Calibrating Verbalized Confidence with Self-Generated Distractors
Authors: Victor Wang
Elias Stengel-Eskin
Venue: ICLR 2026 Poster (2026)
Keywords: LLM uncertainty
confidence calibration
verbalized confidence
Primary area: foundation or frontier models, including LLMs

Abstract: Calibrated confidence estimates are necessary for large language model (LLM) outputs to be trusted by human users. While LLMs can express their confidence in human-interpretable ways, verbalized LLM-generated confidence scores have empirically been found to be miscalibrated, reporting high confidence on instances with low accuracy and thereby harming trust and safety. We hypothesize that this overconfidence often stems from a given LLM’s heightened suggestibility when faced with claims that it encodes little information about; we empirically validate this hypothesis, finding more suggestibility on lower-accuracy claims. Building on this finding, we introduce Distractor-Normalized Coherence (DINCO), which estimates and accounts for an LLM’s suggestibility bias by having the model verbalize its confidence independently across several self-generated distractors (i.e. alternative claims), and normalizes by the total verbalized confidence. To further improve calibration, we leverage generator-validator disagreement, augmenting normalized validator confidence with a consistency-based estimate of generator confidence. Here, we frame the popular approach of self-consistency as leveraging coherence across sampled generations, and normalized verbalized confidence as leveraging coherence across validations on incompatible claims, allowing us to integrate these complementary dimensions of coherence into DINCO. Moreover, our analysis shows that DINCO provides less saturated – and therefore more usable – confidence estimates, and that further sampling alone cannot close the gap between DINCO and baselines, with DINCO at 10 runs outperforming self-consistency at 100. We release our code at https://github.com/victorwang37/dinco.

Published as a conference paper at ICLR 2026
CALIBRATING VERBALIZED CONFIDENCE WITH SELF- GENERATED DISTRACTORS
Victor Wang Elias Stengel-Eskin
The University of Texas at Austin
ABSTRACT
Calibrated confidence estimates are necessary for large language model (LLM) outputs be trusted by human users. While LLMs can express their confidence in human-interpretable ways, verbalized L.LM-generated confidence scores have empirically been found be miscalibrated, reporting high confidence on instances with low accuracy and thereby harming trust and safety. We hypothesize that this overconfidence often stems from a given LLM's heightened suggestibility when faced with claims that it encodes little information about; empirically validate this hypothesis, finding more suggestibility on lower-accuracy claims. Building this finding, introduce Distractor-Normaliiee Coherence (DiNCo). which estimates and accounts for an LLM's s suggestibility bias by having the model ver- balize its confidence independently across several self-generated distractors (i.e. alternative claims), and normalizes by the total verbalized confidence. To further improve calibration, we leverage renerator-validate disagreement, augmenting normalized validator confidence with a consisteney-basee estimate of generator confidence. Here, we frame the popular approach of self consistees as leverag- ing coherence across sampled generations, and normalized verbalized confidence as leveraging coherence across validations on incompatible claims, allowing us to integrate these complementary dimensions of coherence into DINCO. Moreover, our analysis shows that DINCO provides less saturated and therefore more us- able confidence estimates, and that further sampling alone cannot close the gap between DINCO and baselines, with DINCO at 10 inference calls outperforming self-consistency at 100.'
INTRODUCTION
LLMs encode a vast amount of knowledge in their parameters, demonstrating superhuman perfor- mance on knowledge-intensing benchmarks (Comanici et al., 2025; OpenAI, 2023). Users often rely on information obtained from these models to make important decisions, but this information is not always accurate. Thus, we seek to qualify LLM responses with confidence estimates that are calibrated, i.c. match the probability LLMs off the shelf without task-specific or model model-specific tuning (Manakul et al., 2023; Geng al., 2024; Feng et al., 2024; Shorinwa et al., 2025), motivating the development of confidence estimation methods that work in off-the-shelf settings both gray-box settings with logit access, and black-box settings with only textual input and output.
correctness. Users and agentic frameworks often use
In these settings, verbalized confidence is simple and commonly-used approach that prompts the model to report its confidence in an answer (Lin et al., 2022; Xiong et al., 2024; Wei et al., 2024). For blanket term for (1) asking the model to decode numer-
brevity, we use verbalized confidence ical confidence in text like "80%" (Tian et al., 2023) and (2) asking the model whether an answer is correct and taking the token probability P(Truc) (Kadavath et al., 2022). Verbalized confidence appealing for several reasons, including that it resembles one way humans express confidence, mak- ing easy to interpret and integrate into decision -heoreetcc frameworks (Sun et al., 2025; Steyvers et al., 2025). However, verbalized confidence has several drawbacks. First, it empirically tends to exhibit overconfidence (Tian et al., 2023; Xiong et al., 2024; Wei et al., 2024; Xu et al., 2025);
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Verbalized confidence
DINCO
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Interpolation from 0) to (0.24, 0.83] with intermediate thresholds
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Figure 1: Calibration metrics (Expected Calibration Error 4, Brier score area under the ROC curve t; see Appendix B.I) with Qwen3-8B on TriviaQA using P(True) as verbalized confidence. (Left) Verbalized confidence is saturated high confidence, despite a much lower accuracy. For each bar, we label the number of instances whose confidence falls in the interval and we darken larger bins. (Center) DINCO normalizes by the total confidence over candidate answers, relieving saturation and improving calibration. (Right) Since verbalized confidence is saturated at high confidence, it is unable to achieve an acceptable true positive rate (TPR) without incurring a significant false positive rate (FPR) of 0.24. In other words, no rejection threshold can be chosen to reject high proportion of false claims. Meanwhile, DINCO enjoys better granularity, ranking positives above negatives even among instances with verbalized confidence of 1.
Fig. (left) shows that verbalized confidence scores generally outstrip average accuracy within a confidence bin.
We highlight a second underexplored factor that makes verbalized confidence suboptimal: confi- dence saturation, wherein the model's reported scores tend to fall into few bins, making them uninformative. While this might still lead to an acceptable calibration error, it results in "jumpy" curves, as in Fig. (right), where no confidence threshold that accepts least one claim can avoid accepting a substantial proportion of false claims. To address these shortcomings, we introduce DINCo, which leverages incoherence in verbalized confidence across related claims to detect over- confidence. DINCO is motivated by the intuition that incoherent confidence scores, e.g., high verbalized confidence in an answer to a question when other distinct answers also have high ver- balized confidence, should not be taken at face value. In other words, we should discount high confidence if it does not follow rational coherence norms (Hofweber et al., 2024).
To explain and correct for this kind of incoherence, we first define the notion of suggestibility. Some studies indicate that when LLMs are epistemically uncertain, they tend to rely on their context to claim
resolve the uncertainty (Yadkori et al., 2024; Ahdritz et al., 2024), i.e., the confidence increases because it is in the context. We refer to this phenomenon as suggestibilityy and hypothesize that it contributes to the model's assignment of high confidence claims it can neither support refute. We introduce this hypothesis in Section 2.1 and provide empirical support in Section 2.2. To account for this suggestibility bias, we propose a method for calibrating verbalized confidence that normalizes by the total confidence over self-generated distractors (i.e. alternative claims). We generate minimal pair distractors using beam search when available, model for distractors in the black-box setting. Crucially, we use an fff-the-sheit NLI model to downweight distractors that are similar to other distractors or that do not contradict the main claim,
by directly prompting the
The approach above for normalizing verbalized confidence with distractors aims to leverage coher- ence within claim validation, but overlooks another relevant facet of coherence in LLMss. In partic- ular, coherence among sampled generations is correlated with correctness, an observation leveraged by the popular approach of self-consistency (Xiong et al., 2024). Thus, inspired by prior findings on generator-van vallidato disagreement (Li et al., 2024), we integrate these complementary dimensions of coherence into DINCo. Specifically, we use distractor generation and NLI reweighting to estimate and enforce coherence across validations of related claims (e.g. not accepting contradictory claims), while using self-consistency to quantify coherence across sampled generations, upweighting more commonly generated claims.
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We test our method open-source and closed-soure, models, applied short-form (TriviaQA and SimpleQA) (Joshi et al., 2017; Wei et al., 2024) and long-form (FactScore; Min et al., 2023) generation domains. DINCO improves ECE over the best bascline by average of 0.077, 0.092, and 0.055, respectively (note that the best baseline differs between the short-form and long-form settings). DINCO effectively extends to long-form biography generation, where it improves Pearson and Spearman correlation with passage-level FactScore over the best baseline by an average of 0,072 and 0.074, respectively. Further analysis shows that DINCO relieves confidence saturation, and that simply scaling up self-consistency (the strongest baseline overall) does not suffice to match the calibration of DINCO.
COHERENCE (DINCo)
We begin with a motivating hypothesis, supported with preliminary evidence. Then we present the details of our method, illustrated in Fig. 2.
2.1 BACKGROUND AND MOTIVATION
Let C be the set of claims with a binary truth value. We denote the truth value of claim e E C as v(() E {0, 1}. confidence estimation method is a function f C + [0, 1]. which is calibrated it correctly predicts the probability of truth. Verbalized confidence is an approach that prompts an LLM to output its confidence (V((e) (c) claim e.²
For topic that the model knows little about, it may be willing to adopt the information presented in its context as its prior (Yadkori et al., 2024; Ahdritz et al., 2024), a phenomenon we refer to as suggestibility. In other words, the very act of presenting claim for the model to report its confidence on can bias the reported confidence. For example, if the model does not know who Kang Ji-hwan is, it may assign 60% confidence both the claim "Kang Ji-hwan was born in 1980." and the claim "Kang Ji-hwan was born in 1990.". even though this seemingly violates coherence norms (since the claims mutually exclusive). Nonetheless, this behavior may not be strictly irrational, each confidence estimate is conditioned on different information, namely the fact that the respective claim was verbalized in the user prompt. In other words, the very fact that the claim has been provided in the input might lend credence to the claim. We further discuss the connection between this notion, LLM sycophancy, and human suggestibility in Appendix A.
We seek to correct for this bias caused by the model's suggestibility when presented with claims knows little about. Let fVC be the verbalized confidence, and let flat be the latent, inaccessible model confidence. We model the bias as multiplicative scalar (c). which depends on the claim e because the model has varying degrees of uncertainty for different topics: fVC (c) ((c))latt (c). To approximate flat we make the assumption that the biases for logically related (e.g., equivalent or contradictory) claims are approximately cqual, since they rely on a shared, localized set of knowl- edge. Let C' C C be set of mutually exclusive and exhaustive claims, c.g. claims for the year person was born in. Since the claims in C' are logically related, we assume that ((c) is roughly the same for all C and there is scalar 3(C') with ((C) 3(c) for all C. Assuming the latent confidence plat is probabilistically coherent,
(C)(c)
(C
1= 1=Σ
(c) Σ
Σ
(1)
B(c)
CEC 3(C)
CEC
EEC
Thus, we can approximate 3(C') and then clat.
(VC ((C)
B(C) R Σ (C).
p plat(c)
(2)
CEC
In practice, we set 3(C) max 3(C)) to account for the case where C fails to contain true claim, or more precisely, a claim that the model believes to be true.
