Always Tell Me The Odds: Fine-grained Conditional Probability Estimation
Authors: Liaoyaqi Wang
Zhengping Jiang
Anqi Liu
Benjamin Van Durme
Venue: COLM 2025 (2025)
Keywords: Large Language Model
Probabilistic Reasoning
Semantics
Calibration
TL;DR: We present a state-of-the-art model for fine-grained probability estimation of textual outcomes conditioned on context.

Abstract: We present a state-of-the-art model for fine-grained probability estimation of propositions conditioned on context. Recent advances in large language models (LLMs) have significantly enhanced their reasoning capabilities, particularly on well-defined tasks with complete information. However, LLMs continue to struggle with making accurate and well-calibrated \emph{probabilistic} predictions under uncertainty or partial information. While incorporating uncertainty into model predictions often boosts performance, obtaining reliable estimates of that uncertainty remains understudied. In particular, LLM probability estimates tend to be coarse and biased towards more frequent numbers. Through a combination of human and synthetic data creation and assessment, scaling to larger models, and better supervision, we propose a set of strong and precise probability estimation models. We conduct systematic evaluations across tasks that rely on conditional probability estimation and show that our approach consistently outperforms existing fine-tuned and prompting-based methods by a large margin.

Published as a conference paper at COLM 2025
Always Tell Me The Odds:
Fine-grained Conditional Probability Estimation
Liaoyaqi Wang"
Zhengping Jiang"
Anqi Liu Benjamin Van Durme
Johns Hopkins University
[_wang240,zjiang31,, aliu74, bvandur /@jh.edu
Abstract
We present state-of-the-art model for fine-grained probability estimation of propositions conditioned on context. Recent advances in large language models (LLMs) have significantly enhanced their reasoning capabilities, par- ticularly well-defined tasks with complete information. However, LLMs continue to struggle with making accurate and well-calibrated probabilistic predictions under uncertainty partial information. While incorporating uncertainty into model predictions often boosts performance, obtaining reliable estimates of that uncertainty remains understudied. In particular, LLM probability estimates tend to be coarse and biased towards more frequent numbers. Through combination of human and synthetic data creation and assessment, scaling to larger models, and better supervision, we propose a set of strong and precise probability estimation models. We conduct systematic evaluations across tasks that rely on conditional prob- ability estimation and show that our approach consistently outperforms existing fine-tuned and prompting-based methods by a large margin 12,
1 Introduction
While large language models (LLMs) have shown remarkable performance in wide range of well-formulated, clearly solvable reasoning tasks, real-world applications often require them to operate under uncertainty and ambiguity, where purely deductive or deterministic reasoning may fail (McCarthy & Hayes, 1981). Much of real-world and commonsense knowledge is inherently probabilistic (Li al., 2021; Moss, 2018; Glickman et al., 2005). Thus, it is crucial for LLMs to integrate evidence with prior knowledge and to evaluate the plausibility of new information (Jaynes, 2003). We want LLMs that always (and accurately) tell us the odds.3
Various approaches incorporate uncertainty into the LLM reasoning process. For example, recent line of work proposes to create structured reasoning traces that allow uncertainty to propagate (Jung et al., 2022; Feng et al., 2024; Xia et al., 2024; Hou et al., 2024b; Akyürek et al., 2024; Sanders & Durme, 2025). Further, has been argued that uncertainty-aware evaluation provides better insights into the model capabilities (Cheng et al., 2024; Jiang et al., 2024b; Yuan et al., 2024). However, these methods typically rely on estimating conditional probabilities over local structures using prompting stra ttrategies that are often underexplored and poorly optimized, effectively delegating this task to the LLM without sufficient scrutiny. Approaches typically involve examining the logits of the model (Zhao et al., 2021b), which are known to be miscalibrated (Jiang et al., 2021; Xiong et al., 2024; Jiang et al., 2022), or using verbalized confidence (Mielke al., 2022; Tian et al., 2023) which tends stick few common discrete values (Razeghi et al., 2022; Cruz et al., 2024; Feng et al., 2024).
'Equal contribution.
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Reasoning- Ghasoniin-AAugmente Augmented
Chat Template
Prompting
Given the proposition and context, estimate the P(proposition context) with world knowledge and contextual factors. Context: A man is sipping a cup of coffee while reading the news. Proposition:
Given (context1), how likely is it that [propositionl) is true?
Given (context2). how likely it that (proposition2) is true?
The
drinks the morning.
LLM Response
To evaluate the probability of the hypothesis... Final probability: 0.7
Many people drink coffes the morning, but Finol probability. 0.23
Okay, so I need to figure out probobility that Final probability. 0.15
The answer is:
The answer isi 1
Rank Consistency
Agreement- Based Filtering
Discrepancy=
Distribution Match
-a-------
Ak-2
ak-1
4K
Based the probabilistic reasoning and other agents' answer..
My confidence to each onswer: [0.15, 0.95, 0.8]
Expected Label Scoring:
Figure 1: We train decoder-based models for fine-grained probability estimation, going beyond human annotations. We rely on two sources of supervision: synthetic (left), and pairwise ranking (right). For synthetic data, we first collect multiple LLMs' probability estimates with reasoning, then group instances by LLM agreement. For those with signifi- cant discrepancies, we submit them to another LLM judge, which rate the quality of each reasoning process. These ratings are then used to aggregate the probability estimates into distribution over possible bins. For pairwise ranking, we use a margin loss to ensure consistency between the pairwise labels and the fine-grained probability estimates - which are computed via the expected label scoring rule.
Driven by the necessity of accurate, fine-grained probabilistic reasoning, in this work we focus on building strong, LLM-based models for estimating the probability of textual proposition given a context. Direct evaluation against human annotations can be subjective and noisy; therefore, we first construct an objective and comprehensive evaluation suite by designing subproblems from existing datasets and frameworks that admit an intuitive probabilistic interpretation.
We therefore propose to train such conditional probability estimation models with (1) modern LLM backend, (2) much more synthetic data, and (3) diverse training objectives. While most existing approaches rely on regression with smaller encoder-based models to fit human annotation (Chen et al., 2020; Nie et al., 2020b), by leveraging an expected label scoring rule that bridges regression / ranking and calibrated classification (Jiang et al., 2024a), we propose simple techniques that can provide fine-grained probability estimates with modern, large decoder-basee model. This allows us to better utilize the exceptional language understanding capabilities of pretrained LLMs. Also, our formulation allows natural integration with ensemble distillation using synthetic data well training with pairwise likelihood comparison. Due to the challenges in getting high quality data described above, these techniques are crucial to further improve performance.
We also propose suite of tasks to comprehensively evaluate the performance of such models, ranging from their alignment with human and synthetic annotations, to the con- sistency of their plausibility rankings with human perception, and their ability to support Bayesian inference and decision-making, Our best model surpasses existing fine-tuned and prompting-based methods across multiple tasks, revealing untapped potential in improving probabilistic reasoning in LLMs through better probability estimation.
2 Related Work
Uncertainty in LLMs Like other modern neural networks (Guo et al., 2017), transformer- based language models are often overconfident in their predictions (Desai & Durrett, 2020;
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Jiang et al., 2022). This makes LLM bad confidence estimator, while instruction tuning has been reported to further concentrating probability mass (Achiam et al., 2023; Hendrycks et al.,, 2020; Padmakumar& & He, 2023). To address this, various methods have been proposed to improve uncertainty estimation, such as analyzing logits (Zhao et al., 2021b), prompting LLMs to verbalize confidence (Mielke et al., 2022; Tian et al., 2023), or probing internal representations (CH-Wang et al., 2024). Of these, verbalized uncertainty has generally been shown to produce the most accurate probability estimation (Tian et al., 2023), though it can still be overconfident (Tanneru et al., 2023; Xiong et al., 2024; Chen et al., 2023) and often relies on a small set of common discrete outputs (e.g., 10,, 75%) (Cruz et al., 2024).
