Towards Harmonized Uncertainty Estimation for Large Language Models
Authors: Rui Li, Jing Long, Muge Qi, Heming Xia, Lei Sha, Peiyi Wang, Zhifang Sui
Accepted at: acl25

Towards Harmonized Uncertainty Estimation for Large Language Models

Rui Li1,2, Jing Long2, Muge Qi2, Heming Xia3, Lei Sha4, Peiyi Wang1, Zhifang Sui1*
1State Key Laboratory of Multimedia Information Processing, School of Computer Science, Peking University
2 School of Software & Microelectronics, Peking University
3 The Hong Kong Polytechnic University  4Beihang University
o_11ru1@stu.pku.edu.cn

Abstract

To facilitate robust and trustworthy deployment of large language models (LLMs), it is essential to quantify the reliability of their generations through uncertainty estimation. While recent efforts have made significant advancements by leveraging the internal logic and linguistic features of LLMs to estimate uncertainty scores, our empirical analysis highlights the pitfalls of these methods to strike a harmonized estimation between indication, balance, and calibration, which hinders their broader capability for accurate uncertainty estimation. To address this challenge, we propose CUE (Corrector for Uncertainty Estimation): A straightforward yet effective method that employs a lightweight model trained on data aligned with the target LLM’s performance to adjust uncertainty scores. Comprehensive experiments across diverse models and tasks demonstrate its effectiveness, which achieves consistent improvements of up to 60% over existing methods. Resources are available at https://github.com/0-L1RU1/Corrector4UE.

1 Introduction

Uncertainty is the only certainty there is.
- by John Allen Paulos

Large Language Models (LLMs) have demonstrated exceptional capabilities in handling a wide range of downstream tasks (OpenAI, 2023; Touvron et al., 2023a,b; Dubey et al., 2024). They are gradually adopted as general-purpose API interfaces (e.g., ChatGPT1), providing valuable services and assistance in human life. Despite these impressive advancements, concerns persist regarding the tendency of LLMs to generate hallucinations and factual inaccuracies with confidence (Zhang et al., 2023; Wachter et al., 2024), which may mislead users to overestimate the reliability of the information provided by these models. To mitigate this issue, uncertainty estimation (Loquercio et al., 2020) proposed quantifying the reliability of model outputs so as to ensure the robustness and trustworthiness of AI-driven services.

Harmonized uncertainty estimation is expected to encompass three key aspects: 1) Indication. The uncertainty score should clearly reflect the reliability of model responses, with higher scores signaling potential inaccuracies. This can be framed as a classification task, with “reliable” or “unreliable” as the classes. 2) Balance. Within classification framework, it’s critical to strike a balance between recall and precision, ensuring that challenging cases are appropriately flagged while minimizing the resources spent on false positives. 3) Calibration. The uncertainty score should align with human intuition and probabilistic expectations, to facilitate effective calibration. By striking a harmonized balance between these three aspects, uncertainty estimation provides an ideal measure of the model’s reliability, offering both usability and interpretability.

There has been growing interest in developing uncertainty estimation methods tailored for LLMs. However, with a thorough analysis across diverse uncertainty estimation methods, we found that there still remains a large performance gap between existing methods to achieve the harmonized uncertainty estimation. Specifically, methods that excel in one aspect fall short in others. For instance, SAR (Duan et al., 2023), the outstanding and state-of-the-art method in the specific dataset SciQA (Auer et al., 2023), achieves the best performance in indication but performs poorly in the view of calibration. Furthermore, we found that the combination of uncertainty scores obtained by existing methods provides little improvement in uncertainty estimation performance, suggesting that these methods are quite homogeneous. These find-
ings highlight considerable room for refinement in uncertainty estimation.

In this paper, we introduce CUE💡, a simple yet effective framework for adjusting uncertainty scores, which is orthogonal to existing uncertainty estimation methods. Specifically, we begin by curating dataset that is closely aligned with the target LLM’s performance within a particular domain of knowledge. This dataset is then utilized to train an auxiliary lightweight model, which serves as a Corrector to adjust the uncertainty scores. By integrating the Corrector trained on global alignment information with those uncertainty estimation methods that rely solely on the intrinsic logic and linguistic features of LLMs, we can significantly refine the uncertainty scores.

Our main contributions are thus as follows:

• According to an empirical analysis of existing uncertainty estimation methods from both classification and calibration views, we found there is substantial room for improvement in their performance regarding classification indication, precision-recall balance, and calibration.

• We propose CUE💡, an uncertainty score correction framework that employs a classifier, aligned with the model’s task performance, as a Corrector to adjust uncertainty scores. This Corrector allows for seamless integration with existing uncertainty estimation methods.

• Extensive experiments and explorations are also conducted in areas such as generalization. The results demonstrate that our CUE💡 consistently enhances various existing uncertainty estimation methods, showing significant improvements in a harmonized manner across diverse data domains and target models.

2 Related Work

2.1 Uncertainty Estimation for LLMs

As illustrated in Figure 1, uncertainty estimation methods for LLMs can be broadly categorized into logit-based methods, verbalized methods, consistency-based methods and internal state-based methods.

Logit-based methods are the most widely used and effective approaches in uncertainty estimation. Predictive Entropy (PE) (Malinin and Gales, 2020) defined uncertainty as the entropy of the output logits distribution, which is widely adopted and built upon in subsequent research. Following that, Kuhn et al. (2023) introduced semantic entropy (SE) that estimates uncertainty by marginalizing over semantically-equivalent samples in NLG tasks. Duan et al. (2023) proposed Shifting Attention to Relevance (SAR), which focuses on relevant information and assigns significance weights to tokens based on their contributions to the overall response. Yaldiz et al. (2024) introduced a Learnable Response Scoring Function (LARS), which utilizes supervised data to capture complex token-probability dependencies.

Verbalized methods (Xiong et al., 2023; Groot and Valdenegro-Toro, 2024) leverage LLMs’ strong language and instruction-following abilities to express uncertainty, often by prompting the model to provide an uncertainty score. However, studies (Ni et al., 2024; Madhusudhan et al., 2024; Becker and Soatto, 2024) have shown that LLMs struggle with faithfully conveying their uncertainties, particularly due to overconfidence. Consistency-based methods, such as those proposed by Li et al. (2024b) and Becker and Soatto (2024), assess uncertainty through multiple generated answers, using techniques like perturbation and aggregation to improve reliability. Pedapati et al. (2024) further reduced overconfidence by guiding LLMs to justify their answers. Internal state-based methods (Azaria and Mitchell, 2023; Liu et al., 2024) analyze LLM activations to predict errors, with Kadavath et al. (2022) and Ji et al. (2024) exploring self-evaluation and probing estimators to enhance uncertainty estimation.

Due to space limitations, a more detailed discuss-

![A concise overview figure of various uncertainty estimation method categories, including logit-based methods, verbalized methods, consistency-based methods, and internal state-based methods.](page_1012_186_377_246.png)

Figure 1: A concise overview figure of various uncertainty estimation method categories, including logit-based methods, verbalized methods, consistency-based methods, and internal state-based methods.
Figure 2: The performance of existing uncertainty estimation methods, evaluated on the SciQA dataset with the LLaMA-3-8B-Instruct model as the target, and the improvements after applying the Corrector. Note that a lower ECE score indicates better performance, so we report its reduction.

Figure 3: AUROC improvement across uncertainty scores combination from different existing methods.

sion of related work is provided in the Appendix A.1.

3 Preliminary Study

3.1 Limitation of Existing UE Methods
We evaluate existing UE methods from both classification and calibration views, focusing on three key aspects of uncertainty scores: indication, balance, and calibration. From the classification view, uncertainty scores are utilized to guide the classification process. Instances with scores above a threshold are classified as \( c_1 \) (unreliable) and those below as \( c_0 \) (reliable). We employ AUROC to measure how well the scores indicate unreliability and F1 score to evaluate their balance between precision and recall. The calibration view involves a more rigorous assessment and interpretation of uncertainty scores. Well-calibrated scores should align with human probabilistic intuition and provide more precise instance rankings. We use ECE to assess calibration.

Basic methods exhibit poor indication performance. Firstly, we focus on representative but naive methods including Lexical Similarity (LS) (Fomicheva et al., 2020), Verbal Confidence (VC) (Xiong et al., 2023), P(true) (Kadavath et al., 2022), and Predictive Entropy (PE) (Malinin and Gales, 2020) that belong to four categories: consistency-based methods, verbal confidence methods, internal state-based methods, and logit-based methods, respectively. As shown in Figure 2 and Table 1, the AUROC scores for these methods across the target models and datasets exhibit general low performance, which is even close to random guessing.

