Large Language Model Confidence Estimation via Black-Box Access
Authors: Tejaswini Pedapati
Amit Dhurandhar
Soumya Ghosh
Soham Dan
Prasanna Sattigeri
Venue: Accepted by TMLR ()

Abstract: Estimating uncertainty or confidence in the responses of a model can be significant in evaluating trust not only in the responses, but also in the model as a whole. In this paper, we explore the problem of estimating confidence for responses of large language models
(LLMs) with simply black-box or query access to them. We propose a simple and extensible framework where, we engineer novel features and train a (interpretable) model (viz. logistic regression) on these features to estimate the confidence. We empirically demonstrate that
our simple framework is effective in estimating confidence of Flan-ul2, Llama-13b, Mistral-7b and GPT-4 on four benchmark Q&A tasks as well as of Pegasus-large and BART-large on two benchmark summarization tasks with it surpassing baselines by even over 10% (on AU-ROC) in some cases. Additionally, our interpretable approach provides insight into features that are predictive of confidence, leading to the interesting and useful discovery that our confidence models built for one LLM generalize zero-shot across others on a given dataset.

Published in Transactions on Machine Learning Research (06/2025)
Large Language Model Confidence Estimation via Black-Box Access
Tejaswini Pedapati"
Sjjaswinip@@u.ibmmccom
IBM Research, Yorktown Heights, NY
Amit Dhurandhar"
adharan Custraaauuuus..o ibm com
IBM Research, Yorktown Heights, NY
Soumya Ghosh
sowmyycighoshh@@maai.com
Merck Research Labs, Cambridge, MA
Soham Dan
vokandan@micrrsspp com
Microsoft, New York, NY
Prasanna Sattigeri
psattig Soathigguus..om ibm..imm..mmmm....mmm. com
IBM Research, Cambridge, MA
Reviewed on OpenReview: https: // openreview. wtt.:/eeeeeeateeeeeeeeeeeeeeeeee forum? id= WrWYChkyRI
Abstract
Estimating uncertainty or confidence in the responses of model can be significant in eval- uating trust not only in the responses, but also in the model whole. In this paper, explore the problem of estimating confidence for responses of large language models (LLMs) with simply black-box or query access to them. We propose a simple and extensible framework where, we engineer novel features and train a (interpretable) model (viz. logistic regression) on these features to estimate the confidence. We empirically demonstrate that our simple framework is effective in estimating confidence of Flan-ul2, Llama- 13b, Mistral- 7b and GPT-4 four benchmark Q&A tasks well of Pegasus-large and BART-large on two benchmark summarization tasks with it surpassing baselines by even 10% (on AU- ROC) in some cases. Additionally, our interpretable approach provides insight into features that are predictive of confidence, Icading to the interesting and useful discovery that our confidence models built for one LLM generalize zero-shol across others on a given dalaset.
1 Introduction
Given the proliferation of deep learning over the last decade (Goodfellow et al., 2016), uncertainly confidence estimation of these models has been an active research area (Gawlikowski et al., 2023). Predicting accurate confidences in the generations producod by a large language model (LLM) are crucial for cliciting trust. in the model and is also helpful for benchmarking and ranking competing models (Ye et al., 2024). Moreover, LLM hallucination detection and mitigation, which is one of the most pressing problems in artificial intelligence research today (Tonmoy al., 2024), can also benefit significantly from accurate confidence
* EEqul contribution

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Published in Transactions on Machine Learning Research (06/2025)
estimation as it would serve as strong indicator of the faithfulness of LLM response. This applies to even settings where strategies such as retrieval augmented generation (RAG) are used (Gao et al., 2023) to mitigate hallucinations. Methods for confidence estimation in LLMs assuming just black-box or query access have been explored only recently (Kuhn et al., 2023; Lin et al., 2024) and this area of research is still largely in its infancy. However, effective solutions here could have significant impact given their low requirement (i.e. just query access) and consequently wide applicability.
There exists slight difference in what is considered uncertainty versus confidence in literature (Lin et al., 2024) and so to be clear we now formally state the exact problem are solving. Let (x. y) denote an input-output pair, where z is the input prompt and y the expected ground truth response. Let f()) denote an LLM that takes the input I and produces a response f()). Let X(..-) denote a similarity metric (viz. rouge, bertscore, etc.) that can compare two pieces of text and output a value in [0, 1], where 0 implies the texts are very different while implies they are exactly the same. Then given some threshold 0 E [0, we want to estimate the following probability for an input text x:
Probability of correct = PAM,, 2 0|z)
(1)
In other words, we want to estimate the probability that the response outpulted by the LLM for some input is correct. Unlike for classification or regression where the responses can be compared exactly, text allows for variation in response where even if they do not match exactly they might be semantically the same. Hence, we introduce the threshold which will typically be tuned basod on the metric, the datasct and the LLM.
Having black-box access to an LLM limits the strategies one could leverage to ascertain confidence, but if the proposed strategies are effective they could be widely applied. Previous approaches (Kuhn et al., 2023; Lin et al., 2024; Jiang et al., 2023b) predominantly exploit the variability in the outputs for given input prompt or based on an ensemble of prompts computing different estimators. Our approach enhances this idea where we design different ways of manipulating the input prompt and based on the variability of the answers produce values for each such manipulation. We aver to these values as features. Based on these features computed for different inputs we train a model to predict if the response was correct or incorrect. The probability of each such prediction is then the confidence that we output. Since, the models we use to produce such predictions are simple (viz. logistic regression) the confidence estimates are typically well calibrated (Morrison, 2012). Moreover, being interpretable we can also see which features were more crucial in the estimation. This general framework and the features we engineer are shown in Figure 1. The framework is extensible, since more fcatures or prompt perturbations can be casily added to this framework.
We observe in the experiments that we out state-of-art. baselines for black-box LLM confidence estimation on standard metrics such as Area Under the Receiver Operator Characteristic (AUROC), Area Under Accuracy-Rejection Curve (AUARC) and Expected Calibration Error (ECE), where improvements in AUROC are over 10% in some cases. The confidence model being interpretable we also analyze which features are important for different LLM and dataset combinations. We interestingly find that for a given dataset the important features are shared across LLMs. Intrigued by this finding we apply confidence models built for one LLM to the responses of another and further find that they generalize well across LLMs. This opens up the possibility of simply building a single (universal) confidence model for some chosen LLM and zero shot applying it other LLMs on a dataset.