We talk about what the model knows, believes, has confidence in, etc. as short-hand notation for the latent ability to produce language similar to that which a human would to demonstrate such knowledge, etc. (Piantadosi & Hill, 2022; West et al., 2024; Hofweber et al., 2024).
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Downweight redundant distractors with P(entail)
Downweight non-contradictory
LLM
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distractors
Main claim
Kang Ji- Ji-hwan highly acclaimed.
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Kang Ji-hwan is widely ridiculed.
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Distractor
Kang Ji-hwan is notoriously disliked.
0.27
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Kang Ji- Ji-hwan is rarely seen.
1.00
avver age
monsin
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Normalized verbalized confidence:
0.34
2.98
Inverse
VCmain + VCC ? Minique i Weontrall ||
0:93
0.69
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Vunique
Normalizing verbalized confidence with DINCo. (Left) The LLM generates claim along with several distractors and reports its confidences on them independently. To calibrate the main claim's confidence, we divide it by 3, the sum over each distractor's confidence, weighted by uniqueness (center) and counterfactuality (right). We seek distractors that are minimal pairs with the main claim, i.c. similar statements that likely contradict the main claim. The distractors shown here are real examples generated by an LLM, and while they happen to share a lexical structure with the main claim, we do not prescribe precise form for generating distractors; see Section 2.3 for method details, Appendix B.2 for prompts, and Appendix C.1 for more examples and analysis.
Figure 2:
2.2 PRELIMINARY STUDY
To empirically support our motivation above, we plot the distributions of the total confidence (B(C)) in Eq. 2) over correctly and incorrectly answered questions. In this preliminary study, we take cor- rectness proxy for epistemic certainty, although in reality LLMs may still be uncertain about questions they answered correctly. In other words, we treat correct and incorrect instances as epis- temically certain and uncertain instances, respectively. Thus, rect instances, the model would be more suggestible and assign high confidence to more answers, leading to higher total confidences. On the other hand, if the model is calibrated (in particular, on uncertain instances, it exhibits epistemic humility, i.e. recognizing its lack of knowledge), then the total confidence would tend to be around I and 0 for correct and incorrect instances, respectively, so the total confidence would be lower on incorrect instances. If the model is confident in its incor- rect answers (e.g. due to misconceptions), we would expect similar behavior between correct and incorrect instances.
hypothesis predicts that incor-
Experimental Setup. We use TriviaQA (Joshi et al., 2017), a dataset evaluating real-world knowledge with short-form answers. We sample 1000 questions from the validation split of the rc. www.noconttttt subset. We use Qwen3-8B (Yang et al., 2025)² and take verbalized confidence using P(True) (Kadavath et al., 2022), computed P(Yes)/ + P(No)) when asking the model whether a given answer is correct (see Appendix B.2 for prompts). For each question, we generate 10 claims, the total confidence is a number from 0 to 10. In Section 2.3, we specify how the distractors are generated and how we avoid overcount- ing redundant answers in the total confidence (Eq. 4). The total confidence B(C) computed here matches how the NVC method and the NVC component of DINCO (but with fewer distractors) will computed for our main ex- periments in Section 3.
120
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dersity
100
correct (p-1.933
incorrect (u=2.66))
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3
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Total con confidence
Results. In Fig. 3, we first observe that some incorrectly answered questions have to-
Figure 3: Total confidence for correctly and incor- rectly answered questions.
"Trouuhhuu, use the instruction-tuned versions of Qwen3 models.
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tal confidence near O, suggesting some epistemic humility. Even so, the incorrect distribution is heavy-tailed, resulting in a higher mean and median than the correct distribution. These results are consistent with hypothesis that LLMs are more prone to accepting claims that they are epis- temically uncertain about. To explain why the model experiences suggestibility even on correct instances (shown by the substantial proportion of correct instances with a total confidence greater than 1), we note that educated guesses can often be correct while still bearing epistemic uncertainty. Despite our usage of correctness that the model tends to report higher confidence on claims for incorrectly answered questions. We verify the same trend on SimpleQA (Wei et al., 2024) and biography generation (Min et al., 2023) in Appendix D.I.
noisy proxy of epistemic certainty, we identify the clear trend
2.3 MЕТHOО
Our preliminary result (Fig. 3) shows that LLMs can produce incoherent probability judgments (i.e. the total confidence ((C)) exceeds 1), especially when epistemically uncertain, suggesting need to estimate and correct for this bias. As proposed in Section 2.1, we normalize verbalized confidence by the total confidence B(C)) over a distractor set C. We now describe in detail how we generate these distractors, and how we account for redundancy among distractors, a process illustrated in Fig. 2. Finally, we discuss how the phenomenon of generator-validat validator disagreement motivates incorporating self-consistency into DINCO.
Distractor generation. The distractor set should contain enough plausible distractors to avoid un- derestimating the normalization factor (B(C) in Eq. 2), while remaining small enough be tractably computed. Thus, we frame the problem of choosing the optimal distractor set containing an original claim Co as maximizing the total acceptance probability Eeec (VC subject to a size constraint |C| K; shortly we address relaxing the requirement of mutual exclusivity. Unfortunately, the validation probability (c) can only be elicited on a per-claim basis, leading to intractable sam- ple complexity. Motivated by the intuition that LLM tends to generate claims that it would find plausible during validation, as a proxy for the set of claims with high verbalized confidence, we use the set of claims with high generation probability (under an appropriate prompt, explained next).
To encourage mutual exclusivity among claims, we set up the claims to be minimal pairs (Fig. 2 left). For given short-form question, we can simply sample many answers, but independent sampling is inefficient as it overrepresents high probability generations (Gekhman et al., 2025), limiting the number of unique distractors. Instead, we use beam search (Sutskever et al., 2014) when available to efficiently identify unique sequences that have high probability mass coverage. For API-access models, we use top token probabilities if available to implement a pseudo-beam search (see Ap- pendix C.2 for details). Otherwise in the black-box setting, we directly prompt the model to generate a list of candidate answers (see Appendix B.2 for prompts). For long-form QA, we follow Min et al. (2023) in decomposing a long generation into claims. We separately prompt the model to generate one distractor for a given claim, and again use beam search to create multiple distractors. We provide examples and analysis of the distractors generated with each of these methods in Appendix C.I.
Addressing Claim Redundancy. Although the heuristic of generating minimal pairs encourages mutually exclusive claims, we have little guarantee of this mutual exclusivity (1) within the distractor set C' and (2) between distractors and the original claim. Assuming such mutual exclusivity when there are actually redundant claims can lead to overcounting in the normalization factor 3CC). Thus, we use an NLI model to quantify entailment and contradiction relationships between claims." 4 We address (1) and (2) with Wunique and Wcontra. respectively (Fig. 2):
Wunique (c) = Eeec P(entail c',c)' Woontra (c) P(contra Co, c) + P(contra c, co)
(3)
2
Intuitively, Wanique downweights claim if it is entailed by other claims, and Weontra downweights a claim if it is not contradictory with the main claim. In the simplified setting where claims are either fully equivalent or fully contradictory, Wunique for claim is the reciprocal of the size of its equivalence class, so we have invariance to claim duplication. Similarly, Weontra grants invariance to
"Access to an NLI model poses only a minimal departure from the zero-resource setting, since NLI is a generic task for which there are off-the-shelf models. NI. is subset of the tasks that L.LMs are capable of, so the usage of a separate ILI model is motivated merely hy efficiency (Kuhn al., 2023; Lin al., 2024).
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including the original claim as distractor. Beyond this simplified setting, using continuous weights allows us to model partial entailment or contradiction, such as for the claims "Kang Ji-hwan is widely ridiculed." and "Kang Ji-hwan is notoriously disliked." in Fig. 2.
We now normalize the verbalized confidence the original claim as (Co( (c)) = B(C) B(C) max 1, (c)) +
Esvo(e) Wunique(e) (oontra()) -
CEC
(4)
generalizing the mutually exclusive case Eq. 2. The maximization with allows for defaulting back the vanilla verbalized confidence in the case where C fails to contain claims that the model considers plausible.
Combining Coherence within Generation and Validation. Our approach so far (summarized in Fig. 2) for normalizing verbalized confidence across distractors focuses on coherence within claim validation. Previous studies disagree on the question of whether models are better at the discriminative or generative counterparts of given task (West et al., 2024; Gekhman et al., 2025), but generally agree on the presence of venciator-validator inconsistency (Li al., 2024), wherein a model may produce inconsistent results between the generation and validation stages. Indeed, find that in the preliminary study setting in Section 2.2, the answer with the highest generation probability and the answer with the highest validation probability (over 10 answers obtained from beam search) agree' on only 592 out of 1000 questions.
We integrate these complementary aspects of generation and validation into DINCO. In partic- ular, draw distributional view of the generator-validdttor gap (Rodriguez al., 2025), in which the generation probability distribution over candidate answers (which we approximate with self-consistency sampling, fSC) is distinct from the validation probability distribution over them (which we approximate with normalized verbalized confidence, fNVC). DINCO thus in- corporates confidence (i.e. probability mass) in both the generator and validator distributions: (DiNCo(c) (CC(c) 2 JNCCCC). We leave a full description of the self-consistency compo- nent to Appendix C.3.
3 EXPERIMENTS
3.1 EXPERIMENTAL SETUP
Short-form Datasets. Short-form QA serves testbed for evaluating factuality as well as cal- ibration because of its tractable evaluation and adjustable difficulty. The task is relevant in prac- tice because it assesses models' ability respond to information-sseking users. TriviaQA contains trivia questions requiring diverse world knowledge (Joshi et al., 2017). SimpleQA similarly contains short, fact-seeking questions, curated with the criterion of challenging frontier models (Wei al., 2024). We sample 1000 questions from each dataset. In Appendix D.6, we evaluate on BioASQ other
(Krithara al., 2023), dataset demanding biomedical expertise, to show generalization domains. We use LLM-as-a-juue judgee to evaluate binary correctness, following best practices for robust evaluation (Wei al., 2024); in Appendix B.3 we confirm high human agreement.
Long-form Datasets. While short-form settings are appealing for their simple evaluation, many real-world tasks require longer generations, for which calibrated confidence estimation remains crit- ical. The long-form setting comes with the evaluation challenge that responses generally contain both correct and incorrect parts, complicating the assignment of single correctness score. In our experiments, evaluate long-form calibration on biography generation using FactScore (Min et al., 2023). FactScore decomposes generated biography into atomic claims and verifies each claim against Wikipedia (see Appendix B.4 for an example), thus enabling claim level. We use the labeled subset containing 183 entities from Min et al. (2023).
evaluate calibration the
$ consider e and c' equivalent answers to question g if 1 P(entail c, (.:')))+ P(entail | é, c: q) > 0.9. 'We use the validation split of the ra. nocontext subset for TriviaQA. On SimpleQA, Gemini-2.5-Flash produced refusal with no output on 86 questions, we exclude them from all experiments.