Enhanced logit-based methods typically have low F1 scores. Some enhanced methods such as Length-normalized Predictive Entropy (LN-PE) (Malinin and Gales, 2020), SAR-t, SAR-s, SAR, and Semantic Entropy (SE) (Kuhn et al., 2023), make tailored adjustments to refine predictive entropy process, which show improvements over PE in terms of AUROC. However, no one is universally optimal for all target models and datasets. Moreover, as depicted in Figure 2 and Figure 5, the F1 scores of those methods are particularly low. This indicates that although those methods provided uncertainty scores with some potential to indicate the reliability of model response, they still fall short in striking a balance between precision and recall.

Most existing methods fall short in calibration. As shown in Table 1 and Figure 6, it appears that prior methods have overlooked the calibration aspect, resulting in relatively poor performance in terms of ECE scores.

3.2 Inter-method Cooperation
We examined whether the uncertainty scores derived from one uncertainty estimation method could refine the scores obtained from another method. Specifically, we integrated the uncertainty scores from each method using the weighted combination and compared its performance with the top-performing method in the pair. As illustrated
Figure 4: An overview of uncertainty score correction framework. Firstly, we construct a dataset that closely aligns with the target model’s performance. This dataset is then utilized to train a lightweight auxiliary model that serves as a correction module, enabling seamless integration with existing uncertainty estimation methods to produce corrected uncertainty scores.

in Figure 3, these integrations do not enhance overall performance and may lead to a decline. This underscores the limitations in the complementary nature of existing methods.

Our analysis reveals a significant performance gap among existing methods in achieving harmonized uncertainty estimation, as individual methods excel in specific aspects but underperform in others. Furthermore, combining uncertainty scores from different methods yields minimal to no improvement, underscoring their homogeneous and non-complementary nature.

4 Method

In this section, we introduce CUE 🧠, a correction framework featuring an intuitive approach to directly optimizing for uncertainty estimation, where a Corrector is trained using a lightweight model to refine the uncertainty score. Through this method, we provide a more robust solution for uncertainty estimation. As shown in Figure 4, Our method comprises three main steps including dataset crafting, corrector training and uncertainty correcting.

4.1 Dataset Crafting

We begin by extracting data from existing datasets to create an evaluation set for assessing the target model \( M \)’s performance in a specific domain. This set consists of a collection of question-answer pairs, denoted as \( \mathcal{D} = \{(q_i, a_i) \mid i = 1, \ldots, n\} \). We then prompt \( M \) to generate responses \( r_i \) for each question \( q_i \), forming a response set \( \mathcal{R} = \{r_i \mid i = 1, \ldots, n\} \). Subsequently, each response \( r_i \) is subjected to a rigorous evaluation against the ground truth \( a_i \), employing a hybrid approach that combines both rule-based and LLM-based methods. The rule-based method compares response \( r_i \) to the ground truth \( a_i \) using the longest common subsequence (LCS). A response \( r_i \) is considered equivalent to \( a_i \) only if its ROUGE-L score, computed as \( \text{ROUGE-L}(r_i, a_i) = \frac{\text{LCS}(r_i, a_i)}{\min(\text{len}(r_i), \text{len}(a_i))} \), is greater than threshold value, formalized as \( \mathcal{M}_{Rule}(r_i, a_i) = \mathbb{I}_{\text{RougeL}(r_i, a_i) > 0.7} \). Additionally, we utilize GPT-turbo-3.5-0613 (Ouyang et al., 2022) to assess the equivalence between \( r_i \) and \( a_i \) by directly prompting, formalized as \( \mathcal{M}_{LLM}(r_i, a_i) = \mathbb{I}_{\text{True in LLM}(r_i, a_i)} \).

To mitigate false positives, we apply rigorous thresholds and strict prompting rules. The final judgment is determined using an “OR” logic: \( \mathcal{M}(r_i, a_i) = \mathcal{M}_{Rule}(r_i, a_i) \lor \mathcal{M}_{LLM}(r_i, a_i) \), preventing the omission of positive instances.

After that, a binary label \( c_i \) is assigned to each sample, defined as
\[
c_i = \mathcal{M}(r_i, a_i)
\]
By pairing question \( q_i \) with the label \( c_i \), we form a correction dataset \( \mathcal{D}_{cor} = \{(q_i, c_i) \mid i = 1, \ldots, n\} \), which serves as a representation of the target model’s performance in generating correct responses across a particular knowledge domain. To directly associate the questions with uncertainty, we transform the dataset form into \( \mathcal{D}_{cor}^* = \{(q_i, 1 - c_i) \mid i = 1, \ldots, n\} \).

4.2 Corrector Training

Employing the correction dataset \( \mathcal{D}_{cor}^* \), we train a classifier to align with the performance of the target model. Specifically, the classifier integrates a fully connected layer following a lightweight encoder model, such as RoBERTa (Liu, 2019) and Deberta (He et al., 2021a), with the
representation of the special token [CLS] as its input, denote as \( h_{[CLS]} \in \mathbb{R}^d \). The output of the classifier is given by \( \hat{y}_i = \sigma (\mathbf{W} \cdot h_{[CLS]} + b) \), where \( \sigma(z) \) is the sigmoid function, used to compute the likelihood \( y_i \) that a data point belongs to label \( c_1 \). During training, we minimize the binary cross-entropy loss function \( \mathcal{L} = - \sum i = 1^N [y_i \log(\hat{y}_i) + (1 - y_i) \log(1 - \hat{y}_i)] \) across the correction dataset.

This results in a Corrector, an auxiliary component that can be integrated with existing uncertainty estimation methods to enhance their reliability.

4.3 Uncertainty Correcting

We derive the probability that an instance \( x \) belongs to category \( c_1 \) from the Corrector. This probability, denoted as the correction score \( C(x) \), can be utilized to adjust the uncertainty scores to align with the target model’s performance, thereby refining the uncertainty estimation process.

In the refinement process, we first normalize the uncertainty scores generated from existing UE methods to match human probabilistic intuition, ensuring they fall within the range \([0, 1]\). Normalization is achieved via \( U_{\text{norm}}(x) = \frac{U(x) - \min(U)}{\max(U) - \min(U)} \), where \( U(x) \) represents the uncertainty score for a specific instance \( x \), computed by a chosen UE method. The terms \( \min(U) \) and \( \max(U) \) denote the minimum and maximum uncertainty scores across the entire dataset, respectively. Following normalization, we apply our correcting by combining the normalized score \( U_{\text{norm}}(x) \) with the correction score \( C(x) \) generated by the Corrector. The combination employs a weighted approach, where the corrected uncertainty score \( U_{\text{cor}}(x) \) is computed as:

\[
U_{\text{cor}}(x) = w^* \cdot U_{\text{norm}}(x) + (1 - w^*) \cdot C(x)
\]

The optimal weight \( w^* \) is determined through a grid search on the development dataset. This weighted method ensures that the corrected uncertainty scores balance the contributions of both the original and correction scores, thereby enhancing the reliability of the uncertainty estimation.

5 Experiments

5.1 Experiments Setup

5.1.1 Models

Target models We selected the OPT-6.7B\(^2\) (Zhang et al., 2022), a model widely utilized in previous studies (Kuhn et al., 2023; Duan et al., 2023), and the advanced open-source model LLaMA-3-8B-Instruct\(^3\) (Dubey et al., 2024) as the target models for our main experiments.

Base Models We employed lightweight encoder models as the base model to train the Corrector, including models from the RoBERTa series (Liu, 2019) and DeBERTa series (He et al., 2021a,b).

5.1.2 Metrics

AUROC We use the area under the receiver operating characteristic curve (AUROC) to evaluate uncertainty estimation methods from a classification view. In our setting, an AUROC of 1 signifies perfect indicative performance to distinguish between samples the target model can answer reliably and those it cannot, while an AUROC of 0.5 indicates that the estimation is no better than random guessing.

F1 Score F1 score is used to evaluate the balance between precision and recall in classification tasks. It is the harmonic mean of precision and recall, where both are equally important. The F1 score ranges from 0 to 1, with 1 indicating perfect precision and recall.

\[
\text{F1 Score} = 2 \times \frac{\text{Precision} \times \text{Recall}}{\text{Precision} + \text{Recall}}
\]

ECE We use Expected Calibration Error (ECE) to evaluate the performance of calibration, which is calculated by partitioning predicted confidence scores into bins and comparing the average confidence in each bin to the actual fraction of correct predictions, formalized as

\[
\text{ECE} = \sum_{m=1}^{M} \frac{|B_m|}{n} |\text{acc}(B_m) - \text{conf}(B_m)|
\]

In the computing of ECE, we treat the confidence score as 1 minus uncertainty score.