Part of the problem is that existing resources for training are predominantly hard labeled, making model calibration and its evaluation particularly challenging (Kumar et al., 2019; Nixon et al., 2019). Recently, the field has seen growing interest in modeling the complete label distribution (Meissner et al., 2021; Liu et al., 2023; Zhou al., 2021; Cheng et al., 2024), and different protocols have been proposed for curating datasets to enable such training and evaluation (Nie et al., 2020b; Chen et al., 2020). However, these datasets tend to be limited in size and are subject to ongoing debate over whether the distributions reflect true aleatoric uncertainty or merely label noise (Jiang & Marneffe, 2022; Baan et al., 2022; Wang et al., 2022; Weber-Genzel et al., 2024; Baan et al., 2024).
Thus, to mitigate the limitations of existing resources, we propose training our model with synthetic data and further supervising it with pairwise ranking supervision.
LLM-based Probabilistic Reasoning Chain-of-thought prompts effectively elicit reason- ing in LLMs (Wei et al., 2022), but do not ensure accurate or calibrated probability estimates in real-world settings (Paruchuri et al., 2024). To address this, various methods integrate probability into reasoning. Ozturkler et al. (2023) proposed a two-stage framework where LLMs are queried in parallel and aggregated via likelihoods. Feng et al. (2024) and Nafar et al. (2024) align Bayesian Networks with LLM abductions for decision-making and QA tasks. Others decompose statements into sub-claims and aggregate them using confidence scores (Xia et al., 2024; Hou et al., 2024b; Cao et al., 2023; Jung et al., 2022; Akyürek et al., 2024). While LLMs can be tuned to mimic Bayesian models (Qiu et al., 2025), such supervi- sion is rarely feasible in practice, limiting prior work to toy examples (Wong et al., 2023). Most existing methods solely rely on aggregating local probability estimates via prompting or probing (Jung et al., 2022), which can be suboptimal as discussed above. We show that LLMs can be directly tuned to produce fine-grained, accurate conditional probabilities for structured reasoning.
Decoder-based Regression Although primarily trained to predict the next token in textual sequences, large language models (LLMs) have been shown to perform effectively as regression models. Vacareanu et al. (2024) demonstrate that LLMs can conduct both linear and non-linear regression in context. In a series of papers, Song et al. (2024) and Song & Bahri (2025) develop techniques for training LLMs to perform numerical regression using only textual representations, achieving arbitrary precision. In this work, to balance efficiency and usability, we propose a hybrid approach that combines coarse-grained probability estimates via textual representations with fine-grained probability refinement through expected label scoring rule aggregation (Jiang et al., 2024a).
3 Methodology
We aim to develop accurate, fine-grained, and broadly applicable models for estimating the conditional probability P(proposition | context), representing the likelihood that candidate proposition is true or occurs given a textual context. Subsection 3.1 outlines our evaluation suite, which covers intrinsic alignment with labels, consistency of probability rankings, and support for structured reasoning, beyond straightforwardly comparing pre- dictions to noisy human annotations. Subsection 3.2 presents our strategy for scaling up training data using LLM-based pseudo-labeling, despite known inaccuracies and biases in LLM probability estimates. Finally, subsection 3.3 describes our training procedure for fine-
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tuning LLM-based models using combination of human annotations, synthetic probability estimates, and pairwise ranking consistency signals.
3.1 Objective and Evaluation
What makes good model for conditional probability estimation? When accurate probabilis- tic labels are available - e-g., from census data or controlled Bayesian setups evaluation is straightforward. However, such labels are rare in real-world settings, where commonsense reasoning and complex patterns are involved. Human-provided estimates can help (Chen et al., 2020; Nie et al., 2020b) but are often subjective and noisy (Meissner et al., 2021).
Instead of relying solely on ground-truit alignment, we advocate evaluating models based on their effectiveness in real-world decision-making, where probabilistic reasoning aids belief modeling and evidence integration (Feng et al., 2024; Xia et al., 2024; Qiu et al., 2024). Inspired by multi-class calibration (Zhao al., 2021a), we hypothesize that better probability estimates should support more human-aligned decisions.
To this end, we propose three task categories: (1) Intrinsic, comparing model estimates with human or LLM labels; (2) Comparison, ranking plausibility among probability estimates; and (3) Structural, assessing uncertainty propagation and decision-making in structured reasoning (subsection 4.1).
3.2 Synthetic Data Creation
This section introduces our method for generating pseudo-probabilittic labels for propo- sitions given textual context, aiming to expand domain and distribution coverage for conditional probability estimation.
Inspired by the fact that increased inference-time compute improves both performance and confidence (Kojima et al., 2022; Muennighoff et al., 2025; Jurayj et al., 2025), we enhance an- notation quality by generating multiple LLM roll-outs for each probability estimation (Wang et al., 2023). Príor work highlights a discrepancy between LLMs' generative and evaluative capabilities (Gu et al., 2024; Zelikman et al., 2022; Kumar et al., 2024), with models excelling at tasks like judging or verifying outputs. We leverage this by proposing multi-step annotation aggregation process, where probability estimates are judged based on reasoning quality by another LLM-yiedding consistent performance gains. We explored various ways of improving the pseudo-label quality, including one similar the confidence ranking ap- proach proposed by Shrivastava et al. (2025) with LLM-based pairwise comparison, which we detailed in Appendix A. Overall, we find the approach adopted here scales the most efficiently with compute.
quality
wwwtooninnnAAggentee Prompting The first step in our annotation process involves eliciting direct probability estimates from LLMs alongside their corresponding reasoning chains. Our prompting strategy instructs LLMs to leverage world knowledge to assess contextual factors before arriving at final estimation, as incorporating reasoning helps models better approximate human decision-making (Chen et al., 2024). This approach encourages models to decompose ambiguous premises into plausible real-world scenarios and estimate the likelihood of each scenario's 's occurrence (Hou et al., 2024a). Intuitively, multiple rounds of annotation using different models and configurations can provide richer understanding of each data point. We enforce reasoning chains for two main reasons: (1) they tend to enhance performance, and (2) they improve interpretability, enabling more rigorous evaluation and verification (Lightman et al., 2023), while also serving as useful input for subsequent steps.*
Agreement-Basee Filtering After obtaining raw probability scores by directly prompting LLMs, we seek identify low-quality estimates for further refinement. We observe that estimates with high agreement are more aligned with human annotations in the UNLI
4llease refer subsection D.1 for the exact prompts we use.
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validation set (Chen et al., 2020). This is supported by prior findings that ensemble agree- ment can be a reliable proxy for label quality (Deng et al., 2023; Baek et al., 2022), and that targeting uncertain labels can improve annotation quality (Gligorié et al., 2024). To quantify confidence, we define discrepancy as the difference between the maximum and minimum probability estimates across models. For low-discrepancy samples, we retain the original scores; high-discrepancy samples are flagged for further review.
LLM-as-a-Judge In the final step, we employ a large language model (LLM) as judge to adjudicate among candidate reasoning chains (Zheng et al., 2023). Specifically, the LLM evaluates the quality of each reasoning process that leads to conditional probability esti- mate (Du et al., 2024; Chiang & yi Lee, 2023). For reasoning traces produced by reinforcement learning-enhance models, we first summarize the rationale to extract key steps influencing the probability assessment before presenting it the judge. This preprocessing step is necessary because such reasoning traces are often lengthy and exhibit complex reasoning patterns (DeepSeek-A) et al., 2025), which may confuse the LLM judge. After analyzing the context, outcome, and candidate reasoning chains, the judge assigns confidence score between and to reflect the reliability of each chain (Xu et al., 2023). These confidence scores are then used as supervision signals during model fine-tuning, as discussed below.
