setting. Second, we present a novel distributional reinforcement learning algorithm consistent with our theoretical formulation. Finally, we evaluate this new algorithm on the Atari 2600 games, observing that it significantly outperforms many of the recent improvements on DQN, including the related distributional algorithm C51.

The integration of artificial intelligence (AI) in legal judgment prediction (LJP) has the potential to transform the legal landscape, particularly in jurisdictions like India, where a significant backlog of cases burdens the legal system. This paper introduces NyayaAnumana, the largest and most diverse corpus of Indian legal cases compiled for LJP, encompassing a total of 7,02,945 preprocessed cases. NyayaAnumana, which combines the words "Nyay" (judgment) and "Anuman" (prediction or inference) respectively for most major Indian languages, includes a wide range of cases from the Supreme Court, High Courts, Tribunal Courts, District Courts, and Daily Orders and, thus, provides unparalleled diversity and coverage. Our dataset surpasses existing datasets like PredEx and ILDC, offering a comprehensive foundation for advanced AI research in the legal domain. In addition to the dataset, we present INLegalLlama, a domain-specific generative large language model (LLM) tailored to the intricacies of the Indian legal system. It is developed through a two-phase training approach over a base LLaMa model. First, Indian legal documents are injected using continual pretraining. Second, task-specific supervised finetuning is done. This method allows the model to achieve a deeper understanding of legal contexts. Our experiments demonstrate that incorporating diverse court data significantly boosts model accuracy, achieving approximately 90% F1-score in prediction tasks. INLegalLlama not only improves prediction accuracy but also offers comprehensible explanations, addressing the need for explainability in AI-assisted legal decisions.

Recent advances in reinforcement learning (RL)-based post-training have led to notable improvements in large language models (LLMs), particularly in enhancing their reasoning capabilities to handle complex tasks. However, most existing methods treat the training data as a unified whole, overlooking the fact that modern LLM training often involves a mixture of data from diverse distributions-varying in both source and difficulty. This heterogeneity introduces a key challenge: how to adaptively schedule training across distributions to optimize learning efficiency. In this paper, we present a principled curriculum learning framework grounded in the notion of distribution-level learnability. Our core insight is that the magnitude of policy advantages reflects how much a model can still benefit from further training on a given distribution. Based on this, we propose a distribution-level curriculum learning framework for RL-based LLM post-training, which leverages the Upper Confidence Bound (UCB) principle to dynamically adjust sampling probabilities for different distrubutions. This approach prioritizes distributions with either high average advantage (exploitation) or low sample count (exploration), yielding an adaptive and theoretically grounded training schedule. We instantiate our curriculum learning framework with GRPO as the underlying RL algorithm and demonstrate its effectiveness on logic reasoning datasets with multiple difficulties and sources. Our experiments show that our framework significantly improves convergence speed and final performance, highlighting the value of distribution-aware curriculum strategies in LLM post-training. Code: https://github.com/ZhentingWang/DUMP.

Large language models (LLMs) are increasingly used as judges to replace costly human preference labels in pairwise evaluation. Despite their practicality, LLM judges remain prone to miscalibration and systematic biases. This paper proposes SCOPE (Selective Conformal Optimized Pairwise Evaluation), a framework for selective pairwise judging with finite-sample statistical guarantees. Under exchangeability, SCOPE calibrates an acceptance threshold such that the error rate among non-abstained judgments is at most a user-specified level α. To provide SCOPE with a bias-neutral uncertainty signal, we introduce Bidirectional Preference Entropy (BPE), which queries the judge under both response positions, aggregates the implied preference probabilities to enforce invariance to response order, and converts the aggregated probability into an entropy-based uncertainty score. Across MT-Bench, RewardBench, and Chatbot Arena, BPE improves uncertainty quality over standard confidence proxies, providing a stronger selection signal that enables SCOPE to consistently meet the target risk level while retaining good coverage across judge scales. In particular, at α= 0.10, SCOPE consistently satisfies the risk bound across all benchmarks and judge scales (empirical risk approx 0.097 to 0.099), while retaining substantial coverage, reaching 0.89 on RewardBench with Qwen-14B and 0.98 on RewardBench with Qwen-32B. Compared to naïve baselines, SCOPE accepts up to 2.4times more judgments on MT-Bench with Qwen-7B under the same target risk constraint, demonstrating that BPE enables reliable and high-coverage LLM-based evaluation.

Ensuring the reliability and safety of automated decision-making is crucial. It is well-known that data distribution shifts in machine learning can produce unreliable outcomes. This paper proposes a new approach for measuring the reliability of predictions under distribution shifts. We analyze how the outputs of a trained neural network change using clustering to measure distances between outputs and class centroids. We propose this distance as a metric to evaluate the confidence of predictions under distribution shifts. We assign each prediction to a cluster with centroid representing the mean softmax output for all correct predictions of a given class. We then define a safety threshold for a class as the smallest distance from an incorrect prediction to the given class centroid. We evaluate the approach on the MNIST and CIFAR-10 datasets using a Convolutional Neural Network and a Vision Transformer, respectively. The results show that our approach is consistent across these data sets and network models, and indicate that the proposed metric can offer an efficient way of determining when automated predictions are acceptable and when they should be deferred to human operators given a distribution shift.

Although the existing causal inference literature focuses on the forward-looking perspective by estimating effects of causes, the backward-looking perspective can provide insights into causes of effects. In backward-looking causal inference, the probability of necessity measures the probability that a certain event is caused by the treatment given the observed treatment and outcome. Most existing results focus on binary outcomes. Motivated by applications with ordinal outcomes, we propose a general definition of the probability of necessity. However, identifying the probability of necessity is challenging because it involves the joint distribution of the potential outcomes. We propose a novel assumption of monotonic incremental treatment effect to identify the probability of necessity with ordinal outcomes. We also discuss the testable implications of this key identification assumption. When it fails, we derive explicit formulas of the sharp large-sample bounds on the probability of necessity.

This paper presents the first dataset for Japanese Legal Judgment Prediction (LJP), the Japanese Tort-case Dataset (JTD), which features two tasks: tort prediction and its rationale extraction. The rationale extraction task identifies the court's accepting arguments from alleged arguments by plaintiffs and defendants, which is a novel task in the field. JTD is constructed based on annotated 3,477 Japanese Civil Code judgments by 41 legal experts, resulting in 7,978 instances with 59,697 of their alleged arguments from the involved parties. Our baseline experiments show the feasibility of the proposed two tasks, and our error analysis by legal experts identifies sources of errors and suggests future directions of the LJP research.