5.1.3 Datasets

We focus on the question-answering task using two representative datasets in the main experiments: TriviaQA (Joshi et al., 2017), and SciQA (Auer et al., 2023). TriviaQA comprises 95,000 question-answer pairs created by trivia enthusiasts, supplemented with independently sourced evidence documents. SciQA contains 2,565 question-answer

\footnotetext{2huggingface.co/facebook/opt-6.7b.}
\footnotetext{3huggingface.co/meta-llama/Meta-Llama-3-8B-Instruct.}
pairs fetched from the open research knowledge graph, covering several research fields ranging from science and technology like Computer Science, Engineering, Chemistry, and Geology, life sciences like Immunology and Genetics to social sciences like Economics and Urban Studies.

5.1.4 Baselines
We select a variety of representative uncertainty estimation methods as baselines, with a particular focus on logits-based methods.

Among the baselines, we cover multiple categories, including logit-based, verbalized, internal state-based, and consistency-based methods, including: Lexical Similarity (LS) (Fomicheva et al., 2020), which computes the similarity between multiple sentences as a measure of consistency; Verbal Confidence (VC) (Xiong et al., 2023), which requires the target model to respond and provide a confidence score; P(True) (Kadavath et al., 2022), which first asks the target model to propose an answer and then evaluates it using an internal probability mechanism; and Predictive Entropy (PE) (Malinin and Gales, 2020), which calculates uncertainty by measuring the entropy of the predictive posterior.

We also explore a series of advanced logit-based methods including: Length-normalized Predictive Entropy (LN-PE) (Malinin and Gales, 2020), which adjusts PE by normalizing it according to sentence length; Semantic Entropy (SE) (Kuhn et al., 2023), which clusters sentences with equivalent meanings and calculating cluster-wise entropy; and Shifting Attention to Relevance (SAR) (Duan et al., 2023), which encompasses SAR-t, SAR-s and SAR, donated as the token-shifted predictive entropy, sentence-shifted predictive entropy, and both token- and sentence-shifted predictive entropy respectively.

5.1.5 Implementation Details
Dataset Splitting For the TriviaQA dataset, we randomly selected 5,000 samples from the training set for data crafting and corrector training. For datasets with limited data, SciQA, we utilized the entire training set. We then used half of the test set to search for the optimal hyperparameter w, while the other half was employed to evaluate the method’s effectiveness.

Hyperparameter For each dataset and model pair, we train a corresponding Corrector, which is universally applicable across various methods. Additionally, for every method, dataset, and model combination, we derive the weight using the development set respectively. The sensitivity analysis of hyperparameter \( w^* \) and its configuration within the cross-domain experiments are elaborated upon in Appendix A.6.

5.2 Main Result
We has evaluated existing methods in Section 3 and found that there still remains a large performance gap between existing methods to achieve the harmonized uncertainty estimation. In this part, we present the performance of CUE 📈 from both classification and calibration views, demonstrating that integrating a Corrector with existing UE methods significantly enhances uncertainty estimation across multiple dimensions, including classification indication, precision-recall balance, and calibration.

Classification View As illustrated in Table 1, the Corrector has resulted in significant improvements, with an average AUROC score increase of 0.27 for TriviaQA and 0.09 for SciQA. Even when applied to challenging methods such as SE and SAR, the Corrector boosts AUROC scores by 0.01 to 0.03. Since AUROC reflects the UE method’s ability to assign higher scores to instances where the target model responds unreliably compared to those it responds to reliably, these improvements indicate that the deployment of the Corrector enhances the overall indicative capacity of the uncertainty scores, making it more effective for users in determining whether to trust the model’s responses.

Furthermore, as illustrated in Figure 5, the F1 score is also boosted by the Corrector, achieving an average increase of 38.97%. This notable improvement demonstrates the Corrector’s ability to help balance precision and recall, effectively mitigating the polarization tendency in the uncertainty scores observed in previous methods.

Calibration View Although calibration is not the direct training objective of our Corrector, its application yields favorable calibration results. When employing the OPT-6.7B model as the target, we observed average ECE reductions of 0.34 on TriviaQA and 0.21 on SciQA. With the LLaMA-3-8B-Instruct model as the target, the reductions are 0.11 and 0.07, respectively—still considerable. To further illustrate the calibration performance
<table>
  <tr>
    <th rowspan="2">Method</th>
    <th colspan="3">TriviaQA</th>
    <th colspan="3">SciQA</th>
  </tr>
  <tr>
    <th>Vanilla</th>
    <th>+Corrector</th>
    <th>Improv</th>
    <th>Vanilla</th>
    <th>+Corrector</th>
    <th>Improv</th>
  </tr>
  <tr>
    <th colspan="7">OPT-6.7B</th>
  </tr>
  <tr>
    <td>LS</td>
    <td>46.49</td>
    <td>65.11</td>
    <td>+18.62</td>
    <td>72.71</td>
    <td>41.76</td>
    <td>-30.94</td>
    <td>44.12</td>
    <td>49.40</td>
    <td>+5.29</td>
    <td>76.38</td>
    <td>32.78</td>
    <td>-43.60</td>
  </tr>
  <tr>
    <td>VC</td>
    <td>60.41</td>
    <td>70.55</td>
    <td>+10.15</td>
    <td>49.13</td>
    <td>27.61</td>
    <td>-21.52</td>
    <td>51.69</td>
    <td>56.55</td>
    <td>+4.86</td>
    <td>62.65</td>
    <td>38.99</td>
    <td>-23.66</td>
  </tr>
  <tr>
    <td>P(True)</td>
    <td>66.74</td>
    <td>72.29</td>
    <td>+5.54</td>
    <td>45.00</td>
    <td>32.63</td>
    <td>-12.80</td>
    <td>56.12</td>
    <td>59.49</td>
    <td>+3.37</td>
    <td>58.79</td>
    <td>34.52</td>
    <td>-24.27</td>
  </tr>
  <tr>
    <td>PE</td>
    <td>56.36</td>
    <td>66.62</td>
    <td>+10.25</td>
    <td>42.39</td>
    <td>20.28</td>
    <td>-22.12</td>
    <td>50.07</td>
    <td>56.02</td>
    <td>+5.95</td>
    <td>62.05</td>
    <td>36.92</td>
    <td>-25.13</td>
  </tr>
  <tr>
    <td>LN-PE</td>
    <td>78.37</td>
    <td>79.93</td>
    <td>+1.57</td>
    <td>32.29</td>
    <td>20.80</td>
    <td>-11.49</td>
    <td>60.88</td>
    <td>64.23</td>
    <td>+3.35</td>
    <td>49.52</td>
    <td>34.68</td>
    <td>-14.84</td>
  </tr>
  <tr>
    <td>SE</td>
    <td>80.66</td>
    <td>81.00</td>
    <td>+0.34</td>
    <td>36.64</td>
    <td>27.05</td>
    <td>-9.59</td>
    <td>64.52</td>
    <td>66.15</td>
    <td>+1.63</td>
    <td>52.66</td>
    <td>42.23</td>
    <td>-10.43</td>
  </tr>
  <tr>
    <td>SAR-t</td>
    <td>78.24</td>
    <td>80.21</td>
    <td>+1.97</td>
    <td>40.14</td>
    <td>37.85</td>
    <td>-2.30</td>
    <td>60.00</td>
    <td>63.75</td>
    <td>+3.74</td>
    <td>45.33</td>
    <td>44.19</td>
    <td>-1.14</td>
  </tr>
  <tr>
    <td>SAR-s</td>
    <td>51.77</td>
    <td>55.83</td>
    <td>+4.06</td>
    <td>53.78</td>
    <td>49.65</td>
    <td>-4.13</td>
    <td>53.20</td>
    <td>54.15</td>
    <td>+0.95</td>
    <td>76.21</td>
    <td>34.83</td>
    <td>-41.38</td>
  </tr>
  <tr>
    <td>SAR</td>
    <td>75.32</td>
    <td>78.67</td>
    <td>+3.35</td>
    <td>40.61</td>
    <td>31.02</td>
    <td>-9.59</td>
    <td>60.04</td>
    <td>62.72</td>
    <td>+2.68</td>
    <td>49.40</td>
    <td>38.99</td>
    <td>-10.41</td>
  </tr>
  <tr>
    <th colspan="7">LLaMA-3-8B-Instruct</th>
  </tr>
  <tr>
    <td>LS</td>
    <td>19.57</td>
    <td>69.82</td>
    <td>+50.25</td>
    <td>70.25</td>
    <td>7.41</td>
    <td>-62.84</td>
    <td>53.67</td>
    <td>65.38</td>
    <td>+11.71</td>
    <td>38.64</td>
    <td>18.19</td>
    <td>-20.45</td>
  </tr>
  <tr>
    <td>VC</td>
    <td>62.34</td>
    <td>74.89</td>
    <td>+12.55</td>
    <td>23.41</td>
    <td>16.78</td>
    <td>-6.63</td>
    <td>68.22</td>
    <td>72.15</td>
    <td>+3.93</td>
    <td>31.88</td>
    <td>19.47</td>
    <td>-12.36</td>
  </tr>
  <tr>
    <td>P(True)</td>
    <td>57.14</td>
    <td>72.29</td>
    <td>+15.15</td>
    <td>24.67</td>
    <td>19.84</td>
    <td>-4.83</td>
    <td>65.63</td>
    <td>71.41</td>
    <td>+5.78</td>
    <td>34.56</td>
    <td>31.92</td>
    <td>-2.64</td>
  </tr>
  <tr>
    <td>PE</td>
    <td>64.52</td>
    <td>69.76</td>
    <td>+5.25</td>
    <td>21.38</td>
    <td>17.24</td>
    <td>-4.13</td>
    <td>66.54</td>
    <td>67.98</td>
    <td>+1.44</td>
    <td>40.67</td>
    <td>34.07</td>
    <td>-6.60</td>
  </tr>
  <tr>
    <td>LN-PE</td>
    <td>72.55</td>
    <td>74.79</td>
    <td>+2.24</td>
    <td>14.31</td>
    <td>11.53</td>
    <td>-2.79</td>
    <td>69.48</td>
    <td>71.56</td>
    <td>+2.08</td>
    <td>29.38</td>
    <td>23.76</td>
    <td>-5.62</td>
  </tr>
  <tr>
    <td>SE</td>
    <td>80.92</td>
    <td>82.12</td>
    <td>+1.20</td>
    <td>13.07</td>
    <td>12.76</td>
    <td>-0.31</td>
    <td>71.59</td>
    <td>72.93</td>
    <td>+1.34</td>
    <td>30.54</td>
    <td>25.23</td>
    <td>-5.30</td>
  </tr>
  <tr>
    <td>SAR-t</td>
    <td>79.55</td>
    <td>79.93</td>
    <td>+0.38</td>
    <td>16.40</td>
    <td>13.70</td>
    <td>-2.70</td>
    <td>72.26</td>
    <td>73.87</td>
    <td>+1.61</td>
    <td>30.37</td>
    <td>26.81</td>
    <td>-3.56</td>
  </tr>
  <tr>
    <td>SAR-s</td>
    <td>69.87</td>
    <td>77.09</td>
    <td>+2.95</td>
    <td>23.17</td>
    <td>20.00</td>
    <td>-3.17</td>
    <td>74.96</td>
    <td>75.72</td>
    <td>+0.76</td>
    <td>38.54</td>
    <td>36.18</td>
    <td>-2.37</td>
  </tr>
  <tr>
    <td>SAR</td>
    <td>80.92</td>
    <td>81.90</td>
    <td>+0.98</td>
    <td>16.17</td>
    <td>13.76</td>
    <td>-2.41</td>
    <td>73.88</td>
    <td>75.19</td>
    <td>+1.31</td>
    <td>28.97</td>
    <td>25.60</td>
    <td>-3.37</td>
  </tr>
</table>