2 Related Work
The literature studying approaches for estimating the uncertainty in a machine learning model's prediction is large. One organization of this body of work involves dichotomizing it into post-hoc and ab inátío approaches. Post-hoc methods attempt to calibrate outputs of a pre-trained model such that the estimate uncertainties correlate well with the accuracy of the model. These include histogram binning Zadrozny & Elkan (2001); Naeini et al. (2015), isotonic regression Zadrozny & Elkan (2002), and parametric mapping approaches, including matrix, vector, and temperature scaling Platt et al. (1999); Guo et al. (2017); Kull et al. (2019). While variants of these approaches Shen et al. (2024); Desai & Durrett (2020) have been adopted for LLMs they assume a white-box setting where access to the LLM's representations are available. In contrast, our approach quantifies LLM's uncertainties without requiring access to the internals of the LLM. Ab initio
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Table 1: Below we see examples of different prompt perturbations for a prompt from the SQuAD dataset. The color blue and strike outs indicate changes to the input prompt. i) SD does not change the prompt (hence empty cell), but using a stochastic decoding scheme samples multiple responses (four example samplings shown). PP paraphrases the prompt. SP randomly reorders some of the sentences. EFA repeats certain sentences with entities in them. SR removes stopwords. SRC checks for consistency in reasonable size random splits of the LLM response (again prompt is not perturbed). The splitting of the two sentences indicates inconsistency as depicted in red. Thus, the perturbations test an LLM in complementary ways.
Input Prompt
context: The Normans (Norman: Nourmands; French: Normands ; Latin Normanui) the people who, in the 10th and 11th renturies, gave thrir name to Normandy, a region of France. They descended from the Normands ("Norman' comes from 'Norseman") of the raiders and pirates of Denmark, Iceland and Norway who, under their leader Rollo, agreed to swear allegiance to King Charles III of France of the West. During generations assimilation and mixing with the native French and Boman-Gaalest populations, their descendants would gradually merge with the Carolingian cultures of West France. The distinct cultural and ethnic identity the Normans originally emerged in the first half the 10th century, and it continued evolve the conturies that followed. question: In what country is Normandy located?
Prompt Pert.
Perturbed Prompt
Output
SD
France, Demmark, Treland, Norway
PP
context: Normandy, region in France came to bear because of Normans in the 10th and 11th centuries. They drscended from the Normands ("Norman" comes comes-from "Noseman"") of the raiders and pirates of Denmark, Iceland and Norway who, under their leader Rollo, agreed to swear allegiance to King Charles III of France of the West. There was generations of mixing with the Koman-Gaulest ethnic identity of the Normans originally emerged in the first half of the 10th century, and contimued to evolve over the centuries that followed. question: In what country is Normandy located?
Iceland
populations and native French. The distinct cultural and
SP
context: The Normans (Norman Nourmands Prench: : Normands ; Latin Normanui) are the people who, in the 10th and 11th centuries, gave their name to Normandy, a region of France. The distinct cultural and ethinic identity of the Normans originally emerged contimued to evolve over the centuries that followed. They descended from the Normands ("Norman' comes from "Norseman") ') of the raiders and pirates of Demmark, Iceland and Norway who, under their leader Rollo, agreed to swear allegiance to King Charles III of France of the West. During generations of assimilation and mixing with the native French and Roman- Roman-Gaulese Gaulese popula- tions, their descendants would gradually merge with the Carolingian cultures West France. question: what country is Normandy located?
Denmark
first half
10th century, and
EFA
context: The Normans (Norman: Nourmands French: Normands; Latin Normanni) are the people who, in the 10th and 11th conturies, gave their name to Normandy, region of France. The Normans (Norman Nourmands French: Normauds: Latin: Normanni) are the people who, in the 10th and 1ith centuries, gave their name to Normandy, region of France. They scended from the Normands ('Norman' comes from 'Norseman") of the raiders and pirates of Denmark, Iceland and Norway who, under their leader Rollo, agreed
France
allegiance King Charles III of France of the West. During generations of assimilation and mixing with the native French and Roman- Gaulese populations, their descendanta would gradually merge with the Car- olingian cultures of West France. The distinct cultural and ethnic identity of the Normans originally emerged in the first half the 10th century, and it continued to evolve over the centuries that followed. question: In what coun- try is Normandy located?
SR context: The Normans (Norman Nourmands French Normands ; Latin Normanui) ase the people who,-in-the 10th and 11th centuries, gave their name to Normandy, a region of France. They descended thom Normands (*Nor- man' comes from 'Norseman"" ") of-the raiders and pirates of Denmark, Iceland and Norway who under their leader Rollo, agreed to swear allegiance tn King Charles III of France of the West. During gonerations of assimilation and mixing with the native French and Coman-Gaulese populations, their desceu- dants would gradually merge with the Carolingian cultures of West France. The distinct cultural and ethnic identity the Normans originally emergged in the half the 10th century,
Norway
continued to evolve eher-the centuries that followed. question: In what country is Normandy located?
SRC
Normandy is located in Denmark. Nor- mandy is located in Iceland.
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Perturbation
Generator
Perturbed Prompts
Prompt
Stochsski
LLM
Parastracing
Eiiyy Free Ampiliation
Expword
haspense Dorusteccy
Outputs
Feature Extractor
Confidence
Surrogate Model
Semantic Set Stmilartyy
Figure 1: Above our (extensible) framework to estimate confidence of LLM responses. We propose six prompt perturbations which then can be converted to features based on semantic diversity the responses and lexical similarity. The input (tokenized) prompt can optionally be also passed as feature. The output labels for each (imput) prompt are created by checking if the LLM output is correct or not. (interpretable) logistic regression model is then trained these features and outputs that for any imput prompt and LLM response estimate the confidence of it being correct based on our model. Moreover, we can also ascertain the features important in estimating these confidences.
approaches, including, training with mix-up augmentations Zhang et al. (2017), confidence penalties Pereyra et al. (2017), focal loss Mukhoti et al. (2020), label-smoothing Szegedy et al. (2016), (approximate) Bayesian procedures Izmailov et al. (2021), those that involve ensembling over multiple models arrived at by retraining from different random initializations Lakshminarayanan et al. (2017) require substantial changes to the training process or severely increase computational burden, making them difficult to use with LLMs.