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Models. Since TriviaQA and SimpleQA are adequately challenging for smaller and larger models, respectively, we focus their evaluation accordingly. TriviaQA is largely solved by frontier models (Wei et al., 2024), and SimpleQA is too difficult for smaller models." On TriviaQA, we use popular open-source models: Qwen3-32B, Qwen3-8B, and Qwen3-1.7B (Yang et al., 2025), Llama-3.2- 3B-Instruct (Dubey et al., 2024), and Gemma-3-4B-IT (Team al., 2025). On SimpleQA, use popular frontier models: GPT-4.1 (2025-04-14; OpenAI, 2025) and Gemini-2.5-Flass (Comanici al., 2025). For SimpleQA evaluated on frontier models, we also consider the black-box setting where no logit access is assumed; here, rather than using pseudo-beam search to generate distrac- tors, we prompt the model directly to generate diverse distractors. Moreover, we replace P(True) with verbalized numerical confidence to forgo logit access. For the long-form task of biography generation, we limit our evaluation to Qwen3-8B and Gemma-3-4B-IT, due to the cost of FactScore evaluation with GPT-4.1. We use the NLI model RaBBRTa-v- www----aeemeeee memeemnl1 1 i - -fever-annl i (He et al., 2021) for our methods and self-consistency; we confirm robustness to the choice of the NLI model in Appendix D.3.
We evaluate Expected Calibration Error (ECE J: Naeini et al., 2015) with 10 bins, Brier score (BS J; Brier, 1950), and area under the ROC curve (AUC Hanley & McNeil, 1982). See Appendix B. for descriptions. Like accuracy, these calibration metrics are on a scale from 0 to meaning that e-g., an improvement of 0.05 is substantial, as it corresponds to 5% absolute improvement (Tian et al., 2023; Xiong et al., 2024); we report all improvements in absolute rather than relative terms. For biography generation, we also evaluate Pearson and Spearman corre- lation between average claim-level confidence and passage-level FactScore (without length penalty), where the latter measures the proportion of claims that are correct.
Evaluation Metries.
Baselines. Following past work in zero-resource calibration, we compare against training-five methods that produce probabilities without post-hoc calibration (Tian al., 2023; Xiong et al., 2024; Steyvers et al., 2025). We provide prompts for our methods and baselines in Appendix B.2. Verbalized confidence (VC; P(True) from Kadavath et al., 2022) asks the model whether its answer is correct and computes P P(Yes)/(P(Yes) + P(No)). It is straightforward to replace P P(True) with verbalized numerical confidence; in Appendix D.2 we show that the latter similarly benefits from our methods, showing robustness to the format of verbalized confidence. Top-K prompting (K-VC; Verb. IS from Tian et al., 2023) prompts the model to provide its top K guesses along with verbal- ized numerical confidences. For our black-box setting, we use the candidate answers generated using the K-VC prompt as distractors, but we discard the verbalized confidences contained in the same generation, and instead separately collect verbalized numerical confidence on each distractor independently. Maximum sequence probability (MSP; Fadeeva et al., 2023) is the LLM's probabil- ity of generating its answer. Self-consistency (SC; Xiong et al., 2024) samples several answers (we use temperature 1) and computes the proportion that match the main answer. Following Kuhn et al. (2023), we use an NLI model to determine semantic equivalence when grouping together answers for SC.5 For biography generation, we modify self-consistent to sample several biographies and measure entailment of each claim (Zhang et al., 2024a). SC-VC (Xiong et al., 2024) is SC weighted by verbalized confidence (we use P(True) following Taubenfeld et al., 2025).
Inference Budget. As our methods and several baselines (K-VC, SC, SC-VC) operate on a vari- able inference-time budget, we control this budget at K 10. For DINCO, we use 5 samples for self-consistency and 5 distractors for normalized verbalized confidence; we show robustness to the budget split for various budgets in Appendix D.4 and thus recommend equal split for simplicity.
3.2 RESULTS
Short-form QA. Tables 1 and 2 report that on TriviaQA and SimpleQA, DINCO outperforms the best baseline, MSP, by an average ECE of 0.077 and 0.092, respectively. While MSP a competitive baseline (e-g. AUC 0.800 surpasses DINCO at 0.786 on SimpleQA with GPT-4.1), this often does not hold across metrics (c.g. MSP has an BCE of 0.263 in the same setting, heavily underperforming DINCO at 0.089) across settings (c.g. MSP underperforms DINCO AUC in TriviaQA by an average of 0.062). Most importantly, the effectiveness of MSP relies on answers having a canonical form, restricting its usage to multiple-choice or short-form questions and preventing its generaliza- tion to long-form settings (Farquhar et al., 2024). We highlight that NVC outperforms SC (e.g. by
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Table I: TriviaQA results. We evaluate Expected Calibration Error (ECE), Brier score (BS), and area under the ROC curve (AUC). In each column, we bold the best result and underline results not significantly worse under a paired test (a 0.05; see Appendix B.5 for tests). For readability of this table, we leave Qwen3-32B results to Appendix D.5, where we verify that the effectiveness of DINCO extends to larger scales of open-source models.
Qwen3-8B
@wen3-1.77
Llamm-3.2-38-lmnse
Genmu--344B-T
Method
ECE
BS
AUC
ECE
BS
AUC
ECE
BS
A
ECE
BS
AUC
VC (Kadavath et al., 2022)
0.240
0.242
0.817
0.387
0.383
0.720
0.189
0.208
0.826
0.300
0.299
0.702
K-VC (Tian et al., 2023)
0.341
0.348
0.604
0.538
0.524
0.596
0.146
0.228
0.678
0.254
0.262
0.786
MSP (Fadeeva et al., 2023)
0.149
0.203
0.819
0.104
0,186
0.774
0.243
0.253
0.764
0.252
0.268
0.790
SC-VC (Xiong
2024)
0.299
0.325
0.704
0.451
0.474
0.559
0.122
0.211
0.761
0.362
0.378
0.653
SC (Xiong et al., 2024)
0.236
0.244
0.785
0.233
0.229
0,780
0,065
0.177
0.808
0.303
0,304
0.713
NVC
0.171
0.190
0.853
0.084
0.168
0.192
0.845
DINCO
0.097
0.155
0.879
0.177
0.179
0.835
0.044
0.148
0.864
0.121
0.191
0.817
Table SimpleQA results. The black-box variants of our methods assume no logit access. Metries and text styling follow Table 1.
GPT-4.1
Gemini S.5iini-2.5-Faaa 5-5-shh
Method
ECE
AUC 1
ECE 4
BS L
AUC
VC (Kadavath al., 2022)
0.547
0.549
0.644
0.409
0.393
0.617
K-VC (Tian et al., 2023)
0.338
0.337
0.632
0.535
0.511
0.566
MSP (Fadeeva et al., 2023)
0.263
0.255
0,800
0.098
0.177
0.773
SC-VC (Xiong et al.. 2024)
0.223
0.252
0.761
0.186
0.221
0.755
SC (Xiong et al., 2024)
0.220
0.252
0.750
0.170
0.212
0.748
NVC VaCalack box
0.213
0.270
0.607
0.208
0.262
0.595
CNNCOtack-boo
0.161
0,251
0.605
0.079
0.199
0.697
NVC
0.164
0.222
0.729
0.105
0,199
0.662
DINCO
0.089
0.183
0.786
0.088
0.174
0.762
an ECE of 0.049 and 0.060 on TriviaQA and SimpleQA, respectively) despite only leveraging co- herence validation and not generation (Section 2.3). Nonetheless, DINCO is more consistently calibrated than NVC (in particular on AUC, c.g. 0.786 DINCo vs. 0.729 NVC with GPT-4.1 on SimpleQA), empirically supporting our motivation in Section 2.3 for integrating coherence gen- eration (SC) and validation (NVC) into DINCo. In the black-box setting on SimpleQA, DINCO continues to do well (e.g. outperforming the baselines on ECE), but tends to fall behind DINCO with logit access, underscoring the benefit of leveraging token probabilities for calibration. In Ap- pendix D.6, we evaluate Qwen3-32B on BioASQ (Krithara et al., 2023), extending these findings to the biomedical domain where expert knowledge is required.
Long-form QA. Table 3 reports results on FactScore. While VC is extremely miscalibrated (e.g. ECE of 0.433 with Qwen3-8B), DINCO is able leverage incoherence across related claims to normalize verbalized confidence and achieve strong calibration. Whether SC or NVC performs bet- ter varies by the model Qwen3-8B or Gemma-3-4B-IT, but DINCO continues outperform SC (0.076 vs. 0.162 ECE with Qwen3-8B, and 0.1 vs. 0.197 ECE with Gemma-3-4B-IT) Further- more, DINCO is the method most strongly correlated with passage-level FactScore (e.g. improving Pearson and Spearman correlation over SC by an average of 0.072 and 0.074, respectively), demon- strating that the effectiveness of DINCO extends to the long-form setting. Taken together with the short-form results in Tables 1 and 2, these results indicate that DINCO is applicable to open- and closed-soune source models, and crucially can transfer seamlessly between short-form QA and long-form generation settings.
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Table 3: FactScore results. In addition the claim-level metrics, we report Pearson (r) and Spear-
man (p) correlation with passage-level FactScore. Text styling follows Table 1, and we bold the best r and p.
Qwen3-8B
Gemumo-3-4B-IP
Method
ECE
BS 4
AUC
F1
P1
ECE
BS
AUC
yt
p
VC (Kadavath et al., 2022)
0.433
0.431
0.625
0.073
0.122
0.527
0.527
0.683
-0.081
-0.129
SC. (Zhang et al., 2024a)
0.162
0.226
0.771
0.468
0.494
0.197
0.233
0.787
0.629
0.607
NVC
0.191
0.263
0.681
0.444
0.443
0.123
0.230
0.726
0.695
0.704
DINCO
0.076
0.202
0.767
0,518
0.538
0.172
0.210
0.793
0.724
0.712
Table 4: Saturation analysis (higher A= lower saturation). DINCO alleviates saturation.
Table 5:
Ablation of NLI-based weighting shows it is crucial to performance.
Method
Ag
Ap,001
Method
ECB
BS 4 AUC 1
VC
0.670
0.605
NVC
0.171
0.190
0.853
SC
0.734
0.734
w/o NLI
0.358
0.335
0.778
SC@ 100
100
0.832
0.832
DINCO
0.097
0.155
0.879
DINCo
0.998
0.984
w/o NLI
0.130
0.185
0.810
DISCUSSION AND ANALYSIS
We conduct further analysis using Qwen3-8B on TriviaQA with P(True), which appeared in Table in the main experiments.
Scaling Seff-Consistency. While our main experiments in Section 3 were approximately controlled for the inference budget of each method, here we examine whether simply scal- ing the inference budget of self-consistency al- lows to recover the calibration of DINCo. Fig. 4 shows that self-consistency alone is un- able to reach the performance of DINCO (using 5 distractors and 5 self-consistency samples as in Section 3) even when scaling up to 100 sam- ples. Although DINCO uses 32% more FLOPs than SC in the context of Section 3, scaling SC up to match and far exceed this slightly higher FLOP count fails to further improve calibration, demonstrating that the effectiveness of DINCO comes from leveraging coherence in both gen- eration and validation, which not matched by scaling the generation axis alone.