Table 1: AUROC and ECE scores (%) on the TriviaQA and SciQA datasets obtained by applying the Corrector to existing uncertainty estimation methods. LS denotes the Lexical Similarity method. VC denotes the Verbal Confidence method. PE denote the Predictive Entropy method. LN-PE denotes the Length-normalized Predictive Entropy method. SE denote the Semantic Entropy. SAR-t refers to the token-level version of the SAR method, while SAR-s denotes the sentence-level version.

![Bar charts comparing performance gains for various methods on TriviaQA and SciQA datasets](page_1012_1042_420_180.png)

Figure 5: The performance gains of using the Corrector to adjust the uncertainty scores for various methods on the datasets of TriviaQA and SciQA, and the target models of LLaMA-3-8B-Instruct and OPT-6.7B, are evaluated in terms of F1 score.

facilitated by the Corrector, we provide calibration plots in Figure 6.

We also conducted extensive experiments on the generalization performance of the Corrector. Due to space constraints, we put the detailed results and analysis in the Appendix A.4.

In summary, integrating the Corrector helps achieve harmonized uncertainty estimation. With the Corrector, we can improve the reliability of uncertainty scores and alignment with the actual performance of the model. The analysis of the pure Corrector’s performance can be found in Appendix A.7.

5.3 Ablation Study

We conducted ablation studies to scrutinize the impact of the base model, the correction score formats and its acquisition methods.

Formats We compared the efficacy of probabilistic values versus label values for correction. As shown in Table 2, probabilistic correction scores demonstrate clear superiority, as they allow finer-grained adjustments by leveraging a broader spectrum for integration. Conversely, discrete values, such as 0 and 1, tend to introduce significant biases in the corrected uncertainty scores.

Base Model We utilized various encoder models as base models to train the Corrector and assess the impact on correction performance. Specifically, we employed models from the RoBERTa series, including RoBERTa-base4 and RoBERTa-large5, as

4huggingface.co/FacebookAI/roberta-base
5huggingface.co/FacebookAI/roberta-large
Figure 6: Calibration Plots. These plots depict the relationship between predicted confidence and observed frequencies. The diagonal line represents perfect calibration, where predicted confidence aligns precisely with actual outcomes. Bars extending above the diagonal indicate underestimation of confidence, while bars below the diagonal reflect overestimation. The final plot highlights the optimal calibration performance achieved through our Corrector.

well as models from the DeBERTa series, including DeBERTa-base⁶, DeBERTa-v3-large⁷, DeBERTa-v3-base⁸, DeBERTa-v3-small⁹, and DeBERTa-v3-xsmall¹⁰. These models represent different types, series, and sizes. As illustrated in Figure 7, more advanced, later-generation, and larger models yield superior results.

Figure 7: The overall AUROC score gains achieved by Correctors trained on different base models across various UE methods on the SciQA dataset and Llama-3-8B-Instruct target model.

Acquisition We compare correction scores from a lightweight classifier with those estimated using GPT-4o. We attempt not to rigorously assess the target model’s answers but to predict its reliability. Despite GPT-4’s strong performance in question answering, our results show it is less effective than the classifier in directly predicting reliability of target models when faced with questions. Additionally, as detailed in Section 3.2, combining uncertainty scores from different UE methods does not improve and may even degrade performance. This highlights the Corrector’s unique role as a complement to existing UE methods.

We also conducted experiments to compare our Corrector with other supervised methods specifically designed for uncertainty estimation. The experimental results demonstrated that our supervised pipeline offered significant advantages in enhancing uncertainty estimation. Detailed results and analysis can be found in Appendix A.3.

<table>
  <tr>
    <th>Methods</th>
    <th>AUROC (↑)</th>
    <th>ECE (↓)</th>
  </tr>
  <tr>
    <td>Corrector</td>
    <td>69.87</td>
    <td>6.73</td>
  </tr>
  <tr>
    <td>Original Best</td>
    <td>80.92</td>
    <td>11.53</td>
  </tr>
  <tr>
    <td>+Corrector Probability</td>
    <td><b>82.12</b></td>
    <td><b>10.46</b></td>
  </tr>
  <tr>
    <td>+Corrector Label</td>
    <td>80.92</td>
    <td>11.53</td>
  </tr>
  <tr>
    <td>+GPT-4o Score</td>
    <td>80.92</td>
    <td>11.53</td>
  </tr>
</table>

Table 2: Ablation Study. LLaMA-3-8B-Instruct as the target model and TriviaQA as the test dataset. Original Best refers to the peak performance achieved by various baseline when the Corrector is not incorporated.
6 Conclusion

Our study highlights the limitations of current uncertainty estimation methods in terms of classification accuracy, precision-recall balance, and calibration. We introduce an innovative uncertainty score correction framework that utilizes a classifier as a Corrector to refine these scores, ensuring alignment with the model’s true task performance. This Corrector integrates seamlessly with existing methods, enhancing their effectiveness. Extensive experiments validate that the Corrector consistently improves performance across various metrics, data domains, and target models. Furthermore, our ablation study underscores the Corrector’s capacity to provide substantial and heterogeneous improvements to existing uncertainty estimation techniques.

Limitations

Although the CUE method proposed in this paper demonstrates good performance, its dependence on labeled data and its generalization ability across different data domains and target models may be limitations. We only compared our method with works that have open-source code, which are often designed for white-box models. Therefore, the effectiveness of our method on black-box models has not been demonstrated through experiments. However, our method does not necessitate access to the inner states of target models, making it a general enhancement strategy for both black-box and white-box uncertainty estimation.