For LLMs particular, recent works Jiang et al. (2021); Xiao et al. (2022); Chen et al. (2022) have empirically found evidence of miscalibration and had varying degrees success in better calibrating smaller LLMs using mixup Park & Caragea (2022), temperature scaling and label smoothing Desai & Durrett (2020). Others Lin et al. (2022) have employed supervised fino-tuning to produce verbalized uncertainties to be better calibrated
certain tasks. However, this additionally requires the ability compute gradients of the LLM's parameters. Our black-box approach has no such requirement. Another body of work Kadavath et al. (2022); Mielke et al. (2022); Zhang et al. (2021), learns an auxiliary model for predicting whether a LLM's generation incorrect. We also employ an auxiliary model, but rely on only the prompts to the LLM and the generations produced by the L.M to train it.
Similar to us, other recent works have also explored black-box approaches. For instance, in Kuhn et al. (2023), multiple completions from an LLM are generated, grouped based on semantic content, and uncertainty is quantified across these semantic groups. Lin et al. (2024) exploit insights from spectral clustering to further finesse this process. In Tian et al. (2023); Xiong et al. (2024) the authors use carefully crafted prompts for certain more capable LLMs to express bel beller-calibrate uncertainties. However, this approach is less effective for smaller and sourcee LLMs Shen et al. (2024). Others Jiang et al. (2023b) have relied on ensembles of prompts created using templates or reordering of examples in few shot scttings to quantify confidences. We on the other hand propose dynamic variations of the prompt applicable (even) in the zero-shot setting, where for certain of our features only analyze the response without any variation in the prompt.
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3 Methodology
3.1 Elicitation of Variable LLM Behavior
The baselines generated multiple responses from the model and extracted features from these generations quantify how much the responses diverge. This divergence metric then used to estimate the confidence of the model. We conjecture that rather than just eliciting multiple generations using the same imput, perturbing the imput and then generating myriad of responses would provide a better estimate of the model's confidence. To that end, we use six perturbations elicit the model's response. We first propose six black-box strategies that can clicit variable behavior in an LLM indicative of how trustworthy its output is likely to be. Based on this variability we construct features for our confidence model in the next subsection. Note that all strategies may not be relevant in all cases. For instance, some of the strategies require a context in the prompt, while others such as SRC require longer responses (two or more sentences). For all the perturbations but for Stochastic Decoding and Split Response Consisteney, the perturbations are applied to the context if available to the question of the inpul.
Stochastic Decoding (SD): Similar to previous works, sample the LLM to obtain various generations. The Algorithm elaborates it further. As seen in Table first row after sampling one could have four different unique outputs, which could be indicative of the LLM not being confident in its response.
Algorithm Stochastic Decoding (SD): Algorithm to collect decoded features under various strategies.
output + generate one response from the model using greedy decoding
+ one response using beam search decoding
+ three responses using nucleus sampling
In this strategy, we paraphrase the context in the prompt and observe how that changes the output. An example of this shown in Table is illustrated in listing 2 For paraphrasing, we use back translation, where we convert the original prompt into another language and translate it back into English. We machine translation models from Helsinki-NT.P (referred to MT-Model in the Algorithm) Huggingface. One could also prompt LLM to paraphrase the responses, however, in initial experiments, observed that when context is involved, the model does not paraphrase the entire context and parts of it were omitted.
Paraphrasing (PP):
Algorithm Paraphrasing (PP): Algorithm for paraphrasing via back-translation.
translated sentence + Translate from English to French
back translated sentence + Translate the above sentence from French to English
Sentence Permutation (SP): If the input has several named entities, we noticed that when the order of the named entities is changed without changing the meaning of the sentence, the output of the LLM also varied. We explain in algorithm 3 We first use a Named Entity Recognizer (NER) to identify the named entities and then randomly reorder these sentences. An example of this is seen Table third row, where the last sentence in the prompt is now the second sentence. We use NLTK for identifying atomic sentences and spacy for NER
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Algorithm 3 Sentence Permutation (SP): Algorithm for random sentence permutation via NER sampling.
sentences <- split sentences
tagged sentences tag named entities using NER model
NER sentences Sample 5 sentences with named entities
sentences remove the sER_teneee from sentences for sentence in seRe_enetnsss Insert sentence randomly into the sentences
final sentences
Entity Frequency Amplification (EFA): Similar to above, repeating sentences with named entities could also throw off the model's outputs. Again, here too the output of the L.M should be maintained if the LLM is confident. An example of this in Table fourth row, where the first sentence is repeated twice. We use NLTK for identifying atomic sentences and spacy for NER.
Algorithm Entity Frequency Amplification (EFA): Algorithm for amplifying entity frequency via repeated insertion.
senlences + split sentences
togged_sennencee sentences + HER_modae tags the sentences
NER sentence Sample one sentence from tagged_ segged_sentence
sentences repeat the NER sentence more times the sentences array
Stopword Removal (SR): We remove stopwords from the context. Stopwords are commonly occurring words (viz. 'the", 'are', 'to', etc.) that are assumed to have limited context specific information. Removal of such words should ideally not alter the response of an LLM if the LLM is certain of the answer. An example of this is seen in Table fifth row, where the stopwords are striked out. We ensured that the negative words were not removed as they would change the meaning of the sentence.
Algorithm Stopword Removal (SR): Algorithm for removing stopwords from document.
input < remove stopwords from the input sentence
Split Response Consistency (SRC): In this case like in the SD case the prompt is not perturbed. Rather the output is analyzed where it is randomly split such that cach part is at least single sentence. Semantic inconsistency between the two parts is measured using an NLI model's (specifically, deberta-large- uli) contradiction probability, where one part is taken as the premise and the other the hypothesis. An example of this is seen Table 1 last row, where the two sentences are clearly at odds with each other. This strategy though requires that the response is at least a couple of sentences long, thus responses with fewer than 3 sentences removed.

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Algorithm 6 Split Response Consistency (SRC): Algorithm for checking consistency over random splits of generated response.
generated sentences Obtain generation from the model using nucleus_ pacleus_sampling
sentences split the output into atomic sentences
probability_scon scores +
Repeat 5 times:
inder Split output into two parts
probability obtain the probability that chunk2 entails chunks1
probability_sooe append(probabii1ng
As seen in Table 9 in the appendir, the four perburbations above (PP, SP, EFA and SR) that alter the original prompt still maintain the semantics as intended in almost all cases.