0.45
SC ECE
0.40
SC BS
0.35
DINCO ECE [Table
0.30
DINCO BS [Table 1]
9.0 10 FLOPs
0.25
0.20
90. x 10
0.15
12 x 10
0.10
SC Samples
Figure 4: Scaling self-consistency does not close the gap with DINCO. FLOP counts are with Qwen3-8B on 1000 TriviaQA questions. Since each distractor requires verbalized confidence step, DINCO costs 32% more than SC. How- ever, due to diminishing returns, scaling SC up to even 100 samples negligibly improves per- formance (despite costing 7.6 times as much as DINCO), justifying the slightly higher cost of DINCO to achieve stronger calibration. We mark that the cost of the lightweight NLI model (184M parameters) used in DINCO and SC neg- ligible, making up less than 1% of the total FLOP count.
In Appendix D.7, we show that combining distractor generation and validation into one step maintains similar calibration to the origi- nal two-step version while reducing cost only 10% more than SC. In Appendix C.I, we moti- vate the feasibility of generating distractors us- ing a smaller model to reduce cost below even that of SC.
Quantifying Saturation. A core motivation for method is the notion (shown in Fig. 1) that verbalized confidence exhibits saturation at high confidence. To better quantify this notion, we introduce metric A. that measures the absence of saturation. We define A, as the proportion of pairs of distinct instances that have a confidence difference exceeding e. For example, if all confidence scores are the same, then Ao 0, and if all confidence scores are distinet, then Ag 1.

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We consider {0, 0.001). Table 4 shows that DINCO leads to substantially higher rates of distinct confidence, indicating lower saturation. In particular, self-consistency scaled to 100 samples (as above) continues to be more saturated than DINCo. While the absence of saturation alone means little without calibration (as evaluated in Section 3), this analysis helps explain DINCO's calibration improvement, hinted at by Fig. 1. Moreover, we argue that lower saturation leads to more usable confidence estimates: a saturated distribution is inherently less controllable, with large jumps in error between thresholds.
Ablating NLI. Our main experiments in Tables 1 to 3 showed comparisons with SC and NVC, which ablate NVC and SC, respectively, from DINCo. Here, to understand how necessary access to an NLI model is, we ablate the NLI component, which is used to downweight distractors that overlap with the main claim or other distractors (Section 2.3, Fig. 2). In Table 5, we see that performance substantially decreases without NIIIbased weighting. emphasizing the utility of an off-the-shelf NLI model to account for claim redundancy. In Appendix D.3, we show that DINCO is robust to the specific choice of the NLI model.
5 RELATED WORK
Considering Multiple Answers for Verbalized Confidence. The approach most related to DINCO is to have the model consider several candidates within single prompt and assign con- fidences to them (Tian et al., 2023; Kadavath et al., 2022; Zhang et al., 2024b; Chhikara, 2025). A subtle but crucial distinction between these methods and DINCO is that if we present all the candi- dates together, we become unable to gauge the probabilistic coherence of the confidence estimates, i.e. whether they form valid probability distribution, since LLMs can satisfy probabilistic coher- ence via simple arithmetic. In Section 2.2, we show that the probabilistic coherence of confidence estimates correlated with answer correctness. Instruction-tune confidence even when it is undue (OpenAI, 2023; Leng et al., 2025; Sun et al., 2025; Xu et al., 2025), leading joint prompting to suffer similar issues of overconfidence as vanilla prompting. With independent prompting, we can expose and account for inconsistencies in self-declared knowledge. In Section 3, we empirically verify that our method leads to better calibration than joint prompting for verbalized confidence.
LLMs have tendency to assert
Confidence Estimation in Long-form Generation. While historically confidence estimation has mostly been applied to classification (Houlsby et al., 2011), multiple-choice (Jiang et al., 2021), and short-form QA (Xiong et al., 2024), with the advent LLMs it has increasingly been considered for long-form generation. Since token-level uncertainty is often ill-suited for representing claim-level uncertainty, the primary approaches have been self-consistency and verbalized confidence (Manakul al., 2023; Zhang et al., 2024a; 2025). We propose a method normalize verbalized confidence, enabling DINCO to combine the complementary confidence signals in these two prior approaches.
Reconciling Inconsistent LLM Probability Judgments. In Appendix A.2, we diseuss a related line of work demonstrating the benefits of reconciling inconsistent LLM probability judgments. In this work, we propose a zero-resource confidence estimator that normalizes verbalized confidence over self- self-generated distractors, motivated by the suggestibility of LLMs in unfamiliar topics.
6 CONCLUSION
We present DINCO, which estimates LLM confidence by leveraging coherence generation as well as validation (through verbalized confidence). Verbalized confidence tends to be saturated at high confidence. We show evidence that this behavior is correlated with suggestibility, where the LLM is more likely to accept claims that it knows less about. Motivated by this finding, DINCO has the LLM verbalize its confidence independently on several self self-ggnerrted distractors to esti- mate and correct for the bias caused by suggestibility. DINCO outperforms existing methods in the zero-resource setting on short-form QA (TriviaQA, SimpleQA) and long-form QA (FactScore) and mitigates saturation.
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ACKNOWLEDGMENTS
The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing computational resources that have contributed the research results reported within this paper.
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Jian- Mlan-Qiao Zhu and Tom Griffiths. Incoherent probability judgments in large language models. In Proceedings of the Annual Meeting of the Cognitive Science Society, volume 46, 2024.
Jianqiao Zhu, Adam Sanborn, and Nicholas Chater. Bayesian inference causes incoherence in hu- man probability judgments. Proceedings of the Annual Meeting of the Cognitive Science Soci- ery, volume 41, 2019.
A RELATED WORK
A.I RELATED BELIVIIR LLMS AND HUMANS
Humans are also known be susceptible to suggestibility. They can alter their memories match the suggestions of other people, especially at young age (Bruck & Ceci, 1999). Sycophancy is similar behavior observed in LLMs, where an epistemically vacuous prompt such as "Are you sure?" often lcads the model to change its answer (Laban et al., 2024). The user expressing doubt suggests to the model that its answer may be incorrect, since the user has some assumed level of credibility and would be unlikely to ask again if they agreed. Since instruction-tune models aim to adhere to user preferences, it is plausible that they would employ an argument from ignorance (Walton, 1996) to accept user claim that they cannot refute.
Zhu & Griffiths (2024) provide evidence of probabilistic incoherence LLMs and attribute this finding to the prior from the Bayesian Sampler model, which has been used to explain incoherence in human probability judgments (Zhu et al., 2019). In particular, if the same prior is used for every probability judgment, the sum of the probability judgments for mutually exclusive events exceed (Zhu & Griffits,, 2024). This failure to satisfy the axioms of probability is consistent with our empirical evidence in Section 2.2.
A.2 RECONCILING INCONSISTENT LLM PROBABILITY JUDGMENTS.
Prior work has demonstrated the benefits of reconciling inconsistent LLM probability judgments instead of taking them at face value. Jung al. (2022) improve factuality by selecting claims to which the L.IM assigns coherent truth values upon negation. Hou et al. (2025) use belief tree propa- gation with logically related claims to detect hallucinations. Nafar al. (2025) find that independent prompting followed by normalization outperforms joint prompting for Bayesian network parameter estimation. Feng et al. (2025); Xia et al. (2024) optimize probability distribution to approximately satisfy LLM-generated probability constraints. In this work, propose zero-resource confidence estimator that normalizes verbalized confidence over self-generated distractors, motivated by the suggestibility of LLMs in unfamiliar topics.
B EXPERIMENTAL SETUP
B. EVALUATION METRICS
We adopt the notation from Section 2.1. For a claim c, the truth value is v(c) E {0,1) and the assigned confidence is f(c) E 1).
Expected Calibration Error (ECE; Naeini et al., 2015). The confidence space [0, 1] is parti- tioned into K intervals of equal length. Out of the claims in the dataset, let B, be the list of claims assigned confidence in the interval T [
k-1 & (with I including 0). We compute
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bin-level truthfulness and confidence as
Uh
v(c),
f 1
M(e.
f(c).
(5)
B&
Bk
BE
CEBL
ECE is the bin weighte average of the absolute differences:
K
Bg N Uk - fal
ECE
(6)
Brier score (BS; Brier, 1950). For a dataset of claims Cl,...,CN CN, the Brier score is the mean squared error between the truth values and the confidence estimates:
1 Σ ((())
BS
Area under ROC curve (AUC; Hanley & McNeil, 1982) The ROC curve (depicted in Fig. right) captures the tradcoffs between true and false positive rate (TPR. FPR) that we can obtain with selective prediction, i.c. setting confidence threshold above which to accept claims, We take correct and incorrect claims to be labeled positive and negative, respectively. The TPR is the proportion of positive instances that are accepted, and the FPR the proportion of negative instances that are accepted. We want the TPR to be high but the FPR to be low. By setting lower confidence threshold, TPR will be higher,
FPR may also be higher. By setting a higher confidence threshold, FPR will be lower, but TPR may also be lower. As not every TPR (or FPR) in the interval [0, 1] may be achievable, the ROC fills in the gaps between achievable (TPR, FPR) tradeoffs with linear interpolations, as seen with verbalized confidence in Fig. 1. As a summary statistic for the selective predictive power that a confidence estimator grants us, we compute the area under the ROC curve (AUC). For example, if all positive instances assigned higher confidence than all negative instances, the AUC is 1. Mcanwhile, if confidences are sampled independently at random from the same distribution, the expected AUC is 0.5.
The AUC can also be characterized as the probability that a random positive instance is assigned higher confidence than a random negative instance, with ties randomly broken. Denoting C4 and C_ as the list of correct and incorrect claims in the dataset, respectively.
1[f(c+) 2 f(c-))
1(f(c+) f(c-))
AUC
Σ Σ
(8)
C
2
C+ C-
B.2 PROMPTS
B.2.1 SHORT-FORM QA
Prompt to generate main answer. Also used for beam search in DINCO and sampling for self-consistency.
Here are 2 sets of example prompt and answer.
Example Prompt: Which American-born Sinclair won the Nobel Prize for Literature in 1930?
Example Answer: Sinclair Lewis
Example Prompt: Where in England Dame Judi Dench born?
Example Answer: York
Now, here a new prompt to answer. Answer with a concise phrase, as the examples.
Prompt: (question) Answer:
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P(True)
Below is question and candidate answer. Your task is to determine whether the answer is correct or not. Only output "Yes" (correct) or "No" (incorrect).
Question: (question]
Candidate answer:
Verbalized numerical confidence
Below question and candidate answer. State your confidence that the candidate answer is correct. Only output an integer followed by "%"
Question: fquestion)
Candidate answer: (candidate_answa)))
KC (Verb. 1S from Tian et al., 2023). We use K 10.