Ethics Statement

In this study, we introduce a method for improving uncertainty estimation in the context of LLMs, which presents no immediate ethical concerns, but certain considerations must be addressed. Uncertainty estimation has significant potential to evaluate the reliability and safety of LLM outputs. However, this potential benefit comes with the risk that systematic mistakes in the uncertainty assessment could foster unfounded and misplaced confidence. Consequently, even re-calibrated uncertainty estimates should be interpreted cautiously, particularly in critical decision-making scenarios where the consequences of inaccuracies can be profound.

The datasets used in our experiment are publicly released and labeled through interaction with humans in English. In this process, user privacy is protected, and no personal information is contained in the dataset. The scientific artifacts that we used are available for research with permissive licenses. And the use of these artifacts in this paper is consistent with their intended use. Therefore, we believe that our research work meets the ethics of ACL.

Acknowledgements

This paper is supported by NSFC project 62476009.

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A Appendix

A.1 Related Work

Uncertainty estimation methods for LLMs have gained significant attention, with approaches can be broadly categorized into logit-based methods, verbalized methods, consistency-based methods, and internal state-based methods..

Logit-based methods Logit-based methods are the most widely used and effective approaches in uncertainty estimation. As a foundational method, Predictive Entropy (PE) (Malinin and Gales, 2020), defines total uncertainty as the entropy of the output logits distribution. After that, researchers proposed a series of methods based on the inherent characteristics of natural language generation to improve upon PE methods. Kuhn et al. (2023) introduced semantic entropy (SE) that estimates
uncertainty by marginalizing over semantically-equivalent samples in NLG tasks. In the similar framework, Nikitin et al. (2024) employed positive semi-definite kernels and von Neumann entropy to capture semantic similarities. Furthermore, Wang et al. (2024) proposed Word-Sequence Entropy (WSE) to adjust uncertainty proportions at both the word and sequence levels based on semantic relevance, ensuring that uncertainty is aligned with the semantic importance of words within a response. In addition to measuring the similarity between generated responses, Wang et al. (2024) proposed to judge the similarity between the target response and the generations. Duan et al. (2023) proposed Shifting Attention to Relevance (SAR), which focus on relevant components and assigns significance weights to tokens based on their contributions to the overall response. Unlike these carefully designed methods, Yaldiz et al. (2024) introduced a Learnable Response Scoring Function (LARS), which utilizes supervised data to capture complex token-probability dependencies. While effective, the above methods are computationally expensive. To alleviate these computational cost, Kossen et al. (2024) proposed Semantic Entropy Probes (SEPs) to approximate semantic entropy by leveraging hidden states from a single generation.

Verbal confidence methods Due to LLMs’ strong language abilities and adherence to instructions, Verbal confidence methods are proposed. For instance, one may attach the question with a prompt like “Please respond and provide your confidence score ranging from 0 to 100.”. Xiong et al. (2023) constructed a prompting, sampling, and aggregation framework to systematically evaluate various strategies and their integration, enabling LLMs to express their confidence in response. Groot and Valdenegro-Toro (2024) proposed FaR prompting strategy, which improves the confidence calibration of LLMs by separating the fact retrieval and reflective reasoning steps. However, verbal confidence methods face significant challenges with over-confidence. Ni et al. (2024) found that LLMs cannot convey their uncertainties faithfully in natural language. Becker and Soatto (2024) found that combining language confidence and proxy model probability estimation can improve the estimation of uncertainty. Madhusudhan et al. (2024) noted LLMs’ language perception accuracy often lags behind probability perception, especially in specific domains Furthermore, Tao et al. (2024) found that LLMs often exhibit a high degree of overconfidence when expressing their own confidence by comparing language-based methods, consistency-based methods, and their hybrid benchmark testing methods. Their research indicates that some prompt strategies can improve the calibration of verbal confidence.

Internal state-based method Internal state-based methods suggest that the activation of the target model can be analyzed to predict the model errors. Azaria and Mitchell (2023) proposed SAPLMA by training a classifier on the hidden layer activations of an LLM to assess statement truthfulness. Similarly, Liu et al. (2024) also introduced a supervised method by training a model on labeled datasets that analyze hidden layer activations and probability-related features. Focusing on the self-assessment capabilities of LLMs, Kadavath et al. (2022) trained models to explore the LLMs’ ability to evaluate the accuracy of their responses through calibration on multiple-choice and true/false questions. Ji et al. (2024) employed a probing estimator to analyze the internal mechanisms of LLMs across various NLG tasks, assessing uncertainty before response generation. Additionally, some works introduced novel interventions to refine the uncertainty estimation performance during inference. Han et al. (2024) proposed to learn from past experience (LePe) method by leveraging historical performance records and fine-tuning instructions. Li et al. (2024a) presented Inference-Time Intervention (ITI) to adjust model activations selectively during inference across a limited number of attention heads, guided by a predefined set of directions. Ulmer et al. (2024) proposes a method to set confidence targets and train an additional model that predicts an LLM’s confidence based on its textual input and output.

Consistency-based method The consistency-based method is to evaluate the uncertainty of the large model through multiple generated answers. Recently, Li et al. (2024b) employed UQ sampling with perturbation and an aggregation module to quantify sampling uncertainty in text generation tasks. Pedapati et al. (2024) proposed a paradigm to reduce overconfidence in incorrect answers by having LLMs reflect on and justify each candidate answer, then aggregating these justifications to calibrate confidence estimates. Becker and Soatto (2024) proposed extracting semantic diversity and syntactic similarity from perturbed prompts, training a model on these features to estimate confidence. Yang et al. (2024) explored the stability of
explanations generated by LLMs to estimate the model’s confidence in its answers. Lin et al. (2023) discussed combining observed consistency and self-reflection to assess language model uncertainty.

Supervised method Notably, this is not an independent classification. For the convenience of comparison, the four categories above that use supervised methods are also summarized here.

Since our research follows a supervised learning approach, we provide a complementary summary of existing supervised methods. Compared with these supervised methods, our work has fundamentally different setup, scope and application timing.

First, Shen et al. (2024); Chaudhry et al. (2024); Kapoor et al. (2024) primarily focus on specific aspects of uncertainty estimation, such as classification tasks or confidence calibration in verbal or probabilistic forms, while our method aims to enhance overall performance rather than addressing isolated uncertainty measures. Second, Azaria and Mitchell (2023); Liu et al. (2024); Kadavath et al. (2022) rely on accessible internal states for prediction, providing insights into the LLM explainability but limiting their applicability to black-box models. In comparison, our Corrector is agnostic to both response content and probability distributions, enabling broader adaptability across diverse settings. Third, Ji et al. (2024); Tao et al. (2024) involve more fine-grained and complex data creation processes, such as probabilistic alignment and other intricate algorithms. In contrast, our method employs a straightforward data creation and training procedure, enabling broad applicability.

A.2 Background and Theory

In this section, we commence by clarifying the two scales of uncertainty: relative uncertainty and absolute uncertainty. We then formalize the relative uncertainty estimation as a classification task to determine whether the target model can correctly respond to a given question. Subsequently, we delve into the theoretical foundations of widely-used logit-based uncertainty estimation methods, and critically examine the inherent limitations shared by those approaches that rely exclusively on target model outputs.

A.2.1 Relative Uncertainty and Absolute Uncertainty

Research on uncertainty estimation has led to two key concepts (Kamath et al., 2020; Vazhentsev et al., 2023): relative uncertainty and absolute uncertainty, each providing distinct methods for assessing and interpreting levels of uncertainty. Given an input \( x \), a ground truth answer \( y \), and the predictive distribution of \( Y \), the predictive uncertainty for the target model regarding the input \( x \) is denoted as \( \mathrm{UE}(x, \theta) \). Relative uncertainty scores emphasize the accuracy of sample ranking, especially in discerning questions that the target model can correctly respond to from those it struggles with. Ideally, for every pair \( (x_i, y_i) \) and \( (x_j, y_j) \) with their predictive distributions \( Y_i \) and \( Y_j \), we should have
\[
\mathrm{UE}(x_i, \theta) \leq \mathrm{UE}(x_j, \theta) \iff P(Y_i = y_i|x_i, \theta) \geq P(Y_j = y_j|x_j, \theta).
\]
(5)

Stricter than relative uncertainty scores, absolute uncertainty scores support to represent the model’s accuracy. In cases where there is an 80% uncertainty prediction, it implies that the question is expected to be answered correctly only 20% of the time under similar conditions. This relationship can be mathematically expressed as
\[
P(Y = y|\mathrm{UE}(x, \theta) = q) = 1 - q.
\]
(6)

As relative uncertainty concerns solely with the relative rankings of \( h(x) = \mathrm{UE}(x, \theta) \), it can be framed as a classification problem aimed at finding a function \( h \) that minimizes the expected loss of misclassification (Allikivi et al., 2024; Tao et al., 2023). Consider two class labels, \( \mathcal{C} = \{c_0, c_1\} \), indicating whether the target model can correctly answer the question or not, respectively. This leads to the formulation of a decision rule
\[
g(h; \tau) = \begin{cases}
c_0 & \text{if } h(x) \leq \tau \text{ (confident)} \\
c_1 & \text{if } h(x) > \tau \text{ (uncertain)}
\end{cases},
\]
(7)

where \( h(x) \) is a scalar measure of uncertainty and \( \tau \) is the threshold.