3.2 Featurization
Now based on the above strategies we can construct features to train our confidence model. For each of the first five strategies above we create two types of features: i) based on semantics of the outputs and ii) based on lexical overlap. For the SRC these are not relevant so we create a different feature as seen below.
Semantic Set: Based on the responses of the first strategics (run multiple times) we create semantically equivalent sets for each. A semantically equivalent set consists of outputs that semantically the same. If response entails another response and vice-versa, then they both grouped under the same semantic set. The number of such sets is feature for our model. As such, more the number of sets lower the confidence estimate. For example, if from five paraphrasings we get responses excellent, great, bad, subpar and fantastic, then the number semantic sets would be two as excellent, great and fantastic would form one semantic set, while bad and subpar would form the other.
Lexical Similarity: We compute the average lexical similarity for outputs of each of the first five strategies (run multiple times). The similarity can be measured using standard NT.P metrics such as rouge, blue score etc. The higher the lexical similarity higher the estimated confidence. We use rouge score to quantify the lexical similarity. Considering the same five paraphrasings example described above we would compute the average rouge score considering pairs of the responses and use it as feature.
SRC Minimum Value: As mentioned above, semantic inconsistency between the two parts measured using an NLI models contradiction probability,, where one part taken as the premise and the other the hypothesis. The highest contradiction probability amongst multiple such partitions is the feature value for this strategy. In Table 1 last row, there only two sentences only one split would be done and since the sentences contradict each other the NLI contradiction probability would be high or consistency would be low. Note that optionally one can also pass the entire prompt as feature in addition to the above. In the experiments, we saw minimal improvement with such an addition.
Semantic set and lexical similarity were first used by (Kuhn et al., 2023) where they applied only for SD perturbation discussed previously. We describe the psuedocode of the entire end to end algorithm 3.1
3.3 Label Creation and Confidence Estimation
Once we have the input features to our confidence model we now need to determine labels for these inputs. For training the model we compute labels by matching the LLM output to the ground truth response in the datasot, where a match corresponds the label 1, while mismatch corresponds to a label In particular, we use the rouge score compute the similarity between the output and the ground truth and the score is greater than threshold of 0.3, it corresponds to label otherwise it is deemed incorrect and is
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end end end Published in Transactions on Machine Learning Research (06/2025)
Algorithm 7 Pseudocode for generating features and labels
features- t-1 labels +1
for in dalapoinls do
feature t
for in perturbations do
pert perturb datapoint using perturbation p
pert gere___eeeations = obtain model's response for p semantic sets = Set()
setNum =
Compute SRC Minimum Value
if perturbation SRC then
end feature.appendl
for geml, gen& in
do Compute number of semantic sets if genl entails gens? then
semanticSetNum max (semantii set number of genl, gen2 and 0)
if semanticSe/Num semanticSetNum +- sctNum++
0 then
semantic sets + assign semanticSetNum as the semantic number for genl and gen2
end end else
if genl not in semantic sets then semantic sets + assign setNum++ as the semantic number of gen1 end
if gen2 not in semanlic sels then
semantic sets + assign + as the semantic number of gen2
feature.apper (pmend of semantic values) (orppeucroogeeSSoeeeeeeCreeee
end
Compute label using lexical similarity label generaatoss
prompt the model to get 5 generations for d using nucleus sampling rs <- compute rouge score of label generations
if rS > 0.3 then
label
end
else
label
end
labels.append((lbe))
end
labeled similar to previous works (Lin et al., 2024). With the described features and their labels we train a logistic regression model and use it for predicting confidence scores for out-of-sample outputs.
Given that logistic regression is also an interpretable model we can also study which of our features turn out to be most beneficial and if our model trained one LLM is transferable to other LLMs for the same
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dataset. Transfer across datasets can be more challenging some datasets have contexts (viz. SQuAD); while others do not (viz. NQ) amongst other factors such as difference in domains.
4 Experiment Details
4.1 Models
We demonstrate the efficacy of our method on question answering and summarization tasks. For sum- marization, we used BaRT-large (Lewis et al., 2019) and Pegasus-large (Zhang et al., 2019) and for question answering, Llama-2-13b chat version (Touvron et al., 2023), and Flan-ul2 models (Tay et al., 2023). For detecting entailment, we use deberta-large-nli model which is specialized for NLI tasks (He et al., 2021).
used GPT-4 (OpenAI, 2023), Nistral-7B-Insttuut-vo.. (Jiang et al., 2023a),
4.2 Datasets
For question answering we elicited responses from these models on four datasets, namely, CoQA (Reddy et al., 2019), SQuAD (Rajpurkar et al., 2016), TriviaQA (Joshi ct al., 2017) and Natural Questions (NQ) (Kwiatkowski el. al., 2019). CoQA and SQuAD provide the context and expect the model respond to the question based on the context, while TriviaQA and NQ do not have a context and require the model to tap into its learnt knowledge. For our experiments, we use the validation splits for all the datasets as done previously (Lin et al., 2024). CoQA has 7983 datapoints, TriviaQA has 9960 datapoints, SQuAD has 10,600 used CNN Daily Mail (See et al., 2017)
datapoints and NQ has 7830 datapoints. For summarization, and (Hermann et al., 2015) and XSUM (Narayan et al., 2018) datasets. We use a subset of the validation splits of both the datasets comprising of 4000 datapoints.
We follow previous works (Lin al., 2024), which used 1000 datapoints for hyperparameter tuning, to train our Logistic Regression Classificr and the rest of them were used for evaluation. For cach of the prompt perturbations specified above, we use five generations for each perturbation for more robust evaluation. All results averaged over five runs and we rеport standard deviations rounded to three decimal places for our method. We use zero-shot prompting for the datascts with context. For TriviaQA, Flan-ul2, Mistral-7B- Instruct-v0.2 and GPT-4 also worked well with zero shot prompting while Llama-2-13b chat was performant with a two-shot prompt. For NQ, we used a five shot prompt. The details about the prompts used are provided in the Appendix A.
4.3 Compute
We used internally hostod models to generate the responses. Thus, we used V100s GPUs for the feature extraction step once the responses were generated. The logistic regression model was trained on an intel CPU.