3.3 Model Fine-tuning
While there already exists pretraining- model for subjective probabilities (Chen et al., 2020), they are typically small-encoder-bassee and tuned naively to match human scalar annotation. In this section, we describe training recipe for leveraging modern large-scale decoder-based language models, incorporating calibrated human annotations, utilizing synthetic labels for domain generalization, and integrating pairwise ranking consistency signals to enhance model performance.
Decoder-based Regression refers to the practice of training decoder-only language models to perform regression by outputting textual representations of numeric values (Song & Bahri, 2025). While LLM embeddings have been used as features for regression tasks, prior studies report unclear scaling trends (Tang et al., 2024), as these embeddings mostly capture semantic similarity rather than supporting precise numerical prediction (Devlin et al., 2019; Li al., 2020). Besides, we prefer textual-based outputs as users will be able to directly get the outputs from common LLM services without additional processing (Kwon et al., 2023).
we split the interval [0, 1] into N bins of equal width {by, by, - 1h, and assign each bin bj unique special token tj. For an instance (x; y/i E D (where D denotes the dataset), we tune the decoder-only model to predict the token 1 corresponding to the bin bj that contains the target probability y;- However, this conversion y + t is lossy, resulting in a coarse prediction. To recover fine-grained scalar prediction, we consider taking the expectation over all possible token predictions. Suppose we have scoring function ff that converts each bin into scalar f == B IR+
Specifically,
we construct our fine prediction as
N-1 g Σ
f(bp) p(t, x;).
This is called the expected label scoring rule, introduced by Jiang et al. (2024a), shows that both the mean absolute error and the ranking risk regret lower-bound the calibration error of the underlying classifier. In our case, this implies that the better the model approximates the true token distribution, the more accurately it captures the underlying conditional probability. Inspired by their work, we convert the ground truth y/ to Gaussian distribution centered at y; with a small fixed variance o² We then quantize this distribution into discrete distribution with N fixed supports at (f. ((N--)),, as Jiang et al. (2024a) has shown that the Wasserstein-2 distance minimizing quantization is given by
9(1)1x) F( ((b)111) + f(bb)) F
((bb)) + ff(b 1
2
2
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where F(.) is the CDF function of the Gaussian distribution (y), Figure 2 shows some example quantization. We then use the quantized distribution as the target for our model training. This allows us to further discriminate targets that fall in the same bin. While prior work has been shown that enforcing distribution over a set of sequences can be achieved through sampling and reweighting (Zhang et al., 2024), our regression setting only requires a single decoding step. This enables us to directly optimize the model using a forward KL-divergence loss:
Direct (D) E (xy))-DDL (Q(H))))IP((()))
We use forward KL-divergence as we observe more stable training, and it seems that the theoretical mode-seeking behavior of reverse-KL does not always hold for LLMs (Wu et al., 2025).
Gxcetin
Gasetin
Gscettnn
Saucda
D
Pounticol
044 24
0.5
05
0.
Figure 2: Illustration of our distribution quantization process. Notice that how the quantiza- tion preserves fine-grained label ordering and allows for better discrimination of targets that fall in the same bin.
Utilizing Synthetic Data While our approach regression allows to convert any target scalar label y into distribution over bin-associated tokens, our synthetic data is annotated with multiple LLMs to provide more reliable estimate of the true conditional probability. On each synthetic data point x E X, our synthetic data creation process generates paired with K confidence judgments {co c As through the previous discussion, each of the score can be mapped to a distribution over tokens. To increase sharpness of the overall prediction, we construct a mixture distribution over (t)(t) (f/x)), where each component is weighted by power-normalized
probability estimates ty.,...,
confidence score:
(ct)aa
Q(1xx) = Σ オे. (t(x),,, where n² K=
Rank Consistency Training As ground truth probability estimates are only available on very limited data points, we can still leverage other forms of supervision to improve model performance. Similar to how we construct the evaluation suite from defeasible NLI (Rudinger et al., 2020) and choice of plausible alternatives (Dagan et al., 2005; Zellers et al., 2019), supervise rank-consistency using pairwise ranking labels. Given two instance v1), (2), we use the pairwise margin-loss (Weston et al., 1999) to enforce the order consistency:
LRank (DPairwise)
E EE Pairwise max 10, 6 - sgn(y) - y2) (f1 }
as has been shown lead to less over-confident probability estimates (Li et al., 2019). Thus the final loss becomes
L(D) = DDirect (DDreet((Huaaa (DHuman)
+ BB (. + BB2CRank (Prairwieee.
Notice that while Jiang et al. (2024a) empirically show that optimizing for rank-consistency improves the calibration of the underlying classifier, their theoretical results depend the assumption that the predictive distribution is single-peaked. For our case this does not always hold, and in fact our gradient analysis in Appendix B shows that the margin-loss

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as well as sppeectd-scorinn--rll-bbaaas regression loss alone can be under-specified, and does not guarantee that greedy decoding will yield reasonable approximation of the target conditional probability. Therefore, we apply rank-consistency training only when the pairwise labels are available, to improve correspondence of the probability estimates.
4 Experiment Setup
4.1 Dataset
Training We construct our training dataset from mixture of human-annotated, synthetic, and pairwise ranking data. As our tasks are closely related to Natural Language Inference (NL.T), we heavily rely on NLI datasets for training. We use the following datasets: UNLI (Chen et al., 2020), which contains high-quality subjective probabilistic relabeling of subset of SNLI (Bowman al., 2015); ANLI (Nie et al., 2020a), which contains adversarially generated NLI examples that are often longer and involve more challenging reasoning patterns (Williams et al., 2020); and WANLI (Liu al., 2022), which consists of automatically generated NLI pairs seeded from challenging instances in MultiNLI (Williams et al., 2018), identified through data maps (Swayamdipta et al., 2020).
Since UNLI applies logistic transformation to its annotation scale, we invert that trans- formation to make our model more sensitive to probability differences near and 1. As ANLI and WANLI do not provide probabilistic labels, we use our proposed synthetic data annotation pipeline, detailed in subsection 3.2, to generate pseudo-labels for these datasets.
Additionally, we leverage 8-NLI (Rudinger et al., 2020) and HellaSwag (Zellers et al., 2019) for -on--sonssttecy training. For 8-NLI, we train our models' probability estimates to train the
be consistent with the direction of defeasible updates, while for HellaSwag, models to assign the highest probability to the most plausible completion. During training, upsample UNLI and 8-NLI by up to 4 times (Lee al., 2022; Li et al., 2025).
Intrinsic Evaluation Besides evaluating our models the UNLI dataset, we also sam- ple from EntailmentBank (Dalvi et al., 2021) and e-CARE (Du et al., 2022) to directly evaluate the quality of our probability estimates. While EntailmentBank trees are predom- inantly entailment-foocseed we rewrite hypotheses to introduce probabilistic uncertainty by removing modality, increasing vagueness, generating alternatives, perform existential instantiation, and generate abductions to convert the task into probabilistic reasoning challenge. To evaluate the quality of probability estimations on other domains, we follow GNLI (Hosseini et al., 2024) to generate synthetic NLI pairs in more distant domains such as fans forum, blog post, medical texts, and others. We also cast the scalar annotation (Jiang et al., 2024a) Circa (Louis et al., 2020) to the probability of an ambiguous response being affirmative. For all intrinsic evaluation, we report Spearman correlation.
Comparison Evaluation We evaluate accuracy of plausibility ranking on 6-NLI, Hel- laSwag and COPA (Roemmele et al., 2011). Specifically, for COPA, rank the two choices based on the conditional probability of the effect given the cause, regardless of the question type. For each instance, our model is required to predict conditional probability for each of the alternatives independently, which makes the task more challenging than the original multiple-choice setting. We report the accuracy of the model's top-1 ranking.