Drawing from decision theory, we derive the expected loss as conditional risk for the sample \( x \):
\[
\mathrm{Risk}(x) = \lambda_{c_i, c_{1-i}} h_{c_{1-i}}(x),
\]
(8)

where \( c_i, i \in \{0, 1\} \) denotes the true label of the sample \( x \), and \( h_{c_{1-i}}(x) = P(c_{1-i} | x) \) is the posterior probability of misclassifying the sample \( x \) as class \( c_{1-i} \). \( \lambda_{c_i, c_{1-i}} \) represents the loss associated with this misclassification—specifically, a penalty incurred when the sample with the label \( c_i \) is classified as \( c_{1-i} \). Our task is to find \( h^* \) that minimizes the overall risk
\[
\mathrm{Risk}(h) = \mathbb{E}_x \left[ \mathrm{Risk}(h(x)) \mid x \right].
\]
(9)
A.2.2 Theoretical Foundations of Uncertainty Estimation for LLM

LLMs typically generate outputs in an auto-regressive manner, which iteratively predict the probability distribution of the subsequent token based on the evolving context (Gregor et al., 2014). Given an input sequence \( x \) with the objective of generating an output sequence \( y = \{y_1, y_2, \ldots, y_L\} \), the conditional probability of the \( l \)-th token \( y_l \) is denoted as \( P(y_l | y_{<l}, x; \theta) \). This probability depends on all previously generated tokens \( y_{<l} = \{y_1, y_2, \ldots, y_{l-1}\} \) as well as the input \( x \). The probability of generating the entire sequence \( y \) can be expressed as the product of the conditional probabilities of each individual token:

\[
P(y|x; \theta) = \prod_{l=1}^L P(y_l | y_{<l}, x; \theta),
\]

where \( P(y_l | y_{<l}, x; \theta) = \frac{e^{z_l/T}}{\sum_j e^{z_j/T}} \), \( z \) is the raw logit, and \( T \) is the temperature that controls the smoothness of the probability distribution. This posterior probability provides a probabilistic framework for sequence generation. Moreover, according to prior research (Malinin and Gales, 2020), the total uncertainty for the generation of \( y \) is given by the entropy of the predictive posterior:

\[
\begin{align*}
\mathrm{PE}(x) &= \mathcal{H}[P(y \mid x, \theta)] \\
&= \mathbb{E}_{P(y|x,\theta)}[-\ln P(y \mid x, \theta)] \\
&= -\sum_{y \in Y} P(y \mid x, \theta) \ln P(y \mid x, \theta).
\end{align*}
\]

In practice, due to the exponential computational complexity of traversing the entire response set, Monte Carlo approximation method (Papadopoulos and Yeung, 2001) is employed via beam search with a single target model for generation. The approximate entropy is defined as

\[
PE(x) \approx -\frac{1}{B} \sum_{b=1}^B \ln P(y_b|x, \theta),
\]

where \( P(y_b|x, \theta) \) denotes the posterior probability of the \( b \)-th beam search candidate. Base on these, Kuhn et al. (2023) proposed to cluster generations with similar meanings and compute entropy using the probabilities associated with each semantic cluster. This approach is formulated as

\[
SE(x, \theta) = -\frac{1}{C} \sum_{i=1}^C \ln P(c_i|x, \theta),
\]

where \( c_i \) denotes each semantic cluster and \( C \) represents the set of all clusters.

Another form of improvement is to assign weights to each token in the generation when calculating posterior probabilities (Duan et al., 2023; Bakman et al., 2024), either through a manually designed algorithm or a training way, which can be formulated as

\[
\tilde{P}(y \mid x; \theta) = \prod_{l=1}^L P(y_l \mid y_{<l}, x; \theta) \cdot w_l,
\]

where \( w_l \) represents the weight assigned to the \( l \)-th token.

A.3 Compare with Other Supervised UE Method

Following publicly available code and experimental settings, we compare our Corrector’s performance with the supervised UE method provided by Liu et al. (2024), which we refer to as Wb-S. We focused on enhancement after applying the supervised method to the strong unsupervised baselines, targeting model LLaMA-3-8B-Instruct on dataset TriviaQA.

As shown in Table 3, our Corrector achieved harmonized improvements on all strong baselines. However, when we replaced Corrector with the Wb-S method, integrated into our pipeline, only marginal improvements were observed with the Predictive Entropy (PE) method, and no significant effects were noted with other unsupervised methods, and thus the metrics are almost indistinguishable from vanilla’s after using the Wb-S method.

A.4 Generalization

The previous results indicate the Corrector’s effectiveness on the in-distribution evaluation set. In the subsequent analysis, we investigate its cross-dataset and cross-model generalization capabilities.

Cross-Dataset Generalization  In terms of cross-dataset (different domain) generalization, we tested the Corrector’s generalization via cross-training (training on TriviaQA and then testing on SciQA, and training on SciQA and then testing on TriviaQA, with OPT-2.7B as the target model). These datasets differ significantly in question domain (trivia vs. scientific) and size (10,000 vs. 1,000 examples), presenting a significant generalization
<table>
  <tr>
    <th rowspan="2">Method</th>
    <th colspan="3">AUROC (↑)</th>
    <th colspan="3">ECE (↓)</th>
    <th colspan="3">F1 (↑)</th>
  </tr>
  <tr>
    <th>Vanilla</th>
    <th>+Corrector</th>
    <th>+Wb-S</th>
    <th>Vanilla</th>
    <th>+Corrector</th>
    <th>+Wb-S</th>
    <th>Vanilla</th>
    <th>+Corrector</th>
    <th>+Wb-S</th>
  </tr>
  <tr>
    <td>PE</td>
    <td>64.52</td>
    <td>69.76</td>
    <td><b>64.37</b></td>
    <td>21.38</td>
    <td>17.24</td>
    <td><b>20.18</b></td>
    <td>47.54</td>
    <td><b>64.97</b></td>
    <td>52.45</td>
  </tr>
  <tr>
    <td>LN-PE</td>
    <td>72.55</td>
    <td>74.79</td>
    <td><b>72.75</b></td>
    <td>14.31</td>
    <td>11.53</td>
    <td><b>14.31</b></td>
    <td>52.83</td>
    <td><b>61.99</b></td>
    <td><b>52.83</b></td>
  </tr>
  <tr>
    <td>SE</td>
    <td>80.92</td>
    <td>82.12</td>
    <td><b>80.92</b></td>
    <td>13.07</td>
    <td>12.76</td>
    <td><b>13.07</b></td>
    <td>57.14</td>
    <td><b>65.75</b></td>
    <td><b>57.14</b></td>
  </tr>
  <tr>
    <td>SAR-t</td>
    <td>79.55</td>
    <td>79.93</td>
    <td><b>79.55</b></td>
    <td>16.40</td>
    <td>13.70</td>
    <td><b>16.40</b></td>
    <td>51.67</td>
    <td><b>57.60</b></td>
    <td><b>51.67</b></td>
  </tr>
  <tr>
    <td>SAR-s</td>
    <td>69.87</td>
    <td>77.09</td>
    <td><b>69.87</b></td>
    <td>23.17</td>
    <td><b>20.00</b></td>
    <td><b>23.17</b></td>
    <td>53.44</td>
    <td><b>66.22</b></td>
    <td><b>53.44</b></td>
  </tr>
  <tr>
    <td>SAR</td>
    <td>80.92</td>
    <td>81.90</td>
    <td><b>80.92</b></td>
    <td>16.17</td>
    <td>13.76</td>
    <td><b>16.17</b></td>
    <td>51.20</td>
    <td><b>59.54</b></td>
    <td><b>51.20</b></td>
  </tr>
</table>

Table 3: Comparison of performance between the Corrector and the Wb-S method.

challenge. We observed average absolute improvements of 4%-6% over baselines.

To further investigate cross-dataset (same domain) generalization, we conducted additional experiments using MedMCQA (Pal et al., 2022) and MedQA (Jin et al., 2020) (distinct but related medical datasets). Using the same setup as different domain, we observed absolute improvements of 7-11%. This demonstrates promising generalization within the same domain. The results are shown in the Table 4a.