Provide your {K} best guesses and the probability that each is correct (0.0 to 1.0) for the following question. Give ONLY the guesses and probabilities, no other words or explanation. For example:
GI: <fisst most likely guess, as short possible; not complete sentence, just the guess
Pl: <hhh probability between 0.0 and 1.0 that G1 is correct, without any extra commentary whatsoever; just the probability!>
G[K]: <h most likely guess, as short possible; not complete sentence, just the guess!>
P[K]: <the probability between 0.0 and 1.0 that G(K} is correct, without any extra commentary whatsoever; just the probability!>
The question is: {question]
Follow-up up after main answer for SC-VC
your answer correct? Only output "Yes" "No".
B.2.2 BIOGRAPHY GENERATION
Prompt to generate main biography. Also used to sample biographies for self-consistency.
Write me paragraph biography (entity).
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Prompt to generate one distractor. We use beam search to extract multiple.
You will be given a fact about person. Assuming the fact accurate, your task to generate plausible but inaccurate statement of similar nature. The distractor statement should form a minimal pair with the original statement, i.c. the distractor should be similar the original possible while ensuring that the distractor is not factual. The distractor should be crafted so that someone with only superficial
knowledge about the topic is likely to fooled.
Let's examples before the real task.
Topic: Barack Ohama
Fact: Barack Ohama was born Hawaii.
Distractor: Barack Obama was born in Kenya.
Topic: Wright brothers
Fact: Wright airplanes were involved in fatal crashes.
Distractor: Wright airplanes were praised for their safety.
Topic: John Clempert
Fact: John Clempert was inspired by Houdini when developing acts.
Distractor:
John Clempert was inspired by Penn and Teller when developing acts.
Now for the real task. Only output a distractor as the examples.
Topic: {entity)
Fact: {claim)
Distractor:
P(True)
Your task is to determine whether the following claim related to (entity) is correct. Only output "Yes" (correct) or "No" (incorrect).
Claim: {claim}
Yes No:
Verbalized numerical confidence
The claim below was found in a passage about (entity).. State your confidence that the claim is correct. Only output an integer followed by "%"
Claim: {claim}
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Prompt for the LLM to determine whether a sampled biography entails claim for self- consistency
You will be given passage and claim. Your task determine whether the passage supports, refutes, or does not mention the claim. Output only "Support", "Refute",
"No Mention".
Let's some examples before the real task.
Passage:
Barack Obama was the 44th President of the United States, serving from 2009 to 2017. Born on August 4, 1961, in Honolulu, Hawaii, he was the first African American hold the office. Before his presidency, Obama served state senator in Illinois and later as the 47th Governor of Illinois. A former constitutional law professor, he was known for his eloquence, bipartisan approach, and focus on issues such as healthcare reform, climate change, and foreign policy. His presidency was marked by significant legislative achievements, including the Affordable Care Act, and a commitment to diplomacy and international cooperation. After leaving office, he authored memoirs and remained active in public life, advocating for social justice and community engagement.
Claim: Barack Obama was born in Hawaii. Relationship: Support
Passage:
Tiger Woods is one of the most iconic and accomplished golfers in history, known for his extraordinary talent, dominance the course, and global influence on the sport. Born on December 30, 1975, in Cypress, Florida, Woods rose to fame in the mid- 1990s and quickly became household name, winning his first major championship at the 1997 Masters at just 21 years old. Over his career, he has claimed major titles, the most in PGA Tour history, and has consistently ranked among the world's top golfers for over two decades. His aggressive playing style, precision, and mental toughness set him apart, making him a symbol excellence in golf. Despite personal challenges and setbacks, Woods has remained dominant force in the sport, inspiring millions of fans around the world.
Claim: Tiger Woods won major championship 19 years old.
Relationship: Refute
Passage: Albert Einstein was a theoretical physicist renowned for developing the theory relativity, which revolutionized the understanding of space, time, and gravity. Born in 1879 Ulm, Germany, he later moved to Switzerland and eventually to the United States. Einstein's work, including the famous equation E=mc?, laid the foundation for modern physics and contributed to the development of nuclear energy. Despite his scientific achievements, he also passionate advocate for peace, civil rights, and education. His legacy endures as one of the most influential scientists in history. Claim: Albert Einstein became US citizen.
Relationship: No Mention
Now for the real task.
Passage: (sampled_biograggpp
Claim: {claim}
Relationship:
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B.2.3 SHEUDO-BEAN SEARCH
Prompt for the LLM complete the prefix of answer. Used for pseudo- beam search (Appendix C.2).
You will be given prompt along with prefix to begin your answer with. Your answer should start with the given prefix. If the prefix itself is your final answer, you can simply output just the prefix.
Let's look at 2 examples before the real task.
Example Prompt: Which American-born Sinclair won the Nobel Prize for Literature in 1930?
Example Answer Prefix: Sin
Example Answer: Sinclair Lewis
Example Prompt: Where in England Dame Judi Dench born?
Example Answer Prefix: York
Example Answer: York
Now, here is new prompt to answer. Answer with a concise phrase starting with the given prefix, as in the examples.
Prompt: (question)
Prefix: {prefix}
Answer:
B.3 LLM-AS-A-JUDGE
We score model responses for short-form QA (TriviaQA and SimpleQA) using an LLM judge rather than lexical matching for robust evaluation (Wei et al., 2024). Our biography generation task is eval- uated with FactScore (Min et al., 2023), which uses a strong LLM for atomic claim decomposition and verification; we use GPT-4.1. We use 3lama-3.1-88 1.1--B-Instrue (Dubey al., 2024) as the judge on TriviaQA, and GPT-4.1 as the judge on SimpleQA. On a sample of 100 questions from each dataset, we compared LLM judgments with human judgments from one author of this paper. We used responses from Qwen3-8B on TriviaQA, and responses from GPT-4. on SimpleQA. The rate of agreement was 96/100 and 99/100, respectively. Upon reviewing disagreements, we found that in most cases there was genuine room for interpretation. For example, for the question "In The Living Daylights what did Carla keep in her cello case?", the target answer is "A machine gun" while model answer was "A gun". It is unclear whether the model answer has the desired specificity. As another example, for the question "In Charles Dickens' "Great Expectations", " who or what was Abel Magwitch?", the target answer is "Convict" while the model answer main benefactor of Pip". ". The model answer contains the answer but contains more information, and without external information, it is impossible to determine whether the model's answer is correct. Overall, given the high agreement on unambiguously gradable questions, we deem it safe to adopt LLM-as-a-judge as reliable evaluator our experiments.
"A convict and the
B.4 FACTSCORE EXAMPLE
Table 6 presents an example of atomic claim decomposition and verification with FactScore.
B.5
SIGNIFICANCE TESTING
ECE. We subsample 10k subsets of size 0.9 times the original dataset, where sampling is done without replacement. We construct upper one-sided confidence interval with confidence level
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Table 6: Example of FactScore atomic claim decomposition and verification.
Generation
Kang Ji-hwan is renowned South Korean actor and singer, best known for his role as the lead vocalist of the popular K-pop group BE:FIRST. Born April 15, 2001, in Seoul, South Korea, Kang began his career in the entertainment industry at young age, show- casing his talent through various music projects and performances. His distinctive voice and charismatic stage presence quickly earned him loyal fanbase. Beyond his work in music, Kang has also ventured into acting, appearing in television dramas and variety shows, further solidifying his status as a multifaceted entertainer. With his dedication and natural talent, Kang Ji-hwan continues to make a significant impact the K-pop and en- tertainment world.
Extracted claim
Correct?
Kang Ji-hwan is South Korean actor:
Yes
Kang Ji-hwan is South Korean singer.
No
Kang Ji-hwan is renowned.
Yes
Kang Ji-hwan is best known for his role the lead vocalist of BE:FIRST.
No
BE:FIRST is K-pop group.
Yes
BE:FIRST is popular group.
Yes
Kang was born on April 15, 2001.
No
Kang born in Seoul, South Korea.
Yes
Kang began his career in the entertainment industry at young age. Kang has showeased his talent through various music projects.
No
No
Kang has showcased his talent through various performances. He has distinctive voice.
Yes
No
He has charismatic stage presence. His distinctive voice quickly earned him loyal fanbase. His charismatic stage presence quickly eamed him a loyal fanbase. He quickly earned loyal fanbase.
No
No
Yes
Kang has worked in music.
No
Kang has ventured into acting.
Yes
Kang has appeared in television dramas.
Yes
Kang has appeared in variety shows.
Yes
Kang is multifaceted entertainer.
Yes
Kang's 's status as multifaceted entertainer has been further solidified.
Yes
Kang Ji-hwan is dedicated.
Yes
Kang Ji-hwan has natural talent.
No
Kang Ji-hwan continues make sigmificant impact in the K- K-pop world.
No
Kang Ji-hwan continues to make significant impact in the entertainment world.
No
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0.95 for the tested method's ECE minus the best method's ECE and check whether the interval contains O.
BS. As the Brier score is simply the mean squared enror between confidences and truth values, it is well behaved and amenable to bootstrapping. We sample 10k subsets with the same size as the original dataset, where sampling is done with replacement. We construct an upper one-sided confidence interval with confidence level 0.95 for the tested method's BS minus the best method's BS and check whether the interval contains O.
AUC. As AUC is a U-statistic, we use a one-sided DeL.ng test (DeLong et al., 1988; Sun & Xu, 2014) with confidence level 0.95.
C METHODS
C.I EXAMPLES, ANALYSIS, AND DISCUSSION OF DISTRACTORS
Examples. Building on the example in Fig. 2, we provide more examples of distractors gener- ated by LLMs using different methods of self-generating distractors: beam search on vanilla QA prompt (Table 7), beam search on a prompt eliciting one distractor for a given claim (Table 8), pseudo-beam search
vanilla QA prompt (Table 9a), and prompt eliciting list of candidate answers (Table 9b); see Sections 2.3 and 3 for further descriptions and Appendix B.2 for prompts. For each distractor, we also report the verbalized confidence, Wunique: and Weontra- which under- lie the computation of ((C) in Eq. and measure plausibility, uniqueness, and counterfactuality. respectively. We report these quantities under the experimental settings in Section 3.
Quantitative Analysis. We analyze these attributes of self-generated distractors over a full dataset. Fig. 5 shows that beam search on a vanilla QA prompt leads to distractors with high mean values of P(True) (indicating plausibility). Wanique (indicating low repetitiveness), and Wenntra (indicating opposition to the main claim). In other words, beam search on vanilla QA prompt enables efficient and diverse exploration of alternative claims, despite not explicitly prompting for distractors. We summarize this analysis for the other distractor generation methods in Fig. 6, where we consistently observe high average values of verbalized confidence, Wuniques and Weontra, suggesting that many self- self-aeneeated distractors fulfill the desiderata of being plausible, unique, and counterfactual.
Discussion. Identifying entailment and contradiction relationships may be a nontrivial task in gen- eral, such as in cases requiring domain expertise. DINCo does not dictate the choice of the NLI model, so in such cases, it may be worth using a stronger NLI model or even the generator LLM itself. While this reliance on the NLI capabilities of the same LLM we are attempting to calibrate may seem circular, argue that NLI is a tractable and modular task that is largely solved, in con- trast to LLM calibration. In this paper, we empirically show that a lightweight off- fff-the-shelf NLI model suffices to achieve strong calibration across short-form and long-form generation settings. In Appendix D.6, we show that the same lightweight NLI model continues to suffice in the biomedical domain despite its highly technical nature. In Appendix D.3, we show that DINCO is robust the choice of the NLI model.