Structural Evaluation We evaluate our model on reasoning traces from two reasoning frameworks. The first is Maieutic Prompting (Jung et al., 2022), which iteratively expands maieutic tree using abductive and recursive explanations, solving for the most consistent truth value assignment. While the original formulation incorporates multiple methods to estimate uncertainty and applies post-hoc filtering, we make the task more challenging by relying solely our model to estimate all required conditional probabilities. Specifically, we estimate both belief and consistency, and we remove the NLI filtering constraint to further increase difficulty.
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The second framework is BIRD (Feng et al., 2024), which performs Bayesian inference through -LM-generated abductions. Given a BIRD trace, we use our model to score both the conditional probabilities of each factor condition given the context, and the probabil- ities of each final outcome given those conditions. Unlike the original BIRD method, we do not apply any optimization for consistency, aside from renormalizing the aggregated probabilities over final outcomes.
For Maieutic Prompting, we evaluate on publicly available traces from Com2Sense (Singh et al., 2021), CREAK (Onoe et al., 2021), and CSQA2 (Talmor et al., 2021)all binary QA benchmarks focused on commonsense reasoning and fact verification. For BIRD, we use their released dataset, including additional sentence generation for the comparison subset of Com2Sense, and evaluate on Today (Feng et al., 2023) for temporal reasoning. We report overall accuracy across all datasets. Example traces and scoring details can be found in Appendix E.
4.2 Model and Tuning Details
We use four open-source models to generate synthetic datasets: Qwen2.5-32B-Intteen and QwQ-32B (Qwen et al., 2025), -irpseeek-R---iisii----e-e----e
(DeepSeek-Al et al., 2025), (Grattafiori et al., 2024). For all models except QwQ, we apply greedy decoding; for QwQ, we follow the official recommendation of using temperature = 0.6 and MinP = 0 to mitigate repetition.
and 1lama-3.3-700-Inae
For fine-tuning, we consider three base models: Llama-3-8B-Instrun -wen2.55-BBInnttruee and Swee2.5-11B-Instrruce We apply LoRA (Hu et al., 2022) with r 16 and 32, using learning rate of 2 x 10-5 and linear learning rate scheduler. For each configuration, perform a hyperparameter sweep over the number levels E {10,20, 100), B2 {1,10, 100), and (0.01, 0.05,,055000 0.1).
We evaluate our proposed models against a diverse set of baselines to ensure a compre- hensive comparison. These baselines are: 1) Encoder Model: The RoBERTa-L model (Chen et al., 2020), a fine-tuned encoder using UNLI dataset. 2) Zero-Shot LLMs: We prompt both proprietary model (GPT4o (Achiam et al., 2023)) and open-source model DeeesSeekRDistilQQQeee
(DeepSeek-Al et al., 2025)) in a zero-shot inference setting, Due the slow inference speed of GPT-4o on structural reasoning tasks with the Maieutic Prompting method, we employ stratified sampling to create a representative mini-batch for its evaluation. 3) Probe Method: We adopt the true/fales probing technique from Tian al. (2023). This involves extracting probabilities from the "true" and "false" token logits of the Bwen2.5-14B-Instruue model, which are then calibrated with a general temperature scaling method (Shen et al., 2024).
Results and Discussion
Our comprehensive evaluation result is shown in Table 1. Overall, we observe noticeable improvements over previous generation models for subjective probability estimation (Chen et al., 2020), as well as over 0-shot prompting approaches that are widely considered. We discuss particular observations below.
Model performance improves with scale. Even when trained the same dataset as prior work (Chen et al., 2020), our models built on top of modern LLM backend consistently outperform both smaller encoder-based approaches and zero-shot prompting baselines. Notably, Chen et al. (2020) report an aggregated human performance of Spearman's p 0.727, whereas all of our models exceed this benchmark by substantial margin. These results suggest that large language models, with their extensive world knowledge, can be effectively leveraged for probabilistic inference.
$nttps://huggingfacee wttpaee////e//////////////0
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Encoder RoBERTa-L
0-Shot
Probe
Ours
Type
Tasks
6-148
P-8B
6-7B
87-141
+Syn
+Syn+R
UNLI
.707
.629
.699
.681
.804
.802
.813
.812
.812
Intrinsic
circa
.300
.633
.734
.553
.74
.544
.536
.564
.747
GNLI
.866
.755
.7%6
.833
.789
.811
.814
.838
820
EntailmentBank
.503
.783
.760
.558
.639
.687
.735
.789
.787
e-CARE
.871
.888
.856
.855
.870
.888
.884
.905
6-SNLI
77.9
77.2
75.3
81.3
83.3
84.4
85.1
86.0
88.9
Comp.
6-ATOMIC
75.0
69.1
70.7
74.7
78.9
78.6
80.1
81.2
87.6
COPA
83.0
87.7
81.0
86.8
85.1
87.2
86.5
87.9
89.3
HellaSwag
57.8
75.4
74.9
67.2
70.2
75.3
75.5
95.7
C2S-M
50.6
68.5
77.9
49.4
65.5
68.4
73.5
75.2
75.6
Structural
CREAK-M
62.5
68.0
81.1
42.1
83.2
82.8
84.8
85.6
86.5
CSQA2-M
49.5
55.6
71.4
48.5
60.4
63.0
67.9
70.4
72.0
C2S-Sent-B
65.0
73.0
76.8
51.3
72.7
76.3
71.3
73.3
89.8
TODAY-B 56.0
64.0
63.0
65.0
65.0
64.0
68.0
67.0
66.3
Table 1: Evaluation of our models the tasks listed out in subsection 4.1. denotes the -rrpSeek-K--Disstll---ee------
(DeepSeek-Al et al., 2025) and @ GPT-4o (Achiam et al., 2023). A corresponds to I.lama-3-Inste Instructt series (Grattafiori et al., 2024), & the Qwen2.5- Instruct series (Qwen al., 2025). +Syn indicates @wwen255-14B-Instie ansttructuugng augmented with synthetic data, +R indicates the same model with rank consistency training. -M and -B suffixes denote the Maieutic Prompting and Bird frameworks respectively. Best results are in Bold, second-best results are underlined.
Synthetic data enhances domain generalization. Comparing the results in column +Syn with those column 1711 we find that incorporating synthetic data generally improves performance. These improvements are especially notable in probability estimation for more challenging reasoning tasks (e.g,, EntailmentBank) and in generalization (e.g., GNLI, Circa). We hypothesize that this is due to the broader range of inference types covered by ANLI and WANLI, as well as their longer contexts, which may help the model better understand complex scenarios and integrate multiple pieces of evidence.
distant domains
Rank-consistency training facilitates better decision making, The results in column +Syn+R clearly demonstrate that rank-consistency training further improves performance across multiple tasks, particularly in plausibility ranking and structured reasoning. While this outcome intuitive, it is noteworthy that on some taskssuch 8-SNLI and C25-Sent-B our models perform competitively or even outperform previous results, of larger models or systems employing more complex processing pipelines (Srikanth & Rudinger, 2025; Feng et al., 2024). Although the improvements with Maieutic Prompting are less pronounced, this likely due to its lack of fully probabilistic interpretation (Jung et al., 2022).
Our model captures some level of human uncertainty. An additional benefit of our model that, even without explicit training on human label distributions, it still exhibits human-like uncertainty on many data points, as shown in Figure 3. To show this, we compare the token-level distribution of our model with the distributed scalar judgments collected by Pavlick & Kwiatkowski (2019). We observe that our model produces multi- modal distributions when human opinion divides, and agree with human judgment when the opinion is more consistent. A more detailed analysis of model's performance the ChaosNLI (Nie al., 2020b) and ProtoQA (Boratko et al., 2020) datasets is provided in Appendix C.

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Figure Comparison of token-level distribution between our model and human label distri- bution. The x-axis represents the probability bins, while the y-axis indicates the probability density. The distribution learned by our model reflects intrinsic human disagreements.