LLM-as-a-judge models have been used for evaluating both human and AI generated content, specifically by providing scores and rationales. Rationales, in addition to increasing transparency, help models learn to calibrate its judgments. Enhancing a model's rationale can therefore improve its calibration abilities and ultimately the ability to score content. We introduce Self-Rationalization, an iterative process of improving the rationales for the judge models, which consequently improves the score for fine-grained customizable scoring criteria (i.e., likert-scale scoring with arbitrary evaluation criteria). Self-rationalization works by having the model generate multiple judgments with rationales for the same input, curating a preference pair dataset from its own judgements, and iteratively fine-tuning the judge via DPO. Intuitively, this approach allows the judge model to self-improve by learning from its own rationales, leading to better alignment and evaluation accuracy. After just two iterations -- while only relying on examples in the training set -- human evaluation shows that our judge model learns to produce higher quality rationales, with a win rate of 62% on average compared to models just trained via SFT on rationale . This judge model also achieves high scoring accuracy on BigGen Bench and Reward Bench, outperforming even bigger sized models trained using SFT with rationale, self-consistency or best-of-N sampling by 3% to 9%.

Large Language Models (LLMs) are increasingly employed as automated evaluators to assess the safety of generated content, yet their reliability in this role remains uncertain. This study evaluates a diverse set of 11 LLM judge models across critical safety domains, examining three key aspects: self-consistency in repeated judging tasks, alignment with human judgments, and susceptibility to input artifacts such as apologetic or verbose phrasing. Our findings reveal that biases in LLM judges can significantly distort the final verdict on which content source is safer, undermining the validity of comparative evaluations. Notably, apologetic language artifacts alone can skew evaluator preferences by up to 98\%. Contrary to expectations, larger models do not consistently exhibit greater robustness, while smaller models sometimes show higher resistance to specific artifacts. To mitigate LLM evaluator robustness issues, we investigate jury-based evaluations aggregating decisions from multiple models. Although this approach both improves robustness and enhances alignment to human judgements, artifact sensitivity persists even with the best jury configurations. These results highlight the urgent need for diversified, artifact-resistant methodologies to ensure reliable safety assessments.

Reward models are crucial for aligning large language models (LLMs) with human values and intentions. Existing approaches follow either Generative (GRMs) or Discriminative (DRMs) paradigms, yet both suffer from limitations: GRMs typically demand costly point-wise supervision, while DRMs produce uncalibrated relative scores that lack probabilistic interpretation. To address these challenges, we introduce a novel reward modeling paradigm: Probabilistic Reward Model (PRM). Instead of modeling reward as a deterministic scalar, our approach treats it as a random variable, learning a full probability distribution for the quality of each response. To make this paradigm practical, we present its closed-form, discrete realization: the Ordinal Probabilistic Reward Model (OPRM), which discretizes the quality score into a finite set of ordinal ratings. Building on OPRM, we propose a data-efficient training strategy called Region Flooding Tuning (RgFT). It enables rewards to better reflect absolute text quality by incorporating quality-level annotations, which guide the model to concentrate the probability mass within corresponding rating sub-regions. Experiments on various reward model benchmarks show that our method improves accuracy by 2.9%sim7.4% compared to prior reward models, demonstrating strong performance and data efficiency. Analysis of the score distribution provides evidence that our method captures not only relative rankings but also absolute quality.

We study the problem of multi-task learning under user-level differential privacy, in which n users contribute data to m tasks, each involving a subset of users. One important aspect of the problem, that can significantly impact quality, is the distribution skew among tasks. Certain tasks may have much fewer data samples than others, making them more susceptible to the noise added for privacy. It is natural to ask whether algorithms can adapt to this skew to improve the overall utility. We give a systematic analysis of the problem, by studying how to optimally allocate a user's privacy budget among tasks. We propose a generic algorithm, based on an adaptive reweighting of the empirical loss, and show that when there is task distribution skew, this gives a quantifiable improvement of excess empirical risk. Experimental studies on recommendation problems that exhibit a long tail of small tasks, demonstrate that our methods significantly improve utility, achieving the state of the art on two standard benchmarks.

Ensuring equitable treatment (fairness) across protected attributes (such as gender or ethnicity) is a critical issue in machine learning. Most existing literature focuses on binary classification, but achieving fairness in regression tasks-such as insurance pricing or hiring score assessments-is equally important. Moreover, anti-discrimination laws also apply to continuous attributes, such as age, for which many existing methods are not applicable. In practice, multiple protected attributes can exist simultaneously; however, methods targeting fairness across several attributes often overlook so-called "fairness gerrymandering", thereby ignoring disparities among intersectional subgroups (e.g., African-American women or Hispanic men). In this paper, we propose a distance covariance regularisation framework that mitigates the association between model predictions and protected attributes, in line with the fairness definition of demographic parity, and that captures both linear and nonlinear dependencies. To enhance applicability in the presence of multiple protected attributes, we extend our framework by incorporating two multivariate dependence measures based on distance covariance: the previously proposed joint distance covariance (JdCov) and our novel concatenated distance covariance (CCdCov), which effectively address fairness gerrymandering in both regression and classification tasks involving protected attributes of various types. We discuss and illustrate how to calibrate regularisation strength, including a method based on Jensen-Shannon divergence, which quantifies dissimilarities in prediction distributions across groups. We apply our framework to the COMPAS recidivism dataset and a large motor insurance claims dataset.

Survival analysis is the problem of estimating probability distributions for future event times, which can be seen as a problem in uncertainty quantification. Although there are fundamental theories on strictly proper scoring rules for uncertainty quantification, little is known about those for survival analysis. In this paper, we investigate extensions of four major strictly proper scoring rules for survival analysis and we prove that these extensions are proper under certain conditions, which arise from the discretization of the estimation of probability distributions. We also compare the estimation performances of these extended scoring rules by using real datasets, and the extensions of the logarithmic score and the Brier score performed the best.

The evaluation of recommender system fairness has become increasingly important, especially with recent legislation that emphasises the development of fair and responsible artificial intelligence. This has led to the emergence of various fairness evaluation measures, which quantify fairness based on different definitions. However, many of such measures are simply proposed and used without further analysis on their robustness. As a result, there is insufficient understanding and awareness of the measures' limitations. Among other issues, it is not known what kind of model outputs produce the (un)fairest score, how the measure scores are empirically distributed, and whether there are cases where the measures cannot be computed (e.g., due to division by zero). These issues cause difficulty in interpreting the measure scores and confusion on which measure(s) should be used for a specific case. This thesis presents a series of papers that assess and overcome various theoretical, empirical, and conceptual limitations of existing recommender system fairness evaluation measures. We investigate a wide range of offline evaluation measures for different fairness notions, divided based on the evaluation subjects (users and items) and for different evaluation granularities (groups of subjects and individual subjects). Firstly, we perform theoretical and empirical analysis on the measures, exposing flaws that limit their interpretability, expressiveness, or applicability. Secondly, we contribute novel evaluation approaches and measures that overcome these limitations. Finally, considering the measures' limitations, we recommend guidelines for the appropriate measure usage, thereby allowing for more precise selection of fairness evaluation measures in practical scenarios. Overall, this thesis contributes to advancing the state-of-the-art offline evaluation of fairness in recommender systems.