4.4 Baselines
We consider methods proposed in recent works (Kuhn al., 2023; Lin et al., 2024; Xiong et al., 2024) which are state-of-the-art as the baselines. (Kuhn et al., 2023) proposed computing the number of semantic sets, semantic entropy and lexical similarity metrics from the generated outputs. (Lin et al., 2024) use eigen value, eccentricity and degree metrics inspired from spectral clustering to estimate the uncertainty of the model. While (Xiong et al., 2024) used aggregated verbalized confidence scores. We use average verbalized confidence (AVC) as that performed the best in the previous work. To be consistent with our method we average over five estimates. We use the open source code provided by the authors of (Lin et al., 2024) for comparing with the basclines
'https:// http://ttuuuub.mmllitt www.://mm/mn///----/// The results different some cases from those reported their paper passibly because of different random splits and different. L.Mss used, since did provided code.

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Table 2: AUROCs four Q&A and two summarization datascts (CNN, XSUM) using total of six LLMs (Llama, Flan-ul2, Mistral, GPT-4, Pegasus, BART). Higher values are better. Best results bolded.
Dataset(LLM) # of Lexical
EigenValue
Eccentricity
Degree
SE
AVC
Ours
SS
Similarity
(LiviaQA((llama)
0.73
0.76
0.77
0.76
0.77
0.75
0.79
0.88
IriviaQA((llan-uu
0.8
0.86
0.86
0.87
0.85
0.81
0.95
TriviaQA
0.65
0.72
0.76
0.75
0.75
0.68
0.73
0.81
1.003
TriviaQ IriviaQA(GPTT4)) (GPT-4)
0.89
0.91
0.91
0.92
0.91
0.92
0.94
0.96 .07
SQuAD(Llama)
0.65
0.72
0.74
0.58
0.72
0.61
0.61
0.83
.004
SQuAD(Flan-ul2))
0.6
0.7
0.67
0.65
0.67
0.63
0.66
0.8
(QuAA(Mistral)
0.59
0.7
0.67
0.65
0.67
0.62
0.64
0.84
+.008
SQuAD
0.79
0.82
0.84
0.79
0.83
0.81
0.86
0.91 +000
(oQA()
0.61
0.74
0.76
0.76
0.77
0.64
0.78
0.92
CoQA(Flan-ull)
0.61
0.76
0.78
0.78
0.79
0.63
0.76
0.87
#1.001
CoQA (Mistral)
0.74
0.79
0.77
0.79
0.59
0.75
0.81
#..002
CGQA(GPT-4)
0.81
0.86
0.88
0.87
0.88
0.89
0.91
0.95
NQ(Llama)
0.65
0.75
0.75
0.73
0.74
0.68
0.74
0.85
11.003
NQ ull
0.76
0.76
0.86
0.86
0.86
0.81
0.84
0.93
#: #00002
NQ(Mistral)
0.66
0.73
0.77
0.77
0.78
0.68
0.75
0.83
NQ(GPT-4)
0.81
0.85
0.85
0.85
0.88
0.89
0.9
0.93 +..3
CNN (Pegasus)
0.51
0.67
0.73
0.72
0.72
0.55
0.73
0.77
CNN (BART)
0.51
0.60
0.52
0.48
0.54
0.53
0.5
0.57
XSUM (Pegasus)
0.51
0.58
0.69
0.70
0.71
0.54
0.71
0.73
XSUM (BART)
0.59
0.53
0.51
0.52
0.52
0.53
0.57
Table 3: AUARCs four Q&A and two summarization datasets (CNN, XSUM) using total of six LLMs (Llama, Flan-ul2, Mistral, GPT-4, Pegasus, BART). Higher values are better. Best results bolded.
# of Lexical
(Lta))t(LLM)
EigenValue
Eccentricity
Degree
SE
AVC
Ours
SS
Similarity
IriviaQQA(Llam
0.77
0.8
0.8
0.8
0.8
0.79
0.8
0.83 +.01
TriviaQA (FlannFFaaan Fllanull ul2)
0.69
0.72
0.73
0.73
0.73
0.71
0.72
0.74
d: 4.002
TriviaQA Mistral)
0.55
0.63
0.64
0.64
0.64
0.58
0.63
0.64
:..06
(GPT-)
0.8
0.84
0.84
0.84
0.82
0.84
0.85
0. 0.89+.on4
SQuAD(L.lama)
0.3
0.36
0.37
0.28
0.36
0.36
0.31
0.68
.044
SQuAA(FFan--l)
0.73
0.95
0.83
0.82
0.83
0.78
0.83
0.96
(QQuAD(Mistral
0.72
0.93
0.82
0.82
0.82
0.76
0.83
0.96
000
SQuADDGPT--4
0.72
0.72
0.63
0.66
0.69
0.71
0.83
4300000
(Llama)
0.56
0.67
0.67
0.67
0.67
0.61
0.66
0.71
CoQA(FFla FFlan-ul2)
0.7
0.79
0.8
0.79
0.79
0.73
0.77
0.8
+ +0000
CoQA (Mistrr)))
0.46
0.62
0.64
0.63
0.64
0.51
0.62
+.008
CoQA(GPT-N
0.68
0.73
0.72
0.73
0.74
0.72
0.76
0.861
NQ(Llama)
0.37
0.41
0.42
0.41
0.41
0.39
0.42
0.445
40.00
NQ(Flan-ul2)
0.41
0.44
0.47
0.46
0.45
0.44
0.45
0.47
.007
NQ(Mistral)
0.32
0.38
0.40
0.40
0.39
0.36
0.39
0.42
(Q(GPT-4)
0.69
0.73
0.74
0.74
0.74
0.73
0.72
0. 6.799.007
CNN (Pegasus)
0.45
0.51
0.53
0.43
0.52
0.48
0.47
0.74 +: 4.004
CNN (BART)
0.21
0.22
0.21
0.21
0.21
0.23
0.23
0.34
XSUM (Pegasus)
0.16
0.17
0.19
0.17
0.17
0.21
0.19
0.27
XSUM (BART)
0.21
0.22
0.20
0.21
0.22
0.23
0.22
0.35
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Published in Transactions on Machine Learning Research (06/2025)
5 Results
5.1 Confidence Estimation
We use three metrics to evaluate effectiveness of the models: i) Area under the receiver operating character- istic (AUROC) curve which computes the model's discrimination ability for various thresholds. The curve is plotted by varying the thresholds of the prediction probabilities the model and the false positive rale and the true positive rate form the X and the Y axes. The area under this curve called the AUROC. ii) An accuracy rejection curve can also be plotted by increasing the rejection threshold gradually and plotting the model's average accuracy at that threshold. The area under this curve is called AUARC (Lin et al., 2024). iii) Expected calibration error (ECE) is also reported in Table 6, which measures the discrepancy bet accuracy and confidences.