DINCO uses the generator LLM to generate its own distractors to relieve the need for external mod- els. However, one concern is that a model's epistemic uncertainty may limit its ability to generate plausible, high-quality distractors. Fortunately, our quantitative analysis (Figs. 5 and 6) provides empirical evidence of the plausibility of generated distractors. To offer an intuitive explanation of this behavior, we note that the "plausibility" of a claim depends on the model assessing this plau- sibility. If a model is epistemically uncertain, the distractors it generates may be implausible to an expert model that is epistemically certain, but this is acceptable because we are interested in whether the distractors are plausible to the epistemically uncertain model so that we can leverage the phe- nomenon of suggestibility under epistemic uncertainty (Section 2.1). As an extreme case of this, is even acceptable for a distractor to be factual, long it is counterfactual to the main claim. As shown by the examples in Table 9b, we want to know whether the model can distinguish correct and incorrect claims, regardless of whether the distractors happen be correct.
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Table 7: Example distractors and their attributes. Qwen3-8B on TriviaQA. Main answer and dis- tractors generated using beam search on vanilla QA prompt (Appendix B.2.1).
Question
In The Living Daylights what did Carla keep her cello case
True answer
A machine gun
Main answer
VC
B(C)
NVC
A gun
1.00
5.91
0.17
Distractor
VC
Wunique
Woontra
A bomb
1.00
1.00
1.00
A jar of honey
0.12
1.00
1.00
A set of keys
0.92
1.00
1.00
A jar of spiders
0.82
0.99
1.00
A stash of drugs
1.00
1.00
1.00
A pair of binoculars
0.01
1.00
1.00
A jar of pickles
0.99
0.50
1.00
A jar of pickled gherkins
1.00
0.51
1.00
A secret weapon
0.97
0.54
0.10
Table 8: Example distractors and their attributes. Qwen3-8B on FactScore (biography generation). The main claim is extracted from the generated biography, and the distractors are generated using beam search on a prompt eliciting one distractor given the main claim (Appendix B.2.2).
Main claim (entity: Kalki Koechlin)
VC
3(C)
NVC
She played Nandini in the film Chandni Chowk to China.
0.49
1.48
0.33
Distractor
VC
Wunique
Weontra
She played Chandni in the film Chandni Chowk to China. She played Nandini in the film Chandani Chowk China. She played Nandini in the film Chandni Chowk to Russia.
0.07
0.50
0.98
0.81
0.98
0.00
0.45
0.99
1.00
She played Nandini in the film Titanic.
0,00
1.00
1.00
**** She played Chandni in the film *Chandni Chowk to China*
1.00
0.50
0.98
She played Nandini in the film Chandni Chowk to Italy.
0.01
0.99
1.00
She played Nandini in the film Dangal.
0.00
1.00
0.99
She played Nandini in the film Chandni Chowk to Canada.
0.00
0.99
1.00
The discussion above for the well-suitedness model to generate its own distractors also hints the potential for distractors be generated by smaller model reduce computational cost. Although we do not investigate this idea in this paper, we expect the task of generating somewhat plausible distractors to be much easier than knowing the correct answer. Considering the example in Table 9, the information required to correctly answer the question is obscure, but anyone who under- stands the question can produce distractors that would likely be indistinguishable from the correct answer to someone else who does not know the answer to the question. This suggests that model with a smaller capacity than the model we seek to calibrate could be used for efficient distractor generation with little degradation in distractor quality. Outside of calibration, other methods such as contrastive decoding (Li et al., 2023) leverage a similar intuition that weaker models can produce plausible but incorrect candidate generations.
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1
4=0.45
P(True)
-0.30, p -0.31
μ=0.74
W_ u_uiniuue
0.31, p = -0.27
r r= +0.58, p = +0.60
#=0.71
W_ contra
1
P(True)
w w_unique
w_contra
Figure 5: Pairwise correlations (Pearson's r, Spearman's p) between distractor attributes: P(True), Wuniques and Weontra. Due to the large numbers of points, we joiict.jonn distributions by plotting the density using kernel density estimation, with darker shades indicating more points. The diagonals plot the marginal distributions. These results with Qwen3-8B TriviaQA, using beam search on a vanilla QA prompt (Appendix B.2.1) to generate distractors. In other words, we do not explic- itly prompt the LLM generate incorrect answers, motivating uniqueness, and counterfactuality of the distractors. We find that Wunique and Wcontra are positively correlated, indicating that distractors that contradict the main claim tend be repeated fewer times within the distractor set, suggesting a diverse exploration of the set of alternative claims. On the other hand, Wunigue and Wcontra are negatively correlated with P(Truc). Interpreting P(Truc) as measure of plausibility, this means that plausible claims are more repeatedly generated and are more agreement with the main claim. Nevertheless, these negative correlations are weak, the fact that the mean values of P(True) and Weontra are both high indicates that substantial number of plausible distractors are being generated.
investigation of the plausibility,
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Table Example distractors and their attributes. GPT-4.1 on SimpleQA. Table 9a shows examples using pseudo-beam search on vanilla QA prompt to generate distractors. Table 9b shows examples the black-box boo setting using a single generation prompt asking for list of candidate answers, and the verbalized confidence is verbalized numerical confidence rather than P(True) (we still elicit confidence for each candidate answer independently). See Appendix B.2.1 for prompts.
Question
For how many years did Mohamed Abdelaziz Djait serve as the Mufti of the Republic of Tunisia?
True answer
(a) Pseudo-beam search.
Black-box prompting.
Main answer
VC
3(C)
NVC
Main answer
VC
3(C)
NVC
17 years
0.85
6.19
0.14
1.00
9.56
0.10
Distractor
VC
Wanique
Wcontra
Distractor
VC
Wunique
Weontra
Twenty years
0.78
0.32
0.99
1.00
1.00
0.99
Two years
0.62
1.00
1.00
1.00
1.00
0.99
12 years
0.22
1.00
1.00
1.00
1.00
1.00
22 years
0.12
0.54
1.00
0.80
1.00
1.00
Three years
0.90
1.00
1.00
1.00
1.00
1.00
11 years
0.90
1.00
1.00
1.00
1.00
1.00
Six years
0.73
1.00
1.00
8
1.00
1.00
1.00
Eight years
0.78
1.00
1.00
1.00
1.00
1.00
Four years
0.73
1.00
1.00
10
0.80
1.00
1.00
C.2 PSEUDO-BB BEAM SEARCH
DINCO uses beam search generate distinct distractors. However, beam search may be avail- able for API-access models. Here, we describe an approximation of beam search with similar in- ference cost that can be implemented if top token probabilities are provided, as they are in many API-access models such as GPT and Gemini.
We first run one inference pass generate the main answer, which also gives us the highest proba- bility tokens (and their probabilities) at each token position in the main answer. We consider the set of sequences that are a prefix of the main answer followed by a top token that is not the token in the main answer. We sort the sequences by their probabilities (computed with the chain rule). For each of the highest probability sequences, we present it to the LLM as a prefix to complete as an answer to the question (see Appendix B.2.3 for the prompt). This procedure chooses distinct prefixes with relatively high probability, shaping their completions to be distinct as well as plausible answers.
C.3 SELF-CONSISTENCY
coffissnsitency estimates confidence in main answer by sampling many answers and counting the proportion of answers that match the main answer (Xiong et al., 2024; Wang et al., 2023). Given short-form question, a main answer cg, and sampled answers .I,..., CK, we compute the confidence of co as
(C
1 K+ 11 cc).
Following Kuhn et (2023), we use an NLI model to determine semantic equivalence." For the long-form setting, follow Zhang al. (2024a) and decompose long-form response To into claims cI...., Cn and measure entailment from other sampled responses "I....,F'x ,rk- We compute the confidence (SC(c)) of claim e by Eq. 9 with the summand replaced with P(entail rk, G))-
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Bearn search on vanilla QA prompt (Qwen3 88. TriviQA)
Beam search prompt eliciting 1 distractor for a given claim (Qwen3-00, biography)
Pseudo beam search on vanilla QA prompt (GPT-4.1, 4.1, Smpla(A)
Prompt for list of candidates [GPT-4.1, SimpleQA)
Mean:
0.45
0.74
0.71
0.49
69
0.78
0.77
88
0.66
0.73
0.97
0.96
Spearman's p wuuiqque
-0.03
-1.00
-0.04
41.00
Figure 6: Statistics of distractor attributes (verbalized confidence, Wunique: and Weontra) for different methods of self-generating distractors. We report their means and Spearman correlations. Although the second method (beam search on a prompt eliciting one distractor for a given claim) has no correlation between Wunique and Weontra unlike the other methods, we find that all methods have high average values of verbalized confidence, Wunique: and Weontra. suggesting that many self-generated distractors fulfill the desiderata of being plausible, unique, and counterfactual. Notably, the fourth method (prompting for list of candidate answers) has the highest Wunique and Weontra. showing the potential benefit of conditioning on previously generated distractors rather than generating them independently. We note that the fourth method uses verbalized numerical confidence rather than P(True) due to being in the black-box setting (we still elicit confidence for each candidate answer independently).
Since the NLI task here involves long-form text, we use the original LLM instead an NLI model (see Appendix B.2 for prompts).
D ABLATIONS AND RESULTS
D.1 PRELIMINARY STUDY EXTENDED
To extend the preliminary study presented in Section 2.2 comparing confidence levels on correctly and incorrectly answered questions, we provide the same analysis for the other two datasets in our main experiments (Section 3). SimpleQA (Wei et al., 2024) is a short-answer QA dataset similar to TriviaQA except designed challenge frontier models. We accordingly evaluate GPT-4. on Sim- pleQA, thus extending our result to the scale of frontier models. Biography generation is evaluated with FactScore (Min al., 2023). Similar to Fig. 3, Fig. 7 finds higher total confidence (over the main claim and distractors) on incorrectly answered questions, consistent with our hypothesis that LLMs are more susceptible to suggestibility when epistemically uncertain.
GPT-4.1 SimpleQA
Qwen3-88 biography generation
100
coreet (J=55554)
1.0
600
1.0
(u=7.32)
0.8
Curuultive dersity
500
0.8
Comulativeae derail
mean
Count
0.6
400
cornect (u-2.57)
incoreet (p=3.311
0.6
300
meon μ
40
0.4
CDF
0.4
200
0.20
100
0.0
0
0.0
confidence
Figure 7: Distributions of total confidence over correct and incorrect answers, showing the same trends as Fig. 3 of higher total confidence on incorrectly answered questions. For each question, we generate a main answer and 10 distractors, so the maximum possible total confidence is 11.
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Table 10: TriviaQA results. Building on Table report results when replacing P(True) with verbalized numerical confidence. Text styling follows Table 1.