In contrast to classical reinforcement learning, distributional reinforcement learning algorithms aim to learn the distribution of returns rather than their expected value. Since the nature of the return distribution is generally unknown a priori or arbitrarily complex, a common approach finds approximations within a set of representable, parametric distributions. Typically, this involves a projection of the unconstrained distribution onto the set of simplified distributions. We argue that this projection step entails a strong inductive bias when coupled with neural networks and gradient descent, thereby profoundly impacting the generalization behavior of learned models. In order to facilitate reliable uncertainty estimation through diversity, this work studies the combination of several different projections and representations in a distributional ensemble. We establish theoretical properties of such projection ensembles and derive an algorithm that uses ensemble disagreement, measured by the average 1-Wasserstein distance, as a bonus for deep exploration. We evaluate our algorithm on the behavior suite benchmark and find that diverse projection ensembles lead to significant performance improvements over existing methods on a wide variety of tasks with the most pronounced gains in directed exploration problems.

Machine learning models frequently experience performance drops under distribution shifts. The underlying cause of such shifts may be multiple simultaneous factors such as changes in data quality, differences in specific covariate distributions, or changes in the relationship between label and features. When a model does fail during deployment, attributing performance change to these factors is critical for the model developer to identify the root cause and take mitigating actions. In this work, we introduce the problem of attributing performance differences between environments to distribution shifts in the underlying data generating mechanisms. We formulate the problem as a cooperative game where the players are distributions. We define the value of a set of distributions to be the change in model performance when only this set of distributions has changed between environments, and derive an importance weighting method for computing the value of an arbitrary set of distributions. The contribution of each distribution to the total performance change is then quantified as its Shapley value. We demonstrate the correctness and utility of our method on synthetic, semi-synthetic, and real-world case studies, showing its effectiveness in attributing performance changes to a wide range of distribution shifts.

Machine learning shows promise in predicting the outcome of legal cases, but most research has concentrated on civil law cases rather than case law systems. We identified two unique challenges in making legal case outcome predictions with case law. First, it is crucial to identify relevant precedent cases that serve as fundamental evidence for judges during decision-making. Second, it is necessary to consider the evolution of legal principles over time, as early cases may adhere to different legal contexts. In this paper, we proposed a new framework named PILOT (PredictIng Legal case OuTcome) for case outcome prediction. It comprises two modules for relevant case retrieval and temporal pattern handling, respectively. To benchmark the performance of existing legal case outcome prediction models, we curated a dataset from a large-scale case law database. We demonstrate the importance of accurately identifying precedent cases and mitigating the temporal shift when making predictions for case law, as our method shows a significant improvement over the prior methods that focus on civil law case outcome predictions.

Existing LLM-as-a-Judge systems suffer from three fundamental limitations: limited adaptivity to task- and domain-specific evaluation criteria, systematic biases driven by non-semantic cues such as position, length, format, and model provenance, and evaluation inconsistency that leads to contradictory judgments across different evaluation modes (e.g., pointwise versus pairwise). To address these issues, we propose FairJudge, an adaptive, debiased, and consistent LLM-as-a-Judge. Unlike prior approaches that treat the judge as a static evaluator, FairJudge models judging behavior itself as a learnable and regularized policy. From a data-centric perspective, we construct a high-information-density judging dataset that explicitly injects supervision signals aligned with evaluation behavior. Building on this dataset, we adopt a curriculum-style SFT-DPO-GRPO training paradigm that progressively aligns rubric adherence, bias mitigation, and cross-mode consistency, while avoiding catastrophic forgetting. Experimental results on multiple internal and public benchmarks show that FairJudge consistently improves agreement and F1, reduces non-semantic biases, and outperforms substantially larger instruction-tuned LLMs. All resources will be publicly released after acceptance to facilitate future research.

LLM-judged benchmarks are increasingly used to evaluate complex model behaviors, yet their design introduces failure modes absent in conventional ground-truth based benchmarks. We argue that without tight objectives and verifiable constructions, benchmark rankings can produce high-confidence rankings that are in fact largely noise. We introduce two mechanisms to diagnose these issues. Schematic adherence quantifies how much of a judge's overall verdict is explained by the explicit evaluation schema, revealing unexplained variance when judges deviate from their own rubric. Psychometric validity aggregates internal consistency and discriminant validity signals to quantify irreducible uncertainty in any benchmarking run. Applying these tools to Arena-Hard Auto, we find severe schema incoherence and factor collapse across popular judges: for example, unexplained variance exceeding 90 percent for DeepSeek-R1-32B and factor correlations above 0.93 for most criteria. We also show that the ELO-style aggregation used by Arena-Hard Auto collapses and masks genuine ranking uncertainty. Our results highlight design failures that undermine validity and offer actionable principles for building better-scoped, reliability-aware LLM-judged benchmarks. We release our code at https://anonymous.4open.science/r/judgment-to-noise-947D/README.md

We investigate stochastic combinatorial semi-bandits, where the entire joint distribution of outcomes impacts the complexity of the problem instance (unlike in the standard bandits). Typical distributions considered depend on specific parameter values, whose prior knowledge is required in theory but quite difficult to estimate in practice; an example is the commonly assumed sub-Gaussian family. We alleviate this issue by instead considering a new general family of sub-exponential distributions, which contains bounded and Gaussian ones. We prove a new lower bound on the expected regret on this family, that is parameterized by the unknown covariance matrix of outcomes, a tighter quantity than the sub-Gaussian matrix. We then construct an algorithm that uses covariance estimates, and provide a tight asymptotic analysis of the regret. Finally, we apply and extend our results to the family of sparse outcomes, which has applications in many recommender systems.

We develop a method to generate predictive regions that cover a multivariate response variable with a user-specified probability. Our work is composed of two components. First, we use a deep generative model to learn a representation of the response that has a unimodal distribution. Existing multiple-output quantile regression approaches are effective in such cases, so we apply them on the learned representation, and then transform the solution to the original space of the response. This process results in a flexible and informative region that can have an arbitrary shape, a property that existing methods lack. Second, we propose an extension of conformal prediction to the multivariate response setting that modifies any method to return sets with a pre-specified coverage level. The desired coverage is theoretically guaranteed in the finite-sample case for any distribution. Experiments conducted on both real and synthetic data show that our method constructs regions that are significantly smaller compared to existing techniques.