In Table 2, we see that our method quite consistently outperforms all baselines on AUROC. This is also seen for for ECE in Table 6. For estimating the confidence of Llama's responses on TriviaQA, our model is better than the best bascline by 11 percentage points. We are also able to estimate the confidence on the SQuAD dataset using Mistral by 14 percentage points better than the closest competitor. Qualitatively similar results are seen for the SQuAD dataset using Flan-ul2 (better by 10 percentage points) and for the CoQA and NQ datasets using Llama (better by 15 and 10 percentage points respectively). Our results on the summarization datasets using LLMs that excel at summarization (viz. Pegasus and BART) we see again that we are eit better or al. least competitive.
Our performance is also superior to the baselines in most cases on the AUARC metric in Table 3. Our performance on Llama's generations based on the SQuAD dataset exceeds the best baseline's performance by 31 percentage points. In the case of Mistral's performance on TriviaQA and Flan-ul2's generations on CoQA, we are good the baseline. We worse than the baseline on Mistral's generations of CoQA, where our AUROC was also minimally better than the best baselines. In all other instances, our performance is better than others by 1 to 4 percentage points.
We believe these improvements can be attributed to our constructed features and our framework in general. Hence, in the next section we try to ascertain which features for which datasets and LLMs played an important role in predicting the confidences accurately. Note that such an analysis with high confidence is possible because our trained model is interpretable. We also tried to pass the tokenized input prompt additional features (maximum length 256) to our logistic model, however, the improvements were minimal at best and in some cases the performance even dropped possibly because of the model overfitting given that there were now 100s of features. Hence, we do not report these results, although passing the input prompt is still a possibility in general.
Sample Efficiency Though we used 1000 datapoints to train the logistic regression classification, in Table 8, we show that our method is quite performant even with fewer training datapoints. This observation is also consistent when our method uses the same number of queries our baselines reported in 11, 12.
5.2 Confidence Model Interpretability and Transferability
Interpretability: We now study which features in our logistic model were instrumental for accurate confi- dence estimation. In Table 10 included in the appendix, we see the top four features for each dataset- LLM combination. Blanks indicate that there were no features at that rank or lower where their logistic coefficient was greater than 1 As can be seen the simplest feature SD plays role in many cases. This indicates that variability of output for the same input prompt is a strong indicator of response correctness. Moreover, other features such SP and EFA are also crucial in ascertaining confidence as seen in particular for the SQuAD dataset as well as the summarization datascts. This points to order bias when looking at contexts and brittleness to redundant information being also strong indicators of response accuracy. PP and SR also play a role in some cases, where they are more crucial for datasets with no contexts such as TriviaQA and NQ. This makes sense as the specific question is more important here in the absence context and hence the absence of also other features such as SP and EFA. Both the lexical similarity and semantic set featurizations seem to be important in estimating confidence.
11

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Published in Transactions on Machine Learning Research (06/2025)
Table 4: AUROC of the logistic confidence model for one LLM applied to another on a given dataset. As can be seen our confidence models transfer quite well based on AUROC.
Dataset
Source LLM
Self
Target LLM
Target LLM 2
Target LLM 3
Llama
0.88
0.94
(Flan-ul2)
0.80 (Mistral)
0.94
(GPTT4)
Flan-u12
0.94
0.87
(Llama)
0.80 (Mistral)
0.95 (GPT-1)
TriviaQA
Mistral
0.81
0.84 (Llama)
0.91 (Flan-u12)
0.95 (GPT-4)
GPT-4
0.96
0.85 (Llama)
0.92 (Flan-ul2)
0.80 (Mistral)
Llama
0.83
0.81 (Flan-ul2)
0.80 (Mistral)
0.9 (GPT-4)
Flan-ul2
0.8
0.79 (Llama)
0.78 (Mistral)
0.89 (GPT-4)
SQuAD
Mistral
0.84
0.82
(Llama)
0.83 (Flan-ul2)
0.91 (GPT-4)
GPT-4
0.91
0.81
(Tlama)
0.8 (Flan-ul2)
0.81 (Mistral)
Llama
0.92
0.79
(Flan-ul2)
0.78 (Mistral)
0.92 (GPT-1)
Flan-ul2
0.87
0.87
(Llama)
0.81 (Mistral)
0.92 (GPT-4)
CoQA
Mistral
0.81
0.88
(Llama)
0.86 (Flan-ul2)
0.93 (GPT-4)
GPT-4
0.95
0.9 (Llama)
0.86 (Flan-ul2)
0.8 (Mistral)
Llama
0.85
0.91 (Flan-ul2)
0.83 (Mistral)
0.91 (GPT-4)
NQ
Flan-u12
0.93
0.83 (Llama)
0.82 (Mistral)
0.92 (GPT-4)
Mistral
0.83
0.85 (Llama)
0.90 (Flan-ul2)
0.91 (GPT-4)
GPT-
0.93
0.84 (Llama)
0.91 (Flan-ul2)
0.81 (Mistral)
Pegasus
0.77
0.57
(BART)
CNN
BART
0.57
0.77 (Pegasus)
Pegasus
0.73
0.58
(BART)
XSUM
BART
0.57
0.71 (Pegasus)
Table 5: AUARC of the logistic confidence model for one LLM applied to another on a given dataset. As can be seen our confidence models transfer quite well based on AUARC as well.