Conclusion
In this work, we present series of strong and precise probability estimation models, de- veloped through a novel pipeline that combines LLM-based synthetic data generation with decoder-based regression fine-tuning. Our models consistently outperform previous generation approaches for similar tasks, as well as commonly used zero-shot prompting methods with strong LLMs. Through comprehensive evaluation, we demonstrate that improving local probabilistic estimation can significantly enhance the performance of com- plex probabilistic reasoning systems. We hope work inspires future advancements in building efficient, accurate models for real-world probability estimation and, more broadly, in developing general and versatile probabilistic reasoning systems.
7
Acknowledgments
This work is supported by U.S. National Science Foundation under grant No. 2204926, ONR grant (N0001424-1-2089) and Defense Advance Research Projects Agency (DARPA) under Contract No. HR001125C0304. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation and DARPA.
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A The Other Probability Estimation Attempt
In this section, we introduce an alternative approach to probability estimation by scaling compute for synthetic data creation. Inspired by the preference data used Reinforcement Learning from Human Feedback, we hypothesize that pairwise comparison is cognitively easier and more reliable than direct scoring on probability estimation task (Ziegler et al., 2020). Our method consists of three main steps: pairwise comparison, results aggregation and score mapping. We describe each component in detail below.
A.1 Pairwise Comparison
The core of this probability synthesis pipeline is the pairwise comparison of NLI samples using LLMs. Given two given NLI samples and B, each consisting of premise and hypothesis, the LLM is prompted to determine which sample has higher probability of hypothesis conditioned on the premise. We adopt the prompt format logic introduced by Qin et al. (2024), for example:
"Compare the probabilities of two NLI samples. Passage {A}, Passage (B}."
For specific prompts, please refer to Appendix D.1. To mitigate potential positional bi- ases (Zheng et al., 2023; Li et al., 2024; Shi al., 2024), where an LLM might prefer either the first second sample regardless of its content, prompt. A preference for over B, denoted p(A > B) is considered valid only the model consistently ranks A higher across both orderings; otherwise, the comparison is treated draw (Qin et al., 2024).
alternate the order of A and B in the
A.2 Results Aggregation
To construct global ranking of the test dataset, we aggregate pairwise comparison results using the TrueSkill framework (Herbrich al., 2006). In this setup, each NLI sample modeled as a player with a Gaussian skill distribution characterized by mean μt and variance o2. For every comparison between two samples, TrueSkill updates their skill distributions accordingly.
The computational cost of this aggregation is efficient, operating near the theoretical mini- mum of n log(n) comparisons for a dataset size n Herbrich et al. (2006). To further reduce comparison overhead, adopt binning strategy. The test set is divided into in bins based on scores from direct prompting, e.g., with m = 5, bins correspond to levels like "impossible", "technically possible", "plausible", "likely" and "very likely". During ranking, comparisons are limited to samples within the same bin, effectively reducing comparison times converge, as previous researches show that adaptive selection of comparisons can reduce the number of required assessments while maintaining ranking accuracy (Jamieson Nowak, 2011; Yona al., 2022). We iterative updates each samples Gaussian distribution until its variance a falls below predefined threshold 8, indicating stable estimate.
A.3 Score Mapping
In this final step, we map the relative rankings of NLI samples into scalar probability scores. While TrueSkill provides meaningful relative rankings, its outputs are not directly interpretable probabilities. To bridge this gap, we adopt the PlackettLuce approxima- tion (Plackett, 1975). Specifically, we use the ratio ///o from each samples distribution as a proxy "score", and apply softmax transformation obtain a calibrated probability distribution.
A.4 Experiment Setup
We conduct our experiments subset of the UNLI test split (Chen et al., 2020), focusing exclusively on NLI examples labeled as "neutral". This is because samples labeled as "entailment" or "contradiction" typically have ground-truth probabilities close to or 0,
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Correlation Over iterations
Corelattio Vallve
0.5
DeepSoek Coreelciion
QwQ Correlation
0.2
500
1000
1500
2000
2500
Iteration
Figure 4: The Spearman Correlation over Pairwise Comparison Iterations.
respectively, whereas our interest lies in distinguishing uncertainty in the middle range. Since we are working with small dataset, we only adopt the basic strategy to randomly sample compared NLI samples and update their scores using the TrueSkill framework. The random seed is fixed at 42 to ensure the same comparison process for different models. We initialize each player's s mean randomly between 0 and and set the initial variance o =, where larger variance reflects higher uncertainty in the probability estimate.
We evaluate performance using two primary metrics: Spearman correlation between human annotations and the synthesized probability scores, which reflects overall ranking quality; Pairwise comparison accuracy (Compare Acc), which measures agreement between model comparisons and the ground-truth ordering. We also track how the correlation evolves as the number of comparison iterations increases.
For the comparison model, test the performance DeepSeak-R---Ditiill--eeerr 32B (DeepSeek-Ae AT et al., 2025) and QwQ-32B (Qwen et al., 2025).
Method
Correlation Compare Acc
Pairwise Compare
0.6420
0.7562
Direct Prompting
0.6872
/
Pairwise Compare
0.6991
0.7655
Direct Prompting
0.7414
Table 2: Pairwise Compare Results
A.5 Results and Discussion
Main Result Table 2 and Figure 4 presents the results of probability estimation using different methods and models. The final correlations from pairwise comparison process fall short of expectations and do not surpass the direct prompting method. We attribute this to the limited reliability of the pairwise comparison judgments, particularly due to the noisiness in comparing semantically unrelated NLI passages. While pairwise comparison is often considered cognitively easier in human evaluation settings, in our case, the LLMs require comparable amounts of reasoning and contextual understanding for both direct scoring and relative ranking. This undermines the presumed advantage of pairwise compar- ison this task. These results suggest that for probabilistic NLI tasks, pairwise comparison may not offer significant benefit over direct prompting, especially when comparisons are performed over contextually disjoint samples. More structured or semantically clustered comparisons may be necessary to fully realize the benefits of pairwise reasoning.
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Difficulty of NLI Comparison We employ GPT-4o (OpenAI et al., 2024) to do pairwise comparison. The comparison accuracy is evaluated against ground-truth pairwise labels, while entropy, computed from the API logits, serves measure of the model's uncertainty,
As illustrated in Figure 5, the trends in accuracy and entropy align with intuitive expecta- tions. Specifically, the difference in label probabilities increases, the comparison accuracy improves, and the entropy decreases. This pattern suggests that the model becomes more confident in its decisions when the distinction between the samples is more significant. The results highlight GPT-4o can produce reliable judgments in scenarios with greater label separability while reflecting higher uncertainty ambiguous cases.
300
-820
-115
110
Deflrmee
(a) Accuracy and entropy of pairwise comparison as function of label difference.
(b) Distribution of label differences in the pled data.
Figure 5: Analysis of GPT--400 NLI probability pairwise comparison tests. (a) Higher label differences lead to improved accuracy and reduced entropy. (b) Frequency distribution of label differences across the sampled data.
B Analysis of Loss Function Gradients
We analyses the gradients of different losses we adopted to the class-wise probability after the softmax layer. We consider the following losses. Suppose that is the i-th bin, and f(b;) is the corresponding scalar score. Let be the predictive distribution from the model and each p; be the predicted probability for the i-th bin, and and πι the corresponding logits respectively. be the ground truth scalar score.
Mean Square Error takes the form
1
N-1
Pi f(b:) - -s)²
IMSE =
Σ
where the gradient is given by
2
=
alMSE
N-1
Pt f(b,) - p)::((()))
dz;;
Therefore, if the model underpredict the probability, the logits for all bins above the true value will be increased, and the more the differences between the bin value and the true value, the more the logits will be increased. This is not ideal we typically want to model to center its prediction around the true value.