It is well known that accurate probabilistic predictors can be trained through empirical risk minimisation with proper scoring rules as loss functions. While such learners capture so-called aleatoric uncertainty of predictions, various machine learning methods have recently been developed with the goal to let the learner also represent its epistemic uncertainty, i.e., the uncertainty caused by a lack of knowledge and data. An emerging branch of the literature proposes the use of a second-order learner that provides predictions in terms of distributions on probability distributions. However, recent work has revealed serious theoretical shortcomings for second-order predictors based on loss minimisation. In this paper, we generalise these findings and prove a more fundamental result: There seems to be no loss function that provides an incentive for a second-order learner to faithfully represent its epistemic uncertainty in the same manner as proper scoring rules do for standard (first-order) learners. As a main mathematical tool to prove this result, we introduce the generalised notion of second-order scoring rules.

Multivariate probabilistic time series forecasts are commonly evaluated via proper scoring rules, i.e., functions that are minimal in expectation for the ground-truth distribution. However, this property is not sufficient to guarantee good discrimination in the non-asymptotic regime. In this paper, we provide the first systematic finite-sample study of proper scoring rules for time-series forecasting evaluation. Through a power analysis, we identify the "region of reliability" of a scoring rule, i.e., the set of practical conditions where it can be relied on to identify forecasting errors. We carry out our analysis on a comprehensive synthetic benchmark, specifically designed to test several key discrepancies between ground-truth and forecast distributions, and we gauge the generalizability of our findings to real-world tasks with an application to an electricity production problem. Our results reveal critical shortcomings in the evaluation of multivariate probabilistic forecasts as commonly performed in the literature.

Test-time reinforcement learning (TTRL) enables large language models (LLMs) to self-improve on unlabeled inputs, but its effectiveness critically depends on how reward signals are estimated without ground-truth supervision. Most existing TTRL methods rely on majority voting (MV) over rollouts to produce deterministic rewards, implicitly assuming that the majority rollout provides a reliable learning signal. We show that this assumption is fragile: MV reduces the rollout distribution into a single outcome, discarding information about non-majority but correct actions candidates, and yields systematically biased reward estimates. To address this, we propose Distribution-AwareReward Estimation (DARE), which shifts reward estimation from a single majority outcome to the full empirical rollout distribution. DARE further augments this distribution-based reward with an exploration bonus and a distribution pruning mechanism for non-majority rollout exploration and reward denoise, yielding a more informative and robust reward estimation. Extensive experiments on challenging reasoning benchmarks show that DARE improves optimization stability and final performance over recent baselines, achieving relative improvements of 25.3% on challenging AIME 2024 and 5.3% on AMC.

The predictions of small transformers, trained to calculate the greatest common divisor (GCD) of two positive integers, can be fully characterized by looking at model inputs and outputs. As training proceeds, the model learns a list mathcal D of integers, products of divisors of the base used to represent integers and small primes, and predicts the largest element of mathcal D that divides both inputs. Training distributions impact performance. Models trained from uniform operands only learn a handful of GCD (up to 38 GCD leq100). Log-uniform operands boost performance to 73 GCD leq 100, and a log-uniform distribution of outcomes (i.e. GCD) to 91. However, training from uniform (balanced) GCD breaks explainability.

This work demonstrates that Legal Judgement Prediction systems without expert-informed adjustments can be vulnerable to shallow, distracting surface signals that arise from corpus construction, case distribution, and confounding factors. To mitigate this, we use domain expertise to strategically identify statistically predictive but legally irrelevant information. We adopt adversarial training to prevent the system from relying on it. We evaluate our deconfounded models by employing interpretability techniques and comparing to expert annotations. Quantitative experiments and qualitative analysis show that our deconfounded model consistently aligns better with expert rationales than baselines trained for prediction only. We further contribute a set of reference expert annotations to the validation and testing partitions of an existing benchmark dataset of European Court of Human Rights cases.

In the field of algorithmic fairness, significant attention has been put on group fairness criteria, such as Demographic Parity and Equalized Odds. Nevertheless, these objectives, measured as global averages, have raised concerns about persistent local disparities between sensitive groups. In this work, we address the problem of local fairness, which ensures that the predictor is unbiased not only in terms of expectations over the whole population, but also within any subregion of the feature space, unknown at training time. To enforce this objective, we introduce ROAD, a novel approach that leverages the Distributionally Robust Optimization (DRO) framework within a fair adversarial learning objective, where an adversary tries to infer the sensitive attribute from the predictions. Using an instance-level re-weighting strategy, ROAD is designed to prioritize inputs that are likely to be locally unfair, i.e. where the adversary faces the least difficulty in reconstructing the sensitive attribute. Numerical experiments demonstrate the effectiveness of our method: it achieves Pareto dominance with respect to local fairness and accuracy for a given global fairness level across three standard datasets, and also enhances fairness generalization under distribution shift.

Auto-evaluation is crucial for assessing response quality and offering feedback for model development. Recent studies have explored training large language models (LLMs) as generative judges to evaluate and critique other models' outputs. In this work, we investigate the idea of learning from both positive and negative data with preference optimization to enhance the evaluation capabilities of LLM judges across an array of different use cases. We achieve this by employing three approaches to collect the preference pairs for different use cases, each aimed at improving our generative judge from a different perspective. Our comprehensive study over a wide range of benchmarks demonstrates the effectiveness of our method. In particular, our generative judge achieves the best performance on 10 out of 13 benchmarks, outperforming strong baselines like GPT-4o and specialized judge models. Further analysis show that our judge model robustly counters inherent biases such as position and length bias, flexibly adapts to any evaluation protocol specified by practitioners, and provides helpful language feedback for improving downstream generator models.

Categorical distributions are ubiquitous in machine learning, e.g., in classification, language models, and recommendation systems. However, when the number of possible outcomes is very large, using categorical distributions becomes computationally expensive, as the complexity scales linearly with the number of outcomes. To address this problem, we propose augment and reduce (A&R), a method to alleviate the computational complexity. A&R uses two ideas: latent variable augmentation and stochastic variational inference. It maximizes a lower bound on the marginal likelihood of the data. Unlike existing methods which are specific to softmax, A&R is more general and is amenable to other categorical models, such as multinomial probit. On several large-scale classification problems, we show that A&R provides a tighter bound on the marginal likelihood and has better predictive performance than existing approaches.