Dataset
Source LLM
Self
Target LLM
Target LLM 2
Target LLM 3
Llama
0.83
0.74
(Flan-ul2)
0.64 (Mistral)
0.85
(GPT-4)
Flan-u12
0.74
0.83
(Llama)
0.64 (Mistral)
0.83 (GPT-4)
TriviaQA
Mistral
0.64
0.83
(Llama)
0.73 (Flan-ul2)
0.86 (GPT-4)
GPT-4
0.89
0.81
(Llama)
0.71 (Flan-ul2)
0.63 (Mistral)
Llama
0.68
0.62
(Flan-uil
ul2)
0.63 (Mistral)
0.81 (GPT-4)
Flan-u12
0.96
0.89
(Llama)
0.91 (Mistral)
0.82 (GPT-4)
SQuAD
Mistral
0.96
0.90
(Llama)
0.91 (Flan-u12)
0.81 (GPT-4)
GPT-4
0.83
0.65
(Llama)
0.93 (Flan-ul2)
0.94 (Mistral)
Llama
0.71
0.79
(Flan-ul2)
0.61 (Mistral)
0.82 (GPT-4)
Flan-u12
0.80
0.70 (Llama
0.61 (Mistral)
0.81 (GPT-4)
CoQA
Mistral
0.61
0.69
(Llama)
0.79 (Flan-u12)
0.83 (GPT-4)
GPT-4
0.86
0.69 (Llama)
0.79 (Flan-u12)
0.6 (Mistral)
Llama
0.45
0.46
(Flan-ul2)
0.42 (Mistral)
0.77 (GPT-4)
Flan-ul2
0.47
0.45 (Llama)
0.42 (Mistral)
0.76 (GPT-4)
NQ
Mistral
0.42
0.45 (Llama)
0.46 (Flan-ul2)
0.77 (GPT-4)
GPT-4
0.79
0.43 (Llama)
0.46 (Flan-ul2)
0.41 (Mistral)
Pegasus
0.74
0.34 (BART)
CNN
BART
0.34
0.74 (Pegasus)
Pegasus
0.27
0.34 (BART)
XSUM
BART
0.35
0.25 (Pegasus)
12

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Published in Transactions on Machine Learning Research (06/2025)
Looking across the datasets and LLMs we see interesting trend. It seems that for given dataset different LLMs have similar features that appear to be important. For instance, SP lexical similarity is the top feature for all three LLMs on SQuAD, while EFA based feature also appears for Llama and Mistral. For TriviaQA, SD and PP appear for all three models. For CoQA, SD and EFA appear. While for NQ, PP and SD appear as important for all the models. This trend points lowards an interesting prospeci of applying a confidence estimator of one LLM to other LLMs on a given dataset. As such, we could have a universal confidence estimator just built for one of the LLMs that we could apply across others with reasonable assurance. We explore this exciting possibility in the next part.
Transforability: Given the commonality between the important features across LLMs for dataset we now try to test how well does our logistic confidence model for one LLM perform in estimating confidences of another LLM. As seen in Tables 4 and our confidence models are actually quite transferable as they perform comparably or even sometimes better on the other LLMs than the LLM they were built for. This is particularly true for Mistral where, its confidence model performs better for the other two LLMs than itself even coming close in performance to their own confidence models in many cases.
This suggests that we could apply our approach to one LLM and then use the same confidence model to evaluate responses of other LLMs without having to build individual models for them. It would be interesting to further stress test this hepothesis in the future with more L.Ms and datasets. Nonetheless, even in the current setup- of six LLMs and six datasets this observation is interesting and useful.
6 Discussion
In summary, we have provided an extensible framework for black-box confidence estimation of L.MM responses by proposing novel features that are indicative of response correctness. By building interpretable logistic regression model based on these features we were able to obtain statc-of-the-art performance in estimating confidence on six benchmark datasets (CoQA, SQuAD, NQ, TriviaQA, CNN Daily and XSUM) and using six powerful LLMs (Llama-2-13b-chat large). The interpretability of our confidence model aided in identifying features (viz. SD, SP, EFA,PP) that were instrumental in driving its performance for different LLM-dataset combinations. This led to the interesting realization that many of the features crucial for performance were shared across the confidence models of different LLMs for datasct. We thus tested if the confidence models generalized across LLMs for dalasel and found that it indeed was the case leading to the interesting possibility of having universal confidence model trained on just a single LLMs responses, but applied across many others.
Flan-ul2, GPT-4, Pegasus-large and BART-
We used BART and Pegasus for summarization as they are specialized for this task. We tested on Flan-ul2 to validate that our method works on encoder-decoder architectures. Given that our approach worked for GPT-4 we believe is generic enough to extend other powerful LLMs. This includes reasoning models that generate solutions based of on thain-of-thought (COT) traces as these models although on the surface seem more robust suffer from creating incorrect COTs in many cases and sometimes "over think" making mistakes simple problems (Raschka, 2025; Chen et al., 2025). When and how much to "think" is still an active research topic. Our perturbation could alter the COTs affecting the outputs of such models especially when they are uncertain.
Owing to the supervised nature of training the confidence model, one limitation of our approach that at least some of the model's generations must be close to the ground truth for us obtain a reasonable confidence estimator. We used rouge to test accuracy of generations consistent with previous works, however, rouge, like also other NLP metrics, can be error prone. Nonetheless, for summarization with GPT-4 as-a-judge in Tables 13 14 and 15 in the appendix, which are qualitatively similar to those reported in the main paper. In terms of broader impact, our approach can be widely applied as it is simple and works with just black-box access to the LLM. However, our estimates although accurate can be imperfect and this should be taken into account when using our approach in high stakes applications involving LLMs. One should also be cognizant of adversaries aware of our features trying to induce misplaced trust in LLMs they create or prefer.
also report results
13

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Published in Transactions on Machine Learning Research (06/2025)
Given the extensibility of our framework, in the future, it would be interesting to add more features LLMs evolve. One class of such features might be those where the correctness of a response is checked through creating questions that are (causally) related to the original question and context, and seeing how the response varies by asking this question by itself as opposed to in conjunction with the original question and response. Such and other strategies may help in generalizing these confidence estimators also across datasets something that has been seen when we have additional access to logits of L.Mss. Morcover, ideas from selective classification (Bartlett & Wegkamp, 2008; Geifman & El-Yaniv, 2017) could also be adapted for learning a better confidence model.
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Published in Transactions on Machine Learning Research (06/2025)
A Prompt Design
Prompts for TriviaQA:
Flan-ul2 model and GPT-4: Answer the following question in less than 5 words
Q: {question]
A:
Llama-2-13ben model Answer these following question succinctly possible in less than 5
words
Q: In Scotland a bothy/bothie a?
A: House
Q: Who is Posh Spice the spice girls pop band?