Cross-Model Generalization In terms of cross-model generalization, we tested the Corrector’s generalization across models (OPT-2.7B, OPT-6.7B, LLaMA-3-8B) on SciQA, reporting average AUROC improvement over all baselines.

Experimental results show the Corrector generalizes well within the same model family (e.g., OPT-2.7B and OPT-6.7B, with average absolute improvements of 6%-11%), likely due to their similar performance on the original SciQA dataset. We observed limited transferability across LLMs with significantly different performance and architectures (e.g., LLaMA-3-8B and OPT-2.7B). However, even in these scenarios, CUE achieves an average absolute improvement of 3%, which demonstrates the efficacy of our method.

A.5 Statistical Hypothesis Testing

Regarding the performance improvements compared to other robust UE methods, our approach provided harmonized, multi-dimensional enhancements across various aspects of uncertainty estimation, including indication, balance, and calibration. To statistically validate the significance of these improvements across all metrics, we conducted t-tests on the TriviaQA and SciQA datasets, comparing our method against strong baselines (SE, t-SAR, s-SAR, SAR). The results yielded p < 0.05 for each baseline on both datasets, demonstrating that the performance improvements were statistically significant.

It was also essential to clarify that the reported performance of the SE and SAR methods reflected their saturation point. This indicated that further increasing the number of samples—commonly used to enhance their performance—no longer resulted in additional gains. In contrast, our method surpassed this saturation point, effectively addressing the limitations of these methods and delivering continued improvements.

A.6 Detail of Hyperparameter w*

A.6.1 Sensitivity of Hyperparameter w*

To address test the sensitivity of the hyperparameter \( w^* \), we conducted a sensitivity analysis by performing tests on the opt-6.7b and llama3-8b models using the TriviaQA dataset. We adjusted the \( w^* \) values around the original optimal value in thousandth-increments and recorded the AUROC performance under different \( w^* \) values. When the performance change was within 1%, we recorded this range as the “stable range”. Table illustrates the \( w^* \) stable ranges for various UE methods. We observed that, in the majority of cases, performance fluctuations remained below 1% within a \( w^* \) range of 0.107 to 0.442, indicating a degree of robustness in the model to \( w^* \).

A.6.2 Hyperparameter w* Setting in Cross-domain Experiments

By default, we select the optimal weight \( w^* \) using the same (training) domain dev set. However, as shown in the supplementary experiments addressing R1, \( w^* \) exhibits elasticity. Thus, theoretically, using \( w^* \) from a truly cross-domain set would partially retain the Corrector’s enhancement effect.

We validated the cross-domain experiment setup (train on SciQA/TriviaQA and test on TriviaQA/SciQA) mentioned by the reviewer, selecting \( w^* \) using both in-domain and cross-domain
<table>
  <tr>
    <th></th>
    <th colspan="2">TriviaQA</th>
    <th colspan="2">SciQA</th>
    <th colspan="2">MedMCQA</th>
    <th colspan="2">MedQA</th>
    <th colspan="2">OPT-2.7B</th>
    <th colspan="2">OPT-6.7B</th>
    <th colspan="2">LLaMA3-8B</th>
  </tr>
  <tr>
    <th>TriviaQA</th>
    <td>19.59</td>
    <td>4.05</td>
    <th>MedMCQA</th>
    <td>15.54</td>
    <td>10.70</td>
    <th>MedQA</th>
    <td>6.93</td>
    <td>11.21</td>
    <th>OPT-2.7B</th>
    <td>19.59</td>
    <td>11.80</td>
    <td>3.23</td>
    <th>OPT-6.7B</th>
    <td>6.08</td>
    <td>11.21</td>
    <td>3.43</td>
  </tr>
  <tr>
    <th>SciQA</th>
    <td>6.03</td>
    <td>10.20</td>
    <th>MedQA</th>
    <td>6.93</td>
    <td>11.21</td>
    <th></th>
    <th></th>
    <th></th>
    <th></th>
    <th></th>
    <th></th>
    <th></th>
    <th></th>
    <th></th>
    <th></th>
    <th></th>
  </tr>
</table>

(a) Generalization for Domain of Data

(b) Generalization for Target Model

Table 4: Average AUROC scores (%) improvement of after applying our method to baselines. (a) The leftmost column indicates the domains of data used in training, while the topmost row represents the domains of data used for evaluating, with OPT-2.7B serving as the target model. (b) The leftmost column denotes the target model during training, whereas the topmost row signifies the target model during evaluating, with TriviaQA utilized as the target domain of data.

<table>
  <tr>
    <th rowspan="2">Method</th>
    <th colspan="3">TriviaQA AUROC↑</th>
    <th colspan="3">TriviaQA ECE ↓</th>
    <th colspan="3">SciQA AUROC↑</th>
    <th colspan="3">SciQA ECE ↓</th>
  </tr>
  <tr>
    <th>Vanilla</th>
    <th>w/ Corrector</th>
    <th>Improv</th>
    <th>Vanilla</th>
    <th>w/ Corrector</th>
    <th>Improv</th>
    <th>Vanilla</th>
    <th>w/ Corrector</th>
    <th>Improv</th>
    <th>Vanilla</th>
    <th>w/ Corrector</th>
    <th>Improv</th>
  </tr>
  <tr>
    <td>Correcter</td>
    <td>69.87</td>
    <td>-</td>
    <td>-</td>
    <td>6.73</td>
    <td>-</td>
    <td>-</td>
    <td>65.38</td>
    <td>-</td>
    <td>-</td>
    <td>18.19</td>
    <td>-</td>
    <td>-</td>
  </tr>
  <tr>
    <td>LS</td>
    <td>19.57</td>
    <td>69.82</td>
    <td>50.25</td>
    <td>70.25</td>
    <td>7.41</td>
    <td>-62.84</td>
    <td>53.67</td>
    <td>65.38</td>
    <td>11.71</td>
    <td>38.64</td>
    <td>18.19</td>
    <td>-20.45</td>
  </tr>
  <tr>
    <td>VC</td>
    <td>62.34</td>
    <td>74.89</td>
    <td>12.55</td>
    <td>23.41</td>
    <td>16.78</td>
    <td>-66.63</td>
    <td>68.22</td>
    <td>72.15</td>
    <td>3.93</td>
    <td>31.88</td>
    <td>19.47</td>
    <td>-12.36</td>
  </tr>
  <tr>
    <td>P(True)</td>
    <td>57.14</td>
    <td>72.29</td>
    <td>15.15</td>
    <td>24.67</td>
    <td>19.84</td>
    <td>-4.83</td>
    <td>65.63</td>
    <td>71.41</td>
    <td>5.78</td>
    <td>34.56</td>
    <td>31.92</td>
    <td>-2.64</td>
  </tr>
  <tr>
    <td>PE</td>
    <td>64.52</td>
    <td>69.76</td>
    <td>5.24</td>
    <td>21.38</td>
    <td>17.24</td>
    <td>-4.13</td>
    <td>66.54</td>
    <td>67.98</td>
    <td>1.44</td>
    <td>40.67</td>
    <td>34.07</td>
    <td>-6.60</td>
  </tr>
  <tr>
    <td>LN-PE</td>
    <td>72.55</td>
    <td>74.79</td>
    <td>2.24</td>
    <td>14.31</td>
    <td>11.53</td>
    <td>-2.79</td>
    <td>69.48</td>
    <td>71.56</td>
    <td>2.08</td>
    <td>29.38</td>
    <td>23.76</td>
    <td>-5.62</td>
  </tr>
  <tr>
    <td>SE</td>
    <td>80.92</td>
    <td>82.12</td>
    <td>1.20</td>
    <td>13.07</td>
    <td>12.76</td>
    <td>-0.31</td>
    <td>71.59</td>
    <td>72.93</td>
    <td>1.34</td>
    <td>30.54</td>
    <td>25.23</td>
    <td>-5.31</td>
  </tr>
  <tr>
    <td>SAR-t</td>
    <td>79.55</td>
    <td>79.93</td>
    <td>0.38</td>
    <td>16.40</td>
    <td>13.70</td>
    <td>-2.70</td>
    <td>72.26</td>
    <td>73.87</td>
    <td>1.61</td>
    <td>30.37</td>
    <td>26.81</td>
    <td>-3.56</td>
  </tr>
  <tr>
    <td>SAR-s</td>
    <td>69.87</td>
    <td>77.09</td>
    <td>7.22</td>
    <td>23.17</td>
    <td>20.00</td>
    <td>-3.17</td>
    <td>74.96</td>
    <td>75.72</td>
    <td>0.76</td>
    <td>38.54</td>
    <td>36.18</td>
    <td>-2.37</td>
  </tr>
  <tr>
    <td>SAR</td>
    <td>80.92</td>
    <td>81.90</td>
    <td>0.98</td>
    <td>16.17</td>
    <td>13.76</td>
    <td>-2.41</td>
    <td>73.88</td>
    <td>75.19</td>
    <td>1.31</td>
    <td>28.97</td>
    <td>25.60</td>
    <td>-3.37</td>
  </tr>
</table>

Table 5: Comparison of Pure Corrector Performance with the Baseline, and the Performance Gains from Using Corrector Scores as Corrections to the Baseline.