Qwen? 8B
Qwent: 1.78
Llama 3.2-3D 3B Instruct
Gemma-3 3-4B
Method
ECE
BS
AUC
ECE.
BS
AUC
ECE
BS
AUC
ECE
BS
AUC
VCmmetccl (Tian et al., 2023)
0.310
0.304
0.791
0.364
0.342
0.697
0.226
0.226
0.773
0.311
0.321
0.731
VCperno) (Kadavath al,, 2022)
0.240
0.242
0.817
0.387
0,383
0.720
0.189
0,208
0.826
0,300
0.299
0,702
K-VC (Tian al., 2023)
0.348
0.604
0.538
0.524
0.596
0.146
0.228
0.678
0.254
0.262
0.786
MSP (Fadeeva et al., 2023)
0.149
0.203
0.819
0.104
0.186
0.774
0.243
0.253
0.764
0.252
0.268
0,790
SC-VC (Xiong al., 2024)
0.299
0.325
0.704
0,451
0.474
0.559
0.122
0,211
0.761
0,362
0.378
0.653
(Xiong 2024)
0.236
0.244
0.785
0.233
0.780
0.065
0.177
0.808
0.303
0.304
0.713
NVCusecial
0.232
0.225
0,870
0.069
0.162
0.812
0.183
0.202
0.825
0.267
0.272
0.787
DINCOmmeteal
0.117
0.171
0.857
0.170
0.183
0.825
0,037
0.150
0.857
0.101
0.202
0.791
NVCp(Trus)
0.190
0.853
0.084
0.164
0.168
0.192
0.236
DINCOpgrue?
0.097
0,155
0.879
0.177
0.179
0.835
0.044
0.148
0.864
0.121
0.191
0.817
Table 11: FactScore results. Building on Table report results when replacing P(True) with verbalized numerical confidence. Text styling follows Table 3.
Qwen3-8B
Gemma-3-4B-I7
Method
ECE
BS
AUC
P1
PT
ECE
BS 1
AUC
rt
P.
VC Vumerrical (Tian al., 2023)
0.327
0.328
0.749
0.508
0.565
0.482
0,465
0.648
0.186
0.169
VCPeTsue) (Kadavath et al., 2022)
0.433
0.431
0.625
0.073
0.122
0.527
0.527
0.683
-0.081
-0.129
SC (Zhang et al., 2024a)
0.162
0.226
0.771
0.468
0.494
0.197
0.233
0.787
0.629
0.607
NVCmmrial
0.181
0.259
0.693
0.555
0.566
0.090
0.208
0.744
0.683
0.698
0,045
0.193
0.779
0.559
0.574
0.120
0.188
0.808
0.713
0.696
NVC NVCpptru)
0,191
0.263
0.681
0.444
0.443
0.123
0.230
0.726
0.695
0.704
DINCOp(Tre))
0.076
0,202
0.767
0.518
0.538
0.172
0.210
0.793
0.724
0,712
D.2 P(True) vs. VERBALIZED NUMERICAL CONFIDENCE
In Tables 10 and 11, expand on Tables 1 and 3 and report results TriviaQA and biography generation when replacing P(True) with verbalized numerical confidence (see Appendix B.2 for prompts). For SimpleQA, the black-box variants of our methods in Table 2 use verbalized numerical confidence rather than P(True).
We find that the calibration benefits of DINCO generalize to the format of verbalized numerical confidence. For example, DINCO numerical compared to VCnumerical lowers ECE by 0.196 on average on TriviaQA. minConunricalll outperforms the stronger baselines well, e.g. lower ECE than SC by 0. on average on TriviaQA.
In the long-form setting biography generation (Table 11), DINCO numerical continues achieve strong calibration, even outperforming DINCOp(True) in many cases, e.g. 0.045 vs. 0.076 ECE with Qwen3-8B.
D.3 NLI MODEL CHOICE
Our main experiments in Section use the NLI model DeBE-Kia-v--buse-mma mesee--all-eveer-anl (He al., 2021). Here we ablate the particular choice of the NLI model. We choose NLI models that are lightweight (0.2-0.4B parameters) and frequently downloaded from HuggingFace. Table 12 con- firms that DINCO and SC are robust to the choice of the NLI model, with DINCO consistently performing the best.
D.4 BUDGET HYPERPARAMETER,
For our main experiments in Section 3, we gave each method an inference budget of K 10, with DINCo splitting the budget evenly into 5 distractors and 5 SC samples. The equal budget split is straightforward choice requiring no hyperparameter tuning and achieves strong calibration in Section 3. Here examine how robust DINCO is to the budget split and whether tuning
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Table 12: NLI model choice ablation, building on Table I which uses the first NLI model. Text styling follows Table 1. DINCO and SC are robust to the choice of the NLI model, with DINCo consistently performing the best.
Qwen3-8B
Qwen3-1.78
Llama-3.2 -lama-3.2-38-Imae 3.233-nnstrue Gemma-3-4B-I
Method
4
BS
A
ECE
BS
AUC
ECE
BS
A
ECE
BS
A
NIJ model: Mor MoritzLaure wtw.iitttauueeeeeeee www.urreeeDDeeRa---eee Sat-RRTa-----s---ee
-ever-anel
(184M parameters)
SC (Xiong ct al., 2024)
0.236 0.244
0.785
0.233
0.229
0,780
0.065
0.177
0.808
0.303
0.304
0.713
NVC
0.171
0.190
0.853
0.084
0.164
0.806
0.168
0.192
0.845
0.218
0.236
0.791
DINCO
0.097
0,155
0,879
0.177
0.179
0.835
0.044
0.148
0.864
0.121
0,191
0.817
NLI model: sileod/debertee
(184M parameters)
(Xiong et al., 2024)
0.220
0.241
0.778
0.216
0.224
0.771
0.087
0.195
0.774
0.292
0,300
0.709
NVC
0.114
0.160
0.871
0,084
0.163
0.806
0.155
0.188
0.841
0.157
0.203
0.813
DINCO
0.108
0.157
0.879
0.189
0.184
0.835
0.052
0.148
0.864
0.137
0.193
0.821
NIJ model:
1-mu 1
Emgual
-n
(279M
parameters)
SC. (Xiong al,, 2024)
0.225
0.245
0.771
0.212
0.226
0.765
0.098
0.204
0.757
0.289
0.298
0.712
NVC
0.165
0.190
0.855
0.081
0.173
0.788
0.192
0,203
0.836
0.213
0,229
0.807
DINCO
0.088
0.159
0.865
0.173
0.182
0.825
0.058
0.152
0.859
0.106
0.190
0.815
NLI model: microsoit/ /teeberttttteeooo La La-alaaaa aargeremmnn
(406M
parameters)
SC (Xiong et al., 2024)
0.229
0.242
0.783
0.228
0.225
0,782
0.067
0.180
0.801
0.300
0.303
0.713
NVC
0.141
0.169
0.873
0.085
0.167
0.802
0.168
0.191
0.844
0.191
0.218
0.813
DINCO
0.103
0.155
0.878
0.178
0.181
0.831
0.045
0.148
0.864
0.121
0.190
0.818
ECE
BS 4
AUC t
0.225
Budget
0.24
Metric value
0.200
0.22
0.86
12
0.175
16
0.20
0.150
0.82
0.125
0.80
0.16
0.100
0.78
1/4
1/2
3/4
1
0
1/4
1/2
3/4
1
1/4
1/2
3/4
1
Ratio of budget for SC
(0 is NVC: 1/2 is DINCO in Section 3; 1 is SC)
Figure 8: Calibration at different budget splits (Qwen3-8P) -B on TriviaQA). Setting DINCO's (SC sample, distractor) split to the straightforward choice of (1 3 as in Section appears to result in the best performance. Nevertheless, DINCO is robust to the exact budget split, achieving strong calibration across the wide range of budget splits from (}. 3) to t). However, a budget split of (0, 1) or 0) with only distractors or only SC samples leads to suboptimal calibration, which is not remedied by scaling up the budget due to diminishing returns, empirically supporting our motivation (Section 2.3) for combining the confidence signals from generation and validation.
can improve performance. Fig. 8 shows that the budget split can be varied substantially while maintaining performance, revealing DINCO's robustness to the budget split hyperparameters. Thus, we recommend the equal budget split for simplicity. However, we find that regardless of the budget, using no SC samples or no distractors performs worse than allocating at least part of the budget to each component, emphasizing the benefit of jointly scaling the inference budgets for generation and validation.
D.5 OPEN-SOURCE LLM SIZE
Table shows that DINCO achieves strong calibration across small medium-scale (2-8B) open- source models, and Table 2 extends this result to frontier models. Here we consider scaling up within open models, which helps more precisely characterize DINCO's performance across scales due to
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TriviaQA results with Qwen3-32B. Text styling follows Table 1. DINCO extends from the 2-8B scale (Table 1) to the 32B scale.
Table 13:
The effectiveness of
Method
ECE
BS
AUC
VC (Kadavath et al., 2022)
0.153
0.158
0.862
K-VC (Tian et al., 2023)
0.186
0.211
0.737
MSP (Fadeeva al., 2023)
0.113
0.164
0.792
SC-VC (Xiong et al., 2024)
0.166
0.213
0.707
SC (Xiong al., 2024)
0.129
0.190
0.741
NVC
0.194
0.169
0.880
DINCO
0.065
0.131
0.863
known open model sizes. Table 13 confirms that DINCO continues to achieve strong calibration at the 32B model size.
D.6 DINCO IN THE MEDICAL DOMAIN
We test the generalization of DINCO to other domains by evaluating on BioASQ (Krithara et al., 2023), dataset reflecting the information needs of biomedical experts. We sample 1000 factoid questions from the Task B training set of the 2026 challenge edition BioASQ14; see Table 14 for examples. Given the technical expertise required for this task, evaluate Qwen3-32B, thus also building on Appendix D.5 to confirm DINCO's ' effectiveness on large open-source models. Table 15 shows that the calibration of DINCO under the same setup as with TriviaQA generalizes the biomedical domain.
Table 14: Example factoid questions and answers from BioASQ.
Question
True answer
What is the typical alteration of the thy- roid profile metabolism early after coro- nary artery bypass graft surgery? What the prognostic role of alterred thyroid profile after cardiosurgery?
Low T3 syndrome occurs frequently af- ter CABG
Altered thyroid profile after cardio- surgery is associated with several events adults and in children Interferon signature
Which is the most common gene signa- ture Rheumatoid Arthritis patients?
BioASQ results with Qwen3-32B. Text styling follows Table 1. DINCO generalizes Table 15:
setting requiring biomedical expertise.