Marign Loss
For two instance (x),s)), (x* (x*,st) st), the uncropped margin loss takes the form
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N-1
N-1 Σ pff(tt)), 1
Margin ( xz5) -sgnnss/- -sh)
(x),xx)
E p(f(b)))
where
Margin
N E 10
(f(bi)
which similarly encourages the model further updates the logits for the further away bins from the current prediction. To support greedy decoding of reasonable coarse labels, on human annotated data, we still train with the KI.-divereecc loss match the quantized label distribution.
C Aligning with Human Crowd Distributions
To evaluate the instance-level calibration of our model, we assess its ability to produce probability estimates that align with aggregated human judgments for the same instance. This is crucial aspect of building trustworthy models whose confidence reflects human consensus, We conduct two experiments on datasets with rich human annotations to validate this capability.
First, we utilize the ChaosNLI dataset (Nie et al., 2020b), which provides 100-annotator voting distributions for NLI examples. Following similar methodology to Jiang et al. (2024a), we map the distributions onto a single scalar probability score to serve as the ground truth. Specifically, use the mapping rule:
p(entailment) 1.0, p(neutral) 0.2, p(contradiction) = 0.0.
As shown in Table 3, model's predicted probabilities achieve high Spearman's correlation with these human-derived scores and a correspondingly low ranking risk. This indicates strong monotonic relationship between our model's confidence and the consensus of a human crowd.
Dataset
Total Examples
Spearman's p
Ranking Risk 1
ChaosNLI-S
1514
0.9029
0.1959
ChaosNLI-M
1599
0.8031
0.1959
Table 3: Correlation and ranking risk between our model's predictions and aggregated hu- man judgments on ChaosNLI. High correlation and low risk demonstrate strong alignment.
To further validate these findings, we evaluate our model's calibration performance on the ProtoQA dataset (Boratko et al., 2020), commonsense question-answering task where each question has multiple plausible answers annotated with human vote frequencies. For model's predicted probability, p(a Mil4), as
each question-answer pair (q,4;), a)), we treat the output of probabilistic classifier. Table 4 presents the standard calibration metrics. Divergence
The low Expected Calibration Error (ECE), Brier Score, and (JSD) demonstrate that the model's predictive distribution well-calibrated against the distribution of human judgments.
Metric
ECE +
Brier Score |
JSD
Value
0.0105
0.0159
0.0499
Table 4: Calibration metrics on ProtoQA. Low error values across all metrics confirm that the model's predictive distribution aligns well with human answer frequencies.
Taken together, the high rank correlation ChaosNLI and the low calibration error ProtoQA strongly suggest that our model's probability estimates effectively mirror human- annotated label distributions.
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D Prompt Template
Here are the prompts used in our experiments, with input data inserted into the curly brackets.
D.1 Probability Extraction
system:
You are helpful assistant good at probabilistic reasoning in the real world setting,
human: Given a premise and hypothesis, evaluate the probability hypothesis being true based on the information provided in the premise, supplemented by world knowledge and probabilistic reasoning.
Specifically:
1. Use relevant world knowledge to assess contextual factors (e.g., demographics, common practices, or statistical distributions) that may influence the likelihood of the hypothesis given the premise.
2. Perform the probabilistic reasoning estimate the conditional probability P( Hypothesis Premise).
3. Assign probability score between [o, 1] that quantifies P(Hypothesis Premise). Ensure this score reflects the strength of the connection between the premise and hypothesis based on probabilistic reasoning and world knowledge.
Premise: (premise)
Hypothesis: (hypothesis)
Your final probability estimate should be value in the range [0, 1], fine-grained possible, and formatted follows: -* prraa__prbbbilltty" For example, if the estimated probability 0. 068653, output should be: ""0.0653"*
D.2 Reasoning Summary
system: You are helpful assistant good at reasoning summary, human: Simply kNsummarizewx the reasoning process and final estimated probabil lity (a float between 0 and 1). Include key factors, assumptions, and influences on the estimated probability, dismiss repetitive information.
Reasoning: (reasoning)
Begin your summary with "Reasoning:" and end with "Probability:".
D.3
0-shot Probability Scoring
system: You are helpful assistant good at probabilistic reasoning in the real world setting.
human:
Your task to estimate the probability a textual outcome $os given description of the context SCS. Please respond with a probability in the range of [e, 1] that's ss your best estimate of the conditional probability P(OIC).
Cotext (context)
### Outcome {outcome]
Wrap your final answer in triple quotes format. Please don't 't generate any other text.
D.4 Judge with Confidence
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system: You are helpful assistant good at probabilistic reasoning real world setting
human: Define the Random Variables:
Let A( H )) represent the hypothesis: (hypothesis)
N( P represent the premise: {premise)
Below are the estimation of conditional probability P(HIP) based on the given premise and hypothesis from other agents using probabilistic reasoning and real-world knowledge.
Your to assign confidence score (ranging from to 1) each agent's response..
1. (reasoning_1)
2. (reasoning_2)
3. (reasoning_3)
[reasoning_4]
Important Considerations:
Think like a human, go beyond literal semantics by considering context, common sense, and real-world knowledge.
Related premise
hypothesis not necessarily cause high probability.
Assign higher confidence to assumptions that are more commonly observed and reasoning processes that are logically sound, fully justified.
If the premise and hypothesis both refer to entity using indefinite noun phrase (e-g., 'a person'), assume they refer the entity unless there is clear evidence suggesting otherwise.
Errors may arise from disagreements in reasoning, differing assumptions, logical mistakes, overemphasis on corner cases, underconfidence overconfidence, and inaccuracies in estimating P(HIP). Do not blindly trust other agents' probability estimation.
Here are some examples of probability extimation: 1. hypothesis: Three brothers pound on some drums
premise: Three dressed in white shirts and white hats, (two wi th baseball caps, the leader with white construction helmet), pounding sticks on steel plastic drums.
probability: 0. 08000027
2. hypothesis: There is rock currently skipping down pond.
premise: young african boy skipping rocks. probability: 0.058
3. hypothesis: The man is walking into room.
premise: A is standing in the doorway of building. probability: 0.2639
something to do. premise: At least six individuals are on team wearing helmets and knee pads while rollerblading around a skating rink.
hypothesis: People are rollerblading
probability: 0.5
hypothesis: A brown dog is outside it's snowing premise: brown dog plays in deep pile snow.
probability: 7342
hypothesis: Two girls attend convention. premise: Two girls in crowd are dressed up, the cartoon character Wall-E.
probability: 0.94
hypothesis: Some kids splash in the water and interact with each other. premise: many children play in the water.
probability: 0.99
Output Format:
The confidence score for other agents should be decimal value between and 1, formatted as: (eoddeel confidence2, conf confidenced conf sonfidenc)4) Example output: 1,.05.0.
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D.5 Pairwise Compare
system: You helpful assistant, human: Given the premise and hypothesis of two natural inference passages, determine which hypothesis is more likely based on its premise.
Specifically:
Contextual Assessment with World Knowledge
Analyze each pair: Evaluate the premise and hypothesis using relevant world knowledge. Consider contextual factors such as demographics, common practices, or statistical distributions to estimate the likelihood of the hypothesis being true. State assumptions: Explicitly identify any assumptions uncertainties introduced by missing information
premise.
2. Comparison
Compare the likelihood of each hypothesis based on the alignment be twenn the premise and hypothesis.
Justify your reasoning for why hypothesis
likely than the other, considering the degree of alignment and the assumptions made.
If the likelihoods of both hypotheses are sufficiently clase or indistinguishable, return None.
Passage A: (item1}
Passage B: (item2)
3. Output Format Example: In your final decision, strictly output \boxed((Passage A)), \baxed((Passage B)} or boxed{ (oxe{[[Noe]))")))
D.6 EntailmentBank, e-CARE Rewrite
system: You helpful assistant"),
human: Given a natural language inferenc passage: (passage)
Your goal:
Rewrite the original premise and hypothesis
Generate premises related to the passage that the probability of NLI P( hypothesis new premise1, new premise2) ranges from **0.05 to 95**.