<table>
  <tr>
    <th rowspan="2">Method+Cor</th>
    <th colspan="2">OPT-6.7B</th>
    <th colspan="2">LLaMA3-8B</th>
  </tr>
  <tr>
    <th>w* Range</th>
    <th>Range Difference</th>
    <th>w* Range</th>
    <th>Range Difference</th>
  </tr>
  <tr>
    <td>SAR</td>
    <td>0.12 - 0.362</td>
    <td>0.242</td>
    <td>0.367 - 0.809</td>
    <td>0.442</td>
  </tr>
  <tr>
    <td>SAR-s</td>
    <td>0.028 - 0.427</td>
    <td>0.399</td>
    <td>0.539 - 0.844</td>
    <td>0.305</td>
  </tr>
  <tr>
    <td>SAR-t</td>
    <td>0.038 - 0.318</td>
    <td>0.28</td>
    <td>0.447 - 0.733</td>
    <td>0.286</td>
  </tr>
  <tr>
    <td>SE</td>
    <td>0.246 - 0.51</td>
    <td>0.264</td>
    <td>0.701 - 0.808</td>
    <td>0.107</td>
  </tr>
  <tr>
    <td>LN-PE</td>
    <td>0.223 - 0.744</td>
    <td>0.511</td>
    <td>0.759 - 0.926</td>
    <td>0.169</td>
  </tr>
  <tr>
    <td>PE</td>
    <td>0.117 - 0.323</td>
    <td>0.206</td>
    <td>0.759 - 0.945</td>
    <td>0.186</td>
  </tr>
  <tr>
    <td>LS</td>
    <td>0.458 - 1.0</td>
    <td>0.542</td>
    <td>0.993 - 1.0</td>
    <td>0.007</td>
  </tr>
</table>

Table 6: Stable Ranges of \( w^* \) for Different UE Methods on TriviaQA (within 1% AUROC change).

validation sets, and recorded the resulting average AUROC improvements across methods. The experimental results are shown in Table 7.

<table>
  <tr>
    <th>Training => Testing Domain</th>
    <th>Same (training) domain \( w^* \)</th>
    <th>Cross-domain \( w^* \)</th>
  </tr>
  <tr>
    <td>SciQA => TriviaQA</td>
    <td>6.03</td>
    <td>3.92</td>
  </tr>
  <tr>
    <td>TriviaQA => SciQA</td>
    <td>4.05</td>
    <td>2.75</td>
  </tr>
</table>

Table 7: Average AUROC improvements (%) across methods when selecting \( w^* \) using in-domain and cross-domain validation sets for cross-domain experiments.

A.7 Pure Corrector Performance

To provide more insights about pure Corrector performance, we combine and present relevant data from Table 1 and Table 2 of our paper, as shown in the table 5.

On TriviaQA, the pure performance of the Corrector outperformed LS, VC, P(True), and PE, but underperformed SE, SAR-t, SAR-s, and SAR. On SciQA, the pure performance of the Corrector outperformed LS, performed similarly to VC, P(True), PE, and LN-PE, but underperformed SE, SAR-t, SAR-s, and SAR. Across all “w/ Corrector” settings, we saw universal improvement, showing Corrector’s great complementarity with unsupervised methods. Regarding the ECE metric, we observed that the calibration of the lightweight model is significantly better than strong UE methods, which makes the combination of the two even more advantageous.

Reviewer paper summaries: The paper proposes combining uncertainty quantification methods with a BERT-like classifier through a weighted sum approach. It provides a complete pipeline and method validation on TriviaQA and SciQA datasets.

This paper focuses on the uncertainty quantification (UQ) for LLMs. The authors propose to incorporating an additional "corrector" to assist the UQ for LLMs. The correctors are small classification models, individually trained to predict the correctness of the answers of the LLMs on in-domain datasets before testing. After get the dataset-specific LLM-specific corrector, the model's uncertainty is combined with the correctness probability from the corrector, leading to a better estimation of the correctness prediction of the model answer. The authors also performed empirical experiment to study the effectiveness of the additional corrector for more accurate uncertainty quantification.

The paper presents CUE, a Corrector for improving the performance of Uncertainty Estimation techniques.

By learning a classifier from the dataset used in the task, the authors show that they can achieve better performance than baseline white box UE methods.

They mention that this helps harmonize indication (reliable or unreliable classification), balance (precision recall balance) and calibration. These claims, though interesting, are perhaps secondary to the corrector CUE introduced by the authors.
Summary of strengths: - The paper is well-written, easy to follow, and offers valuable educational content through its methodical exploration of uncertainty estimation methods
- All components of the method pipeline are reasonable, thoroughly validated, and clearly presented
- The method, despite its simplicity, demonstrates significant performance improvements

1. The idea is straightforward and reasonable. The predictor is designed to predict a particular LLM's answer correctness on a particular dataset. Therefore, it is expected that the proposed method can slightly improve the uncertainty quantification for LLMs.
2. The experiment is comprehensive, and the proposed method demonstrate improvements over some weak baselines.

1. A Corrector for UE methods seems like a neat idea. Especially one that is agnostic to the UE methods.
2. The experiments and the two ablations, the two datasets (TriviaQA, SciQA), the target (OPT 6.7b, LLAMA 3-8b) and base models (Roberta and DeBerta ones) are all reasonable.
3. The discussion around harmonizing indication, balance and calibration is somewhat interesting.
Summary of weaknesses: The core drawback is methodological - the study compares a supervised method against unsupervised baselines, giving the presented approach an inherent advantage. Given the existence of numerous supervised approaches in the research community, the comparison would be more meaningful if it:
Included other supervised methods as baselines
At minimum, presented standalone performance metrics for the corrector classifier
This omission makes it difficult to assess the true contribution of the method relative to the existing state of the art in supervised uncertainty estimation.

1. **Impractical assumption and method design.** The main concern about this paper is the method design, which assumes we can always access the distribution of the testing dataset. More specifically, the corrector requires training on the training set of each evaluation task. However, this is impractical in realistic settings, where the uncertainty quantification is performed on any data. The authors should evaluate the effectiveness of the corrector (and the proposed method) when it is trained on the traning set of one task but evaluated on another unseen dataset, such as train on TriviaQA and testing the proposed method on SciQA. Also, the proposed method is LLM-specific. The authors should also demonstrate the effectiveness when the corrector is trained with one LLM but applied to another LLM for uncertainty quantification.
2. **The performance improvement against strong UQ methods are not significant.** When applied to UQ methods such as SE and SAR, the improvement is not significant enough. 

I would also suggest the authors including the discussion of other related work:

[1] Shen, Maohao, et al. "Thermometer: Towards Universal Calibration for Large Language Models." arXiv preprint arXiv:2403.08819 (2024).

[2] Chaudhry, Arslan, Sridhar Thiagarajan, and Dilan Gorur. "Finetuning language models to emit linguistic expressions of uncertainty." *arXiv preprint arXiv:2409.12180* (2024).

1. Conducting experiments on more tasks besides QA will be useful.
2. I could be wrong, but I don't think the corrector model (a classifier) trained on one dataset can be used on other datasets even if the task is the same. A corrector based on task properties rather than data properties will be more useful.
Comments, suggestions and typos: i feel like eq.1 could be simplified
line 515 missing reference

Please refer to the weakness above. The ablation study on the corrector training is definitely necessary (train on different task/LLMs from testing). Also, the related work I listed is also closely connected with this paper, such as Thermometer for better uncertainty quantification and fine-tuning LLMs for better verbalized uncertainty.

Typos

1. Line 125, follow that should be following that?
2. Line 237, directly optimized should be directly optimize?
3. Line 414, donated should be denoted?
4. Line 515, missing section reference
Overall assessment scores: 2

2.5

3.5
Metareview score: 3
Metareview: The paper presents CUE which is a corrector model that can improve the performance of uncertainty estimation techniques in large language models. These correctors are small classification models, individually trained to predict the correctness of LLM answers. The model's uncertainty is combined with the correctness probability from the corrector, leading to a better estimation of uncertainty. Experiments show the both the simplicity and effectiveness of this corrector model on uncertainty quantification.
Summary of reasons to publish: The reviewers found the paper well-written and the idea straightforward and reasonable. They also found the experiments comprehensive, and the proposed method demonstrates improvements over baselines with solid analysis.