In the real world, the frequency of occurrence of objects is naturally skewed forming long-tail class distributions, which results in poor performance on the statistically rare classes. A promising solution is to mine tail-class examples to balance the training dataset. However, mining tail-class examples is a very challenging task. For instance, most of the otherwise successful uncertainty-based mining approaches struggle due to distortion of class probabilities resulting from skewness in data. In this work, we propose an effective, yet simple, approach to overcome these challenges. Our framework enhances the subdued tail-class activations and, thereafter, uses a one-class data-centric approach to effectively identify tail-class examples. We carry out an exhaustive evaluation of our framework on three datasets spanning over two computer vision tasks. Substantial improvements in the minority-class mining and fine-tuned model's performance strongly corroborate the value of our proposed solution.

Distribution inference, sometimes called property inference, infers statistical properties about a training set from access to a model trained on that data. Distribution inference attacks can pose serious risks when models are trained on private data, but are difficult to distinguish from the intrinsic purpose of statistical machine learning -- namely, to produce models that capture statistical properties about a distribution. Motivated by Yeom et al.'s membership inference framework, we propose a formal definition of distribution inference attacks that is general enough to describe a broad class of attacks distinguishing between possible training distributions. We show how our definition captures previous ratio-based property inference attacks as well as new kinds of attack including revealing the average node degree or clustering coefficient of a training graph. To understand distribution inference risks, we introduce a metric that quantifies observed leakage by relating it to the leakage that would occur if samples from the training distribution were provided directly to the adversary. We report on a series of experiments across a range of different distributions using both novel black-box attacks and improved versions of the state-of-the-art white-box attacks. Our results show that inexpensive attacks are often as effective as expensive meta-classifier attacks, and that there are surprising asymmetries in the effectiveness of attacks. Code is available at https://github.com/iamgroot42/FormEstDistRisks

We study the problem of estimating the distribution of the return of a policy using an offline dataset that is not generated from the policy, i.e., distributional offline policy evaluation (OPE). We propose an algorithm called Fitted Likelihood Estimation (FLE), which conducts a sequence of Maximum Likelihood Estimation (MLE) and has the flexibility of integrating any state-of-the-art probabilistic generative models as long as it can be trained via MLE. FLE can be used for both finite-horizon and infinite-horizon discounted settings where rewards can be multi-dimensional vectors. Our theoretical results show that for both finite-horizon and infinite-horizon discounted settings, FLE can learn distributions that are close to the ground truth under total variation distance and Wasserstein distance, respectively. Our theoretical results hold under the conditions that the offline data covers the test policy's traces and that the supervised learning MLE procedures succeed. Experimentally, we demonstrate the performance of FLE with two generative models, Gaussian mixture models and diffusion models. For the multi-dimensional reward setting, FLE with diffusion models is capable of estimating the complicated distribution of the return of a test policy.

Courts must justify their decisions, but systematically analyzing judicial reasoning at scale remains difficult. This study tests claims about formalistic judging in Central and Eastern Europe (CEE) by developing automated methods to detect and classify judicial reasoning in decisions of Czech Supreme Courts using state-of-the-art natural language processing methods. We create the MADON dataset of 272 decisions from two Czech Supreme Courts with expert annotations of 9,183 paragraphs with eight argument types and holistic formalism labels for supervised training and evaluation. Using a corpus of 300,511 Czech court decisions, we adapt transformer LLMs to Czech legal domain through continued pretraining and we experiment with methods to address dataset imbalance including asymmetric loss and class weighting. The best models can detect argumentative paragraphs (82.6% Bal-F1), classify traditional types of legal argument (77.5% Bal-F1), and classify decisions as formalistic/non-formalistic (83.8% Bal-F1). Our three-stage pipeline combining ModernBERT, Llama 3.1, and traditional feature-based machine learning achieves promising results for decision classification while reducing computational costs and increasing explainability. Empirically, we challenge prevailing narratives about CEE formalism. We demonstrate that legal argument mining enables promising judicial philosophy classification and highlight its potential for other important tasks in computational legal studies. Our methodology can be used across jurisdictions, and our entire pipeline, datasets, guidelines, models, and source codes are available at https://github.com/trusthlt/madon.

Recent studies show that large language models (LLMs) exhibit self-preference bias when serving as judges, meaning they tend to favor their own responses over those generated by other models. Existing methods typically measure this bias by calculating the difference between the scores a judge model assigns to its own responses and those it assigns to responses from other models. However, this approach conflates self-preference bias with response quality, as higher-quality responses from the judge model may also lead to positive score differences, even in the absence of bias. To address this issue, we introduce gold judgments as proxies for the actual quality of responses and propose the DBG score, which measures self-preference bias as the difference between the scores assigned by the judge model to its own responses and the corresponding gold judgments. Since gold judgments reflect true response quality, the DBG score mitigates the confounding effect of response quality on bias measurement. Using the DBG score, we conduct comprehensive experiments to assess self-preference bias across LLMs of varying versions, sizes, and reasoning abilities. Additionally, we investigate two factors that influence and help alleviate self-preference bias: response text style and the post-training data of judge models. Finally, we explore potential underlying mechanisms of self-preference bias from an attention-based perspective. Our code and data are available at https://github.com/zhiyuanc2001/self-preference.

Predicting case outcomes is useful but still an extremely hard task for attorneys and other Law professionals. It is not easy to search case information to extract valuable information as this requires dealing with huge data sets and their complexity. For instance, the complexity of Brazil legal system along with the high litigation rates makes this problem even harder. This paper introduces an approach for predicting Brazilian court decisions which is also able to predict whether the decision will be unanimous. We developed a working prototype which performs 79% of accuracy (F1-score) on a data set composed of 4,043 cases from a Brazilian court. To our knowledge, this is the first study to forecast judge decisions in Brazil.

Rapid advancements in large language models have unlocked remarkable capabilities when it comes to processing and summarizing unstructured text data. This has implications for the analysis of rich, open-ended datasets, such as survey responses, where LLMs hold the promise of efficiently distilling key themes and sentiments. However, as organizations increasingly turn to these powerful AI systems to make sense of textual feedback, a critical question arises, can we trust LLMs to accurately represent the perspectives contained within these text based datasets? While LLMs excel at generating human-like summaries, there is a risk that their outputs may inadvertently diverge from the true substance of the original responses. Discrepancies between the LLM-generated outputs and the actual themes present in the data could lead to flawed decision-making, with far-reaching consequences for organizations. This research investigates the effectiveness of LLMs as judge models to evaluate the thematic alignment of summaries generated by other LLMs. We utilized an Anthropic Claude model to generate thematic summaries from open-ended survey responses, with Amazon's Titan Express, Nova Pro, and Meta's Llama serving as LLM judges. The LLM-as-judge approach was compared to human evaluations using Cohen's kappa, Spearman's rho, and Krippendorff's alpha, validating a scalable alternative to traditional human centric evaluation methods. Our findings reveal that while LLMs as judges offer a scalable solution comparable to human raters, humans may still excel at detecting subtle, context-specific nuances. This research contributes to the growing body of knowledge on AI assisted text analysis. We discuss limitations and provide recommendations for future research, emphasizing the need for careful consideration when generalizing LLM judge models across various contexts and use cases.