Steps to follow:
1. Rewrite the original premise and hypothesis for clarity and precision
Ensure both the premise and hypothesis are clear, precise, and logically sound.
Removed unnecessary modal verbs (e.g., "can," " "might") and hedging language (e.g., possibly," "somewhat").
If needed, specify concrete example for clarity.
2. Generate new premises that modify the likelihood of the hypothesis being inferred:
Ensure all generated premises are factually correct and logically consistent. Here are strategies you may consider adjust the probability of inference: Al Aetenativve Explanation (Misattribution): Provide different cause for the phenomenan
weakening shifting inference.
Increase Vagueness: Make the premise general, requiring additional inference. Observer-Deeendent Effects: Frame the premise a way that makes the inference more subjective or situational.
Instantiation: Provide specific example that either supports or challenges the inference.
Categorize Premises into Four Bins Based Probability
highly oreely(pobability"o..o 9)) Premises that strongly support the hypothesis but may introduce slight variation or broader interpretations.
moderately 1ikell(probability"" 7.7): Premises that are related to the passage but are more general, potentially requiring additional context to confirm the hypothesis. neutral(probability 0.55): Balances support and doubt, making the inference uncertain. unlikely @orobability"0.33 Premises that introduce alternative mechanisms or are only tangentially related to the hypothesis.
(ontrradct 1)) Challenges the hypothesis by shifting the explanation, observer perspective, causal factorwithout introducing factual errors.
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4. Format the output valid JSON object with the following structure: {(
"premise": "Your revised premise here." "hypothesis": "Your revised hypothesis here.", "highly likely": [premisel, premise2], "moderately likely": [premisel, premise2], "neutral": [premisel, premise2],
"unlikely": : [premise1, premise2], "contradict"; [premisel, premise2]
}}
Recheck that output statements factually accurate and format valid JSON.
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E Example Structural Reasoning Traces
In this section, we provide example traces constructed to evaluate local scoring models. The example reasoning trace corresponds to an instance from the BIRD dataset C25-Sent-B (Feng et al., 2024). The block on the left describe situation with an additional sentence supporting an outcome, and the block on the right describes potential outcomes. In this case BIRD infers the outcome the additional sentence support by reasoning over range of factors and conditions probabilistically. The scores produced by our model are shown in gray blocks. Note that we do not perform NLI-based edge filtering, and the scores not coherent.
0% 0.244
[0] 0.857 [1] 0.458
Lecatio and Accouat iyy [e] Living untan santing easy
(9) 0.060
( 0.950
groonyy
[0) 0.733
it more convenient to buy vegetables at the grocery store than to grow your own vegetables.
(1) 0.389
(0) 0.633
(0) 0.622 (1) 0.418
[% 0.442
Meringia Tomillo silvoltee with surficrent free time gardening:
(0) 0.382 (1) 0.168
The convenlence buying vegetables gracery
(3) 0.595
0.452
(") Heving large besikyer willileliee anal we lnnigated se
Eesttd
growing them home is being compared. The grocery store is located far away from where you live.
(0)
(1) 0.662
[0]
(1) 0.778
gendenim von skilbs
(11 Pussessing seneerrin stitss knvueledie:
(0) 0.081
[1) 0.682
and Vaapetabls Hauinng necess w varieyy engeriaties the grouery sterts
0:728
0.571
gennery narrying
(0) 0.202 (1) 0./60
uegetaillec
Il convenient to grow your own vegetables than to buy vegetables at the grocery store.
Cancre Growing Conditioss
(0) 0,060
(0) 0.455
[R) Uning
(T) 0.416
(1) 0.629
vegetabler:
(1) Havingg sutsabie weather canditiee for uroeing veuntation:
(0) 0.435
[1) 0.800
[0)
(1)
29


Reviews:
[{"review_id": "PWqoI0UA12", "reviewer": "Reviewer_YjWc", "summary": "The paper presents an approach for training LLMs to predict probability estimations. The paper uses LLMs to modify existing datasets by relying on LLMs to generate probability estimates for examples in a suite of NLI-related datasets. The paper then fine-tunes LLMs to perform regression, specifically using a loss function that is a linear combination of the loss of the models' prediction on the original example, the loss of the modified probabilistic label, and a ranking loss. \nThe experiments indicate that typically using this loss fine-tuning the largest possible model outperforms RobERTa-Large and non fine-tuned models.", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "The paper tackles the important problem of using LLMs for regression problems. \nThe paper is thorough and includes conving experiments and results.", "reasons_to_reject": "I started reading this paper thinking that the paper was about how to get better calibrated probabilistic predictions from LLMs, especially based on this sentence in the abstract: \"LLMs continue to struggle with making … well-calibrated probabilistic predictions under uncertainty or partial information\". I was very interested in this paper's approach to creating well-calibrated probabilistic predictions. \n\nHowever, it became clear that the paper is about using LLMs for regression problems that are inspired by probabilistic reasoning rather than actually making well-calibrated probabilistic estimates. \nBuilding models to make well-calibrated probabilistic estimates is very important problem. I wish the paper had either tackled this head-on or framed the paper as using LLMs for regression (which is valid, and worthy work too).", "questions_to_authors": "Are \"human-annotated conditional probabilities” actually “well-calibrated probabilistic predictions”? The intrinsic evaluation, where the model's estimates are compared to human or LLM assigned probabilities, might not reflect well-calibrated probabilistic predictions, since human assigned probabilities might not be well-calibrated (maybe I am just getting hung up on “well-calibrated probabilistic predictions”). \n\n“We advocate evaluating models based on their effectiveness in real-world decision-making, where probabilistic reasoning aids belief modeling and evidence integration.” I was hoping to see some “real-world” task where the quality of the probability predicted by the models was evaluated, rather than on NLI, which is not a real-world task. \n\nIn 3.1, mentioning what tasks will be used would be helpful. In 128 textual outcomes is mentioned, by what are these outcomes? Until 157 I wanted to know what tasks are being used. \nKnowing the task would make the paper clearer. For example, on 213, what is the universe of tokens when converting “any target scalar label y into a distribution over tokens”?\n\n139: missing space: “LLMyielding\"\n\nTable 1: Outside of pearson's correlation for the intrinsic tasks, what are these metrics? What do bold and underline indicate? Is bold first place and underline second? At a minimum, this info should be mentioned in the caption\n\nI like the approach of \"eliciting direct probability estimates from LLMs,” but it would be nice to see how “well-calibrated” these probability estimates are. \nUsing discrepancy makes sense for agreement-based filtering, but it would be nice to see some statistics or analysis about the agreement filtering.", "extra_scores": {}}, {"review_id": "OBKPdzb4xN", "reviewer": "Reviewer_oXxq", "summary": "This paper focuses on fine-grained probability estimation, where the LLMs are trained to produce a probability of the given context being true. The core contribution of this paper lies in the data construction process and model fine-tuning method. The authors first present their data construction pipeline, where LLMs are prompted to generate reasoning chains and probabilities of the context being true given the input context. Then LLM-as-a-judge will further re-evaluation those examples where the generated probabilities are discrepant with each other.\n\nAfter the data construction, the model will be fine-tuned on these data (context + probability). The authors additionally propose a rank consistency training method to mitigate the data scarcity issue. Experiment results on a large number of benchmarks and across multiple models demonstrate the effectiveness of the proposed method.", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "1. The data construction process is rigorous and reasonable. The method of model fine-tuning is also effective.\n2. Strong performance across multiple benchmarks and models. Compared to small-scale RoBERTa model, the model delivered in this paper significantly improve the performance. The ablation study (+ Syn and +Syn+R) further demonstrate the effectiveness of the proposed fine-tuning method.", "reasons_to_reject": "The reviewer does not fine significant issue with this paper. Please refer to the following minor issues:\n\n1. Some writing is unclear, especially for the data construction process:\n   - Reasoning-Augmented Prompting. It is better to show the prompt the authors used to elicit such probabilities. Otherwise the readers cannot know what is acutally the model is generating. Also, how many LLMs are used here? Just one? But the Agreement-Based Filtering implies there are multiple LLMs generating their reasoning and the probabilities individually.