Method
ECE
BS 4
AUC
VC (Kadavath et al., 2022)
0.255
0.257
0.801
K-VC (Tian et al., 2023)
0.256
0.270
0.749
MSP (Fadeeva et al., 2023)
0.215
0.249
0.743
SC-VC CC-VC (Xiong et al., 2024)
0.155
0.233
0.710
SC (Xiong al., 2024)
0.120
0.217
0.727
NVC
0.107
0.167
0.828
DINCO
0.071
0.160
0.822
D.7 DISTRACTOR GENERATION AND VALIDATION IN ONE STEP
Our scaling analysis in Section 4 shows that, while DINCO incurs slightly higher cost than SC, SC cannot match the calibration of DINCO even when scaled arbitrarily. Here, we consider further optimizing efficiency by addressing the main reason for DINCO's higher cost: the separate step
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elicit verbalized confidence for each distractor. To combine the generation and validation of a dis- tractor into a single conversation, we add a user response containing Output "Yes" if your answer is correct or "No" if not. after the LLM'ss answer. Thus, after generating the distractor, only one more input sentence and one more output token need to he processed to ob- tain verbalized confidence. We remark, however, that this approach is not applicable to API-access models for which we cannot seamlessly switch between beam search and ordinary decoding, which motivated our two-step design that can be shared between open and closed models. Furthermore, decoupling the generation and validation of distractors allows for other variants, such as using smaller model generate distractors for efficiency, as discussed in Appendix C.I. Nonetheless, here we investigate the potential for a one-step design to reduce cost in certain settings while main- taining performance. Table 16 shows that the one-pass variant achieves similar calibration to the two-pass variant, while reducing the relative cost over SC from 32% to 10%.
Table 16: Comparison of DINCO's s original two-step design (separate calls for distractor generation and validation) and the one-step design (combining both parts into one conversation for efficiency). With Qwen3-8B
1000 TriviaQA questions, SC uses 9.0 10 FLOPs, DINCo (2-step) uses 1.2 x 10 (32% more than SC), and DINCO (1-step) uses 9.9 x 10 (10% more than SC). The similar calibration between the two-step and one-step variants of DINCO suggests that distractor can be generated and validated jointly to minimize overhead without compromising calibration.
Qwen3-8B
Qwen3-1.78
Gemma-3-4BBil
Method
ECE 4
1
AUC ECE 5
AUC
ECE
BS
AUC
ECE
BS
AUC 1
(Xiong al., 2024)
0.236
0.244
0.785
0.233
0.229
0.177
0.808
0.303
0.304
0.713
DINCO (2-step)
0.097
0.155
0.879
0.177
0.179
0.835
0.044
0.148
0.864
0.121
0.191
0.817
DINCo (1-step)
0.109
0.171
0.852
0.165
0.191
0.798
0.051
0.157
0.851
0.089
0,210
0.772
30


Reviews:
[{"review_id": "dqzRf0Gg8I", "reviewer": "Reviewer_wbh4", "summary": "The paper introduces Distractor-Normalized Coherence (DINCO), a method for calibrating verbalized confidence in LLMs. The authors argue that standard verbalized confidence is often overconfident and saturated due to LLM suggestibility, that is, the tendency to assign high confidence to claims in the context, especially when the model is epistemically uncertain. The paper presents empirical results that suggest that DINCO leads to an improvement in calibration compared to baselines.", "questions": "- Is the proposed method sensitive to hyperparameters, especially to the choice of $K=5$ for distractors and $K=5$ for SC samples?\n- Can you provide a cost analysis (latency/FLOPs) comparing DINCO to SC@10 and SC@100? The claim that DINCO is more efficient than scaling SC needs concrete validation given the multi-stage pipeline (generate distractors -> validate each -> NLI pairs -> SC samples).\n- What happens when the distractor generation fails to produce mutually exclusive claims, even after NLI filtering?\n- Have you evaluated how the choice of NLI model impacts downstream calibration?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The overall idea of the proposed method is creative and explores a new direction for uncertainty quantification in LLMs. I see the main strengths of the paper as follows:\n\n- originality: to the best of my knowledge, the connection between overconfidence and \"suggestibility\" is new, well-justified idea and appear to be empirically supported by the empirical results\n- empirical results: DINCO appears to consistently outperform strong baselines across diverse tasks (both short-form and long-form) and models (open and closed weight) and that scaling up self-consistency to 100 samples still doesn't match DINCO (at 10 inference calls), which is compelling\n- clarity: the paper is very well-written and clear", "weaknesses": "I believe there are a few weaknesses:\n\n- heavy reliance on oracles\n- efficiency: the paper positions DINCO as a \"zero-resource\" method, but the method requires multiple inference passes to generate distractors, multiple passes to validate each one, and calls to an NLI model, which is more expensive than standard SC\n- distractor quality: the method assumes the LLM can generate high-quality, \"minimal pair\" distractors, but if a model is epistemically uncertain, it might fail to generate plausible alternatives, breaking the normalization assumption; beam search or prompting is used, but the paper doesn't rigorously analyze the quality or diversity of these distractors and how they directly impact the $\\beta(C)$ factor\n- hyperparameters: the choice of $K=5$ for distractors and $K=5$ for SC samples seems arbitrary. Is the method sensitive to these?", "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": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1DlOavtvKI", "reviewer": "Reviewer_ESx7", "summary": "**Summary**: this paper proposes a method to improve the calibration of verbalized confidence approach (dubbed DiNCo), through the use of distracting contexts. Analysis across 6 models, reveals average ECE improvements of up to 0.099 over the best baseline in short-form QA datasets, and pearson correlation improvements (with passage-level FActScore) of 0.072 in long-form QA datasets (in biography generation).\n\n**Main Contribution**: \n- the proposal of a calibration method for verbalized confidence using distracting contexts, enabling more diverse and trustworthy confidence estimates.\n- experiments concerning both short-form QA and long-form QA, and across various models.\n- interesting analysis comparing proposed method to self-consistency baseline at different sampling budgets", "questions": "**Questions**:\n1. The exact setup for the preliminary study (in Section 2.2) is slightly unclear to me. Can you please clarify how you generate the set of C claims for the TriviaQA dataset and how many claims there are? I.e., what’s the upper bound in total confidence ? Is the number of claims the same across every question?\n2. The preliminary study in Section 2.2 reveals that incorrect instances are indeed more suggestible (as evidenced by the average higher total confidence). However, I’m a bit surprised that the total confidence for correct answers is not unimodal and centered at 1. Do you have any intuition on why that is?\n3. Table 1 caption mentions “underline results not significantly worse” but there is no underline. Does it mean that all of the results are significantly equal or better than the bold results?\n4. Line 355 claims that “On TriviaQA, DiNCo outperforms the best baseline SC, by an average of ECE OF 0.099”. Can you clarify what is the criteria for selecting SC as the best baseline? If I understand the table correctly, Table 1 shows that MSP has lower ECE than SC in 3 out of the 4 evaluated models (avg ECE: 0.187 vs 0.209, respectively), which would suggest that MSP is a better baseline in the TriviaQA dataset. \n\n**Clarity**:\n1. May be worth defining what is a valid minimal pair, since in Figure 2 left, examples may differ up to 2 words. Are minimal pairs restricted to those that differ in at most 2 words?\n2. It’s worth clarifying when discussing improvements over baselines, whether these concern relative or absolute improvements.\n\n**Formatting**: \n1. Is there a reason why the bins for correct and incorrect are not centered at integer numbers and have width 1? What does the count for total confidence [0, 0.33] mean in this context? Additionally, I recommend adding the corresponding means as vertical lines to ease comparison.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. **Relevant and interesting work in uncertainty quantification in LLMs**, focusing on improving calibration of self-confidence approaches. \n2. **Novel findings** showing that it is possible to calibrate verbalized confidence estimates in a training-free manner, while alleviating confidence saturation and achieving better calibration than the best baseline irrespective of the compute budget.\n3. **Empirically and theoretically grounded**: the proposed improvement follows from the concept of _suggestibility_, which states that LLMs’ generations are highly influenced by the context when LLMs lack sufficient information.\n4. **Extensive experimentation including 6 models and concerning both short-form (i.e., TriviaQA, SimpleQA) and long-form QA datasets (i.e., biography generation)**.", "weaknesses": "1. Small improvements in calibration method, leading to questions about the significance and real impact of such improvements (e.g., one of the main results states that DiNCo improves ECE by an average of up to 0.099 (line 107)). \n2. Results are difficult to generalize, due to experimental changes (e.g., methods and configurations differ from Table 2 to Table 3).\n3. However, it would be nice to test generalization of the proposed approach to other domains, especially those requiring domain expertise (e.g., medical domain or legal domain).", "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": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "4O2Ohz1eYw", "reviewer": "Reviewer_SPC2", "summary": "The paper introduces a method called Distractor-Normalized Coherence (DINCO) to enhance the calibration of confidence estimates in LLMs. It addresses the prevalent issue of overconfidence in LLM outputs, particularly in low-accuracy claims, which can compromise user trust. By prompting LLMs to verbalize their confidence across multiple self-generated distractors, DINCO normalizes these confidence scores and mitigates biases stemming from suggestibility when the model encounters unfamiliar topics. Overall, the paper contributes a novel approach to calibrating LLM confidence, thereby improving the usability and trustworthiness of their outputs.", "questions": "See above", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The authors provide a thorough analysis of the proposed method's performance compared to existing approaches, showcasing its advantages in both short-form and long-form generation tasks.\n\n2. By addressing the critical issue of overconfidence in LLM outputs, the paper contributes to enhancing the trustworthiness and usability of these models in real-world applications. The findings have implications for various domains where LLMs are deployed, emphasizing the need for reliable confidence estimates to ensure user safety and informed decision-making.", "weaknesses": "1. In the experiments, all evaluated open-source models are relatively small. It remains unclear whether the conclusions generalize to larger models (e.g., those exceeding 30B parameters).\n\n2. In Appendix LLM-as-a-Judge, the description of the human evaluation process is unclear. Did the authors conduct human judgments themselves, and if so, how many participants were involved?", "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": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "3WZRkQemnB", "reviewer": "Reviewer_tbbM", "summary": "This paper studies the overconfidence issue in LLM verbalized confidence when faced with claims that they have little knowledge about, and proposes Distractor-Normalized Coherence (DINCO) to address such suggestibility bias. The proposed approach normalizes the verbalized confidence by considering multiple self-generated distractors. Extensive experiments have shown the effectiveness of DINCO by demonstrating how it outperforms the existing baselines even when they scale up.", "questions": "Please see the weaknesses.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. This paper provides a novel insight that the overconfidence of LLMs comes from suggestibility and shows empirical evidence.\n2. Clear motivation and preliminaries are given to support the proposed method.\n3. The proposed solution DINCO is effective with extensive experiments. Comparison across multiple datasets, models, and baselines has shown that DINCO generally improves calibration metrics.", "weaknesses": "1. Since the entire method builds upon the observation that the model will provide higher confidence on incorrect samples, it would be better if more analysis is provided. For e.g., current results are only derived from TriviaQA. Results from other datasets will make the observation more convincing.\n2. Table 5 shows that accessing an NLI model significantly improves the results. However, only one NLI model has been tested in the results. This is not general enough, and more ablation studies should be provided. How sensitive are the results to the choice of the NLI model? Are there any requirements on the NLI model?\n3. Sometimes the model could generate high confidence for an incorrect answer due to false training data or hallucinations. How can we distinguish suggestibility from these cases?", "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": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)