Applying a machine learning model for decision-making in the real world requires to distinguish what the model knows from what it does not. A critical factor in assessing the knowledge of a model is to quantify its predictive uncertainty. Predictive uncertainty is commonly measured by the entropy of the Bayesian model average (BMA) predictive distribution. Yet, the properness of this current measure of predictive uncertainty was recently questioned. We provide new insights regarding those limitations. Our analyses show that the current measure erroneously assumes that the BMA predictive distribution is equivalent to the predictive distribution of the true model that generated the dataset. Consequently, we introduce a theoretically grounded measure to overcome these limitations. We experimentally verify the benefits of our introduced measure of predictive uncertainty. We find that our introduced measure behaves more reasonably in controlled synthetic tasks. Moreover, our evaluations on ImageNet demonstrate that our introduced measure is advantageous in real-world applications utilizing predictive uncertainty.

Recently, rubrics have been used to guide LLM judges in capturing subjective, nuanced, multi-dimensional human preferences, and have been extended from evaluation to reward signals for reinforcement fine-tuning (RFT). However, rubric generation remains hard to control: rubrics often lack coverage, conflate dimensions, misalign preference direction, and contain redundant or highly correlated criteria, degrading judge accuracy and producing suboptimal rewards during RFT. We propose RRD, a principled framework for rubric refinement built on a recursive decompose-filter cycle. RRD decomposes coarse rubrics into fine-grained, discriminative criteria, expanding coverage while sharpening separation between responses. A complementary filtering mechanism removes misaligned and redundant rubrics, and a correlation-aware weighting scheme prevents over-representing highly correlated criteria, yielding rubric sets that are informative, comprehensive, and non-redundant. Empirically, RRD delivers large, consistent gains across both evaluation and training: it improves preference-judgment accuracy on JudgeBench and PPE for both GPT-4o and Llama3.1-405B judges, achieving top performance in all settings with up to +17.7 points on JudgeBench. When used as the reward source for RFT on WildChat, it yields substantially stronger and more stable learning signals, boosting reward by up to 160% (Qwen3-4B) and 60% (Llama3.1-8B) versus 10-20% for prior rubric baselines, with gains that transfer to HealthBench-Hard and BiGGen Bench. Overall, RRD establishes recursive rubric refinement as a scalable and interpretable foundation for LLM judging and reward modeling in open-ended domains.

Reliable probability estimation is of crucial importance in many real-world applications where there is inherent (aleatoric) uncertainty. Probability-estimation models are trained on observed outcomes (e.g. whether it has rained or not, or whether a patient has died or not), because the ground-truth probabilities of the events of interest are typically unknown. The problem is therefore analogous to binary classification, with the difference that the objective is to estimate probabilities rather than predicting the specific outcome. This work investigates probability estimation from high-dimensional data using deep neural networks. There exist several methods to improve the probabilities generated by these models but they mostly focus on model (epistemic) uncertainty. For problems with inherent uncertainty, it is challenging to evaluate performance without access to ground-truth probabilities. To address this, we build a synthetic dataset to study and compare different computable metrics. We evaluate existing methods on the synthetic data as well as on three real-world probability estimation tasks, all of which involve inherent uncertainty: precipitation forecasting from radar images, predicting cancer patient survival from histopathology images, and predicting car crashes from dashcam videos. We also give a theoretical analysis of a model for high-dimensional probability estimation which reproduces several of the phenomena evinced in our experiments. Finally, we propose a new method for probability estimation using neural networks, which modifies the training process to promote output probabilities that are consistent with empirical probabilities computed from the data. The method outperforms existing approaches on most metrics on the simulated as well as real-world data.

Calibrated uncertainty estimates in machine learning are crucial to many fields such as autonomous vehicles, medicine, and weather and climate forecasting. While there is extensive literature on uncertainty calibration for classification, the classification findings do not always translate to regression. As a result, modern models for predicting uncertainty in regression settings typically produce uncalibrated and overconfident estimates. To address these gaps, we present a calibration method for regression settings that does not assume a particular uncertainty distribution over the error: Calibrating Regression Uncertainty Distributions Empirically (CRUDE). CRUDE makes the weaker assumption that error distributions have a constant arbitrary shape across the output space, shifted by predicted mean and scaled by predicted standard deviation. We detail a theoretical connection between CRUDE and conformal inference. Across an extensive set of regression tasks, CRUDE demonstrates consistently sharper, better calibrated, and more accurate uncertainty estimates than state-of-the-art techniques.

LLM-as-a-Judge models generate chain-of-thought (CoT) sequences intended to capture the step-bystep reasoning process that underlies the final evaluation of a response. However, due to the lack of human annotated CoTs for evaluation, the required components and structure of effective reasoning traces remain understudied. Consequently, previous approaches often (1) constrain reasoning traces to hand-designed components, such as a list of criteria, reference answers, or verification questions and (2) structure them such that planning is intertwined with the reasoning for evaluation. In this work, we propose EvalPlanner, a preference optimization algorithm for Thinking-LLM-as-a-Judge that first generates an unconstrained evaluation plan, followed by its execution, and then the final judgment. In a self-training loop, EvalPlanner iteratively optimizes over synthetically constructed evaluation plans and executions, leading to better final verdicts. Our method achieves a new state-of-the-art performance for generative reward models on RewardBench (with a score of 93.9), despite being trained on fewer amount of, and synthetically generated, preference pairs. Additional experiments on other benchmarks like RM-Bench, JudgeBench, and FollowBenchEval further highlight the utility of both planning and reasoning for building robust LLM-as-a-Judge reasoning models.

Large Language Models (LLMs) are increasingly being used to simulate human-like decision making in agent-based financial market models (ABMs). As models become more powerful and accessible, researchers can now incorporate individual LLM decisions into ABM environments. However, integration may introduce inherent biases that need careful evaluation. In this paper we test three state-of-the-art GPT models for bias using two model sampling approaches: one-shot and few-shot API queries. We observe significant variations in distributions of outputs between specific models, and model sub versions, with GPT-4o-Mini-2024-07-18 showing notably better performance (32-43% yes responses) compared to GPT-4-0125-preview's extreme bias (98-99% yes responses). We show that sampling methods and model sub-versions significantly impact results: repeated independent API calls produce different distributions compared to batch sampling within a single call. While no current GPT model can simultaneously achieve a uniform distribution and Markovian properties in one-shot testing, few-shot sampling can approach uniform distributions under certain conditions. We explore the Temperature parameter, providing a definition and comparative results. We further compare our results to true random binary series and test specifically for the common human bias of Negative Recency - finding LLMs have a mixed ability to 'beat' humans in this one regard. These findings emphasise the critical importance of careful LLM integration into ABMs for financial markets and more broadly.