\n   - Agreement-Based Filtering: Is this performed on example-level? Given a specific input example (context + generation), multiple LLMs generate reasoning chains and the final probabilities. The author also mention if the highest probabilities - lowest probabilities > threshold, this example, along with all generated chains, are flagged for further review. However, what is \"further review\" is not explained. According to the context, this should be the LLM judgement, but the writing is confusing. The reader may interpret it as \"further review\" for human evaluation, while the examples that is not filtered will be judged again by the LLM\n\n2. Missing discussion with model calibration methods. To predict the probability of the context being true using an LLM, a more straightforward way is to first elicit a probability from the LLM (e.g., get the next token prediction probability of `True` and `False`) then calibrate such confidence scores. In this line of work, temperature scaling [2] and other calibration methods [1] can achieve the same goal as this paper. These methods can be a strong baseline but are missing in the comparison.\n\nTypos:\n\nLine 113: availablee -> available\n\n[1] Tian, Katherine, et al. \"Just ask for calibration: Strategies for eliciting calibrated confidence scores from language models fine-tuned with human feedback.\" *arXiv preprint arXiv:2305.14975* (2023).\n\n[2] Shen, Maohao, et al. \"Thermometer: Towards Universal Calibration for Large Language Models.\" *arXiv preprint arXiv:2403.08819* (2024).", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "UkxVjy9tGk", "reviewer": "Reviewer_C4re", "summary": "This paper proposes a suite of training methods that improve a model's probability estimation capabilities. At its core, this paper proposes a synthetic data generation method that curates probability labels from multiple LLMs with careful considerations on their confidence and quality. Following the data, the authors propose a discretized probabilistic training scheme with multiple objectives (i.e., direct prediction and ranking consistency) to utilize the synthetic data, and show that the resulting model outperforms vanilla LLMs across several probabilistic benchmarks and evaluation settings.", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "This paper discusses an important aspect of LLM usage: probability and uncertainty estimations. The proposed methods are effective to some extent, especially on in-domain data.", "reasons_to_reject": "1. The contribution is limited; a major part of the proposed methods (i.e., discretized probabilistic training and ranking-consistency training) have been extensively discussed in existing works that use ML models for probability estimation. As I see it, the main novelty is the synthetic data generation part. However, this part itself is flawed because of the next point. \n2. The experiments are insufficient for readers to tell exactly how and why the synthetic data is helpful. The authors claim that the improvement using the synthetic data is because of its high-quality and high-confidence probability estimations from existing LLMs. However, there could be many other reasons for models to improve, such as some data leakage (as the authors are using in-domain training data to create these synthetic labels). Furthermore, it is hard to tell exactly how the synthetic data are generated based only on the description of the main sections of the paper, and the entire approach seems more like a distillation process rather than synthetic data generation process. \n3. The improvements from the synthetic data alone are compared with the in-domain supervised baselines, which further diminishes the contribution of this paper.", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "kYF0yLHflY", "reviewer": "Reviewer_eG4a", "summary": "Probabilistic inference from LLMs can be useful when used in mission-critical applications to account for ambiguity. However, LLMs are known to be calibrated poorly when compared to true distributions. This work addresses this problem by proposing a new pipeline that uses ideas from methods for decoder-based regression, synthetic data generation, and rank-consistency training.\n\nThe proposed pipeline of fine-tuning open-weight decoder LLMs for fine-grained probability estimation includes three parts: 1) discretizing scalar ground truth estimates into quantized bins, and using this discrete distribution for supervised fine-tuning using the KL loss. The model is trained to predict for tokens corresponding to these bins, 2) supervised training using synthetic data generated with a combination of LLM-ensemble and judge framework. Specifically, probability estimates are prompted for an instance from multiple LLMs. The discrepancy between these estimates is compared, and high discrepancy instances are sent to an LLM judge for confidence of each probability estimate along with its reasoning chain. The synthetic estimate is then computed using the Expected Label Score technique, and 3) indirect supervision from rank consistency, which enforces the same order between predicted and true probability estimates between two instances. \n\nThe authors compare their fine-tuning approach with other zero-shot LLMs (Deepseek-R1-Distill-Qwen-32B and GPT-4o) and a fine-tuned encoder-based regression model (RoBERTa-large). The authors evaluate the models across three aspects: a) Intrinsic, which evaluates the quality of the estimates, b) Comparison, which evaluates the plausibility ranking among estimates, and c) Structural, which evaluates the estimates in a structured reasoning setting. The results show that the introduction of both synthetic data and rank-consistency training significantly improves over the baseline models and the models fine-tuned with human supervision.", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "1. The proposed method is largely cogent and useful in scenarios with limited human-annotated data. Experimental choices are driven by insights from literature.\n2. The multi-pronged evaluation setup is commendable. In addition, it is nice to observe the clear individual improvements from the synthetic data and rank-consistency", "reasons_to_reject": "1. The setup of the baseline encoder model is not fully clear. My assumption is that the model is a fully fine-tuned RoBERTa with a regression head (l.176). If that is the case, this model would be an unfair baseline since it compares a categorical model with a regression model. An encoder model trained to predict a categorical distribution over the quantized bins would be a fairer baseline comparison.\n2. While I am happy with the process taken for synthetic data creation, it would have been useful to test the importance of the LLM-judge confidences and the corresponding normalization. Does a simple aggregation of model estimates provide a good-enough distribution? \n3. I would have been interested to see how the pipeline proposed works without any ground truth data. That is, in the equation below line 229, can the first loss term be removed? This setting is essential to contextualize the role human supervision plays in the gains, especially when some of the baselines are considered in the zero-shot setting.\n4. The writing at certain places (esp. Section 3) can be improved. For example, I am not completely sure how the distribution gets aggregated when an instance passes the discrepancy filter. Figure 1 suggests that only the instances that do not pass the discrepancy test are sent to the LLM judge. However, lines 212-220 only explain how the distribution of these instances is computed.  Similarly, Figure 1 suggests that the expected label score for the rank-consistency loss comes from the synthetic data, although my understanding is that this is the score from the model that is fine-tuned. Several important details are not explicitly stated (e.g., what is the judge model?).", "questions_to_authors": "Questions:\n1. L.167: “summarize the rationale to extract key steps”. Is there a chance that this step introduces errors that propagate into subsequent steps?\n2. L.193: Can you provide details on what the function f might correspond to in practice? My intuition suggests that this should be the mean of the bin. \n3. L.219: How are the mixture distributions, Qi(t|k) ‘s , computed? Are these the quantized distributions of the estimates? If so, they should be denoted by small q’s to be consistent with the notation.  \n\nSuggestions:\n- L.162: “high-discrepancy …further review.” I would encourage the authors to mention what “further review” means here.\n- L.261-264: “removing modality, …., reasoning challenge”. I am unsure what this means. Can you provide examples in the appendix?\n\nTypos:\n- L.28: “depends” -> “depend”\n- L.30: “involves” -> “involve”\n- L.113: “setupsevaluation” -> “setup, evaluation”\n- L.187: (0,1) -> [0,1]\n- Equation below l.229: “(D)_synthetic” -> (D_synthetic)\n- L.330: “taskssuch” -> “tasks such”", "extra_scores": {}}]

Author rebuttal:

Meta-review: The overall evaluation is positive (as reflected in most reviewers' comments, and I agree). Toward acceptance, please make sure to carefully address the reviewers’ questions and concerns, and refine the camera-ready version accordingly.
Decision: Accept