Automated LLM-as-a-Judge frameworks have become the de facto standard for scalable evaluation across natural language processing. For instance, in safety evaluation, these judges are relied upon to evaluate harmfulness in order to benchmark the robustness of safety against adversarial attacks. However, we show that existing validation protocols fail to account for substantial distribution shifts inherent to red-teaming: diverse victim models exhibit distinct generation styles, attacks distort output patterns, and semantic ambiguity varies significantly across jailbreak scenarios. Through a comprehensive audit using 6642 human-verified labels, we reveal that the unpredictable interaction of these shifts often causes judge performance to degrade to near random chance. This stands in stark contrast to the high human agreement reported in prior work. Crucially, we find that many attacks inflate their success rates by exploiting judge insufficiencies rather than eliciting genuinely harmful content. To enable more reliable evaluation, we propose ReliableBench, a benchmark of behaviors that remain more consistently judgeable, and JudgeStressTest, a dataset designed to expose judge failures. Data available at: https://github.com/SchwinnL/LLMJudgeReliability.

Recently, there has been a growing trend of utilizing Large Language Model (LLM) to evaluate the quality of other LLMs. Many studies have employed proprietary close-source models, especially GPT4, as the evaluator. Alternatively, other works have fine-tuned judge models based on open-source LLMs as the evaluator. In this study, we conduct an empirical study of different judge models on their evaluation capability. Our findings indicate that although the fine-tuned judge models achieve high accuracy on in-domain test sets, even surpassing GPT4, they are inherently task-specific classifiers, and their generalizability and fairness severely underperform GPT4.

Language models (LMs) are increasingly used as simulacra for people, yet their ability to match the distribution of views of a specific demographic group and be distributionally aligned remains uncertain. This notion of distributional alignment is complex, as there is significant variation in the types of attributes that are simulated. Prior works have underexplored the role of three critical variables -- the question domain, steering method, and distribution expression method -- which motivates our contribution of a benchmark explicitly addressing these dimensions. We construct a dataset expanding beyond political values, create human baselines for this task, and evaluate the extent to which an LM can align with a particular group's opinion distribution to inform design choices of such simulation systems. Our analysis reveals open problems regarding if, and how, LMs can be used to simulate humans, and that LLMs can more accurately describe the opinion distribution than simulate such distributions.

LLM-based judges have emerged as a scalable alternative to human evaluation and are increasingly used to assess, compare, and improve models. However, the reliability of LLM-based judges themselves is rarely scrutinized. As LLMs become more advanced, their responses grow more sophisticated, requiring stronger judges to evaluate them. Existing benchmarks primarily focus on a judge's alignment with human preferences, but often fail to account for more challenging tasks where crowdsourced human preference is a poor indicator of factual and logical correctness. To address this, we propose a novel evaluation framework to objectively evaluate LLM-based judges. Based on this framework, we propose JudgeBench, a benchmark for evaluating LLM-based judges on challenging response pairs spanning knowledge, reasoning, math, and coding. JudgeBench leverages a novel pipeline for converting existing difficult datasets into challenging response pairs with preference labels reflecting objective correctness. Our comprehensive evaluation on a collection of prompted judges, fine-tuned judges, multi-agent judges, and reward models shows that JudgeBench poses a significantly greater challenge than previous benchmarks, with many strong models (e.g., GPT-4o) performing just slightly better than random guessing. Overall, JudgeBench offers a reliable platform for assessing increasingly advanced LLM-based judges. Data and code are available at https://github.com/ScalerLab/JudgeBench .

Reinforcement learning from human feedback (RLHF) has become a key method for aligning large language models (LLMs) with human preferences through the use of reward models. However, traditional reward models typically generate point estimates, which oversimplify the diversity and complexity of human values and preferences. In this paper, we introduce Quantile Reward Models (QRMs), a novel approach to reward modeling that learns a distribution over rewards instead of a single scalar value. Our method uses quantile regression to estimate a full, potentially multimodal distribution over preferences, providing a more powerful and nuanced representation of preferences. This distributional approach can better capture the diversity of human values, addresses label noise, and accommodates conflicting preferences by modeling them as distinct modes in the distribution. Our experimental results show that QRM outperforms comparable traditional point-estimate models on RewardBench. Furthermore, we demonstrate that the additional information provided by the distributional estimates can be utilized in downstream applications, such as risk-aware reinforcement learning, resulting in LLM policies that generate fewer extremely negative responses. Our code and model are released at https://github.com/Nicolinho/QRM.

As LLM-based judges become integral to industry applications, obtaining well-calibrated uncertainty estimates efficiently has become critical for production deployment. However, existing techniques, such as verbalized confidence and multi-generation methods, are often either poorly calibrated or computationally expensive. We introduce linear probes trained with a Brier score-based loss to provide calibrated uncertainty estimates from reasoning judges' hidden states, requiring no additional model training. We evaluate our approach on both objective tasks (reasoning, mathematics, factuality, coding) and subjective human preference judgments. Our results demonstrate that probes achieve superior calibration compared to existing methods with approx10x computational savings, generalize robustly to unseen evaluation domains, and deliver higher accuracy on high-confidence predictions. However, probes produce conservative estimates that underperform on easier datasets but may benefit safety-critical deployments prioritizing low false-positive rates. Overall, our work demonstrates that interpretability-based uncertainty estimation provides a practical and scalable plug-and-play solution for LLM judges in production.

The current focus of AI research is shifting from emphasizing model training towards enhancing evaluation quality, a transition that is crucial for driving further advancements in AI systems. Traditional evaluation methods typically rely on reward models assigning scalar preference scores to outputs. Although effective, such approaches lack interpretability, leaving users often uncertain about why a reward model rates a particular response as high or low. The advent of LLM-as-a-Judge provides a more scalable and interpretable method of supervision, offering insights into the decision-making process. Moreover, with the emergence of large reasoning models, which consume more tokens for deeper thinking and answer refinement, scaling test-time computation in the LLM-as-a-Judge paradigm presents an avenue for further boosting performance and providing more interpretability through reasoning traces. In this paper, we introduce J1-7B, which is first supervised fine-tuned on reflection-enhanced datasets collected via rejection-sampling and subsequently trained using Reinforcement Learning (RL) with verifiable rewards. At inference time, we apply Simple Test-Time Scaling (STTS) strategies for additional performance improvement. Experimental results demonstrate that J1-7B surpasses the previous state-of-the-art LLM-as-a-Judge by 4.8\% and exhibits a 5.1\% stronger scaling trend under STTS. Additionally, we present three key findings: (1) Existing LLM-as-a-Judge does not inherently exhibit such scaling trend. (2) Model simply fine-tuned on reflection-enhanced datasets continues to demonstrate similarly weak scaling behavior. (3) Significant scaling trend emerges primarily during the RL phase, suggesting that effective STTS capability is acquired predominantly through RL training.

Large language models (LLMs) are now widely used to evaluate the quality of text, a field commonly referred to as LLM-as-a-judge. While prior works mainly focus on point-wise and pair-wise evaluation paradigms. Rubric-based evaluation, where LLMs select a score from multiple rubrics, has received less analysis. In this work, we show that rubric-based evaluation implicitly resembles a multi-choice setting and therefore has position bias: LLMs prefer score options appearing at specific positions in the rubric list. Through controlled experiments across multiple models and datasets, we demonstrate consistent position bias. To mitigate this bias, we propose a balanced permutation strategy that evenly distributes each score option across positions. We show that aggregating scores across balanced permutations not only reveals latent position bias, but also improves correlation between the LLM-as-a-Judge and human. Our results suggest that rubric-based LLM-as-a-Judge is not inherently point-wise and that simple permutation-based calibration can substantially improve its reliability.

An automated system that could assist a judge in predicting the outcome of a case would help expedite the judicial process. For such a system to be practically useful, predictions by the system should be explainable. To promote research in developing such a system, we introduce ILDC (Indian Legal Documents Corpus). ILDC is a large corpus of 35k Indian Supreme Court cases annotated with original court decisions. A portion of the corpus (a separate test set) is annotated with gold standard explanations by legal experts. Based on ILDC, we propose the task of Court Judgment Prediction and Explanation (CJPE). The task requires an automated system to predict an explainable outcome of a case. We experiment with a battery of baseline models for case predictions and propose a hierarchical occlusion based model for explainability. Our best prediction model has an accuracy of 78% versus 94% for human legal experts, pointing towards the complexity of the prediction task. The analysis of explanations by the proposed algorithm reveals a significant difference in the point of view of the algorithm and legal experts for explaining the judgments, pointing towards scope for future research.

Reliable estimation of predictive uncertainty is crucial for machine learning applications, particularly in high-stakes scenarios where hedging against risks is essential. Despite its significance, there is no universal agreement on how to best quantify predictive uncertainty. In this work, we revisit core concepts to propose a framework for information-theoretic measures of predictive uncertainty. Our proposed framework categorizes predictive uncertainty measures according to two factors: (I) The predicting model (II) The approximation of the true predictive distribution. Examining all possible combinations of these two factors, we derive a set of predictive uncertainty measures that includes both known and newly introduced ones. We extensively evaluate these measures across a broad set of tasks, identifying conditions under which certain measures excel. Our findings show the importance of aligning the choice of uncertainty measure with the predicting model on in-distribution (ID) data, the limitations of epistemic uncertainty measures for out-of-distribution (OOD) data, and that the disentanglement between measures varies substantially between ID and OOD data. Together, these insights provide a more comprehensive understanding of predictive uncertainty measures, revealing their implicit assumptions and relationships.

Large language models (LLMs) are widely used to evaluate the quality of LLM generations and responses, but this leads to significant challenges: high API costs, uncertain reliability, inflexible pipelines, and inherent biases. To address these, we introduce PAJAMA (Program-As-a-Judge for Automated Model Assessment), a new alternative that uses LLMs to synthesize executable judging programs instead of directly scoring responses. These synthesized programs can be stored and run locally, costing orders of magnitude less while providing interpretable, and auditable judging logic that can be easily adapted. Program-based judges mitigate biases, improving judgment consistency by 15.83% and reducing biased responses by 23.7% on average compared to a Qwen2.5-14B-based LLM-as-a-judge. When program judgments are distilled into a model, PAJAMA outperforms LLM-as-a-judge on the challenging CHAT-HARD subset of RewardBench, outperforming metrics by 2.19% on Prometheus and 8.67% on the JudgeLM dataset, all at three orders of magnitude lower cost.

We present a novel Transformer-based multi-agent system for simulating the judicial rulings of the 2010-2016 Supreme Court of the United States. We train nine separate models with the respective authored opinions of each supreme justice active ca. 2015 and test the resulting system on 96 real-world cases. We find our system predicts the decisions of the real-world Supreme Court with better-than-random accuracy. We further find a correlation between model accuracy with respect to individual justices and their alignment between legal conservatism & liberalism. Our methods and results hold significance for researchers interested in using language models to simulate politically-charged discourse between multiple agents.

To keep pace with the increasing pace of large language models (LLM) development, model output evaluation has transitioned away from time-consuming human evaluation to automatic evaluation, where LLMs themselves are tasked with assessing and critiquing other model outputs. LLM-as-judge models are a class of generative evaluators that excel in evaluating relatively simple domains, like chat quality, but struggle in reasoning intensive domains where model responses contain more substantive and challenging content. To remedy existing judge shortcomings, we explore training judges with reinforcement learning (RL). We make three key contributions: (1) We propose the Equivalent Initial State Group Relative Policy Optimization (EIS-GRPO) algorithm, which allows us to train our judge to be robust to positional biases that arise in more complex evaluation settings. (2) We introduce ReasoningJudgeBench, a benchmark that evaluates judges in diverse reasoning settings not covered by prior work. (3) We train Judge for Reasoning (J4R), a 7B judge trained with EIS-GRPO that outperforms GPT-4o and the next best small judge by 6.7% and 9%, matching or exceeding the performance of larger GRPO-trained judges on both JudgeBench and ReasoningJudgeBench.

Machine learning models are often used to inform real world risk assessment tasks: predicting consumer default risk, predicting whether a person suffers from a serious illness, or predicting a person's risk to appear in court. Given multiple models that perform almost equally well for a prediction task, to what extent do predictions vary across these models? If predictions are relatively consistent for similar models, then the standard approach of choosing the model that optimizes a penalized loss suffices. But what if predictions vary significantly for similar models? In machine learning, this is referred to as predictive multiplicity i.e. the prevalence of conflicting predictions assigned by near-optimal competing models. In this paper, we present a framework for measuring predictive multiplicity in probabilistic classification (predicting the probability of a positive outcome). We introduce measures that capture the variation in risk estimates over the set of competing models, and develop optimization-based methods to compute these measures efficiently and reliably for convex empirical risk minimization problems. We demonstrate the incidence and prevalence of predictive multiplicity in real-world tasks. Further, we provide insight into how predictive multiplicity arises by analyzing the relationship between predictive multiplicity and data set characteristics (outliers, separability, and majority-minority structure). Our results emphasize the need to report predictive multiplicity more widely.