A: Victoria Beckham
Q: (question)
A:
Mistral-7B--Istrrutt-vo model <<> (NNST] Answer the following question as suceinctly as pos- sible in plain text and in less than words. question 

Prompts for CoQA
Flan-ul2 model, -lama-2-13--chr model and GPT-4: for the following question based on the context below: context: [context] Question: fquestion Answer:
Provide an answer in less than 5 words
Mistral-7B-Innn --nstuutt-ttttt model <S>
 Provide an answer in less than 5 words for the following question based on the context below:
context: [context]
Question: fquestion}
Answer: 

Prompts for SQuAD
Flan-u12 model, Llama-2-13b-chat model and GPT-4: Provide an answer for the following question based on the context below, in less than words:
Mistral--B-Inssructvt02 model [INST Provide an answer for the following question based on the context below, in less than 5 words: context: {context]
Question: {question}
Answer: 

Prompts for NQ:
For all the models we used the following prompt:
Here are Example Question Answer pairs:
Question: who makes up the state council in russia
Answer: governors and presidents
Question: when does real time with bill maher come back
Answer: November 9, 2018
Question: where did the phrase american dream come from
Answer: the mystique regarding frontier life
Question: what do you call a group of cels
Answer: bed
Question: who wrote the for mission impossible fallout.
18

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Published in Transactions on Machine Learning Research (06/2025)
Table 6: RCFs on four Q&A and two summarization datasets (CNN, XSUM) using total of LLMs Pegasus, BART). Lower values are better. Best results bolded.
(Llama, Flan-ul2, Mistral, GPT-4,
# of SS
Lexical Similarity
Datase((LLM)
EigenValue
Eccentricity
Degree
SE
AVC
Ours
TriviaQA(llama)
0.13
0.12
0.11
0.11
0.1
0.12
0.09
0.04
TriviaQA (Flan-ul2)
0.06
0.07
0.05
0.05
0.05
0.07
0.06
0.01
(MistaQa(Mistral)
0.17
0.12
0.1
0.1
0.11
0.16
0.11
0.05
(GPT-4
0.07
0.08
0.09
0.09
0.08
0.09
0.03
0.01
(LuuA(Llana))
0.15
0.12
0.1
0.24
0.13
0.18
0.18
0.04
SQuAADFFlan-ul
0.17
0.09
0.13
0.14
0.14
0.17
0.16
0.06
SQuAD(Mistral)
0.2
0.12
0.14
0.15
0.14
0.17
0.15
0.04
SQuAD(GPT-4)
0.11
0.09
0.08
0.19
0.07
0.1
0.11
0.02
(LQam)
0.16
0.1
0.08
0.09
0.09
0.18
0.09
0.02
0.15
0.11
0.09
0.09
0.09
0.17
0.08
0.03
CoQA BBQQAMiiss (Mistral)
0.18
0.1
0.07
0.09
0.07
0.21
0.09
0.05
CGQA(GPT--4
0.11
0.09
0.08
0.08
0.08
0.06
0.05
0.02
NQ(Llama)
0.13
0.08
0.08
0.09
0.09
0.12
0.08
0.04
NQ(Flan- ul2)
0.1
0.09
0.06
0.06
0.06
0.06
0.05
0.02
NQ(Mistral)
0.15
0.09
0.11
0.1
0.09
0.12
0.09
0.05
(Q(GPT-1)
0.1
0.05
0.05
0.06
0.06
0.09
0.06
0.02
CNN (Pegasus)
0.19
0.16
0.11
0.12
0.12
0.19
0.09
0.07
CNN (BART)
0.51
0.19
0.26
0.29
0.25
0.26
0.24
0.19
XSUM (Pegasus)
0.21
0.2
0.15
0.13
0.11
0.21
0.11
0.09
XSUM (BART)
0.26
0.22
0.24
0.27
0.26
0.25
0.23
0.2
Answer: Lorne Balfe
Now answer the following Question succinctly, similar to the above examples: Question: (question)
Answer:
Prompt for GPT-4 as-a-judge: Please provide a score between and of how similar the summaries are. indicating very similar and 0 indicating very different.
B Results
19

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Published in Transactions on Machine Learning Research (06/2025)
Table 7: Results with different number of decodings (for each the features) using our method. Five decodings correspond results the paper. can be seen reducing to three decodings our approach still maintains performance.
Our AUROC
Our AUROC
Our AUARC decodings
Our AUARC decodings
Our ECE decodings
Our ECE decodings
Dataset (LLM)
5 decodings
3 decodings
TriviaQA
0.88
0.86
(188
0.81
0.04
0.05
TriviaQA (FFan-ul2)
0.95
0.91
0.74
0.72
0.01
0.02
IriviaQA(Mistraa
0.81
0.81
0.64
0.65
0.05
0.05
(GPT-4
0.96
0.93
0.89
0.86
0.01
0.02
(QuAA(T.lmma)
0.83
0.81
0.68
0.65
0.04
0.06
SQuAD(Flan Flan-ul2) ul2)
0.8
0.8
0.96
0.94
0.06
0.08
SQuAD (Mistral)
0.84
0.82
0.96
0.93
0.04
0.05
SQuAD GPT-4)
0.91
0.02
0.03
CoQQ(Llannn
0.92
0.91
0.71
0.69
0.02
0.03
(FQnnFla)-n12)
0.87
0.85
0.8
0.78
0.03
0.05
CoiAsMistrral
0.61
0.05
0.06
CGQA(GPT--)
0.95
0.92
0.86
0.84
0.02
0.03
NQ(Llama)
0.85
0.83
0.45
0.44
0.04
0.06
NQ((lan-ul22)
0.93
0.91
0.47
0.45
0.02
0.03
NQ(Mistral)
0.83
0.81
0.42
0.4
0.05
0.06
NQ(GPT--4)
0.93
0.91
0.79
0.77
0,02
0.03
CNN (Pegasus)
0.77
0.75
0.74
0.72
0.07
0.09
CNN (BART)
0.57
0.55
0.34
0.33
0.19
0.21
XSUM (Pegasus)
0.73
0.71
0.27
0.25
0.09
0.11
XSUM (BART)
0.57
0.55
0.35
0.33
0.2
0.22
Below we see how the AUROC, AUARC and ECE values vary with different mumber of samples used to train our logistic regression model for the Q&& datasets. As can be seen our uncertainty estimation procedure is performant even with fower samples for training.
Table 8:
Dataset
LLM
250 samples
500 samples
1000 samples
Mlama
0.83, 0.80, 0.05
0.86, 0.81, 0.04
0.88, 0.83, 0.04
Flan-ul2
0.95, 0.73, 0.03
0.95, 0.74, 0.02
0.95, 0.74, 0.01
TriviaQA
Mistral
0,80, 0.63, 0.06
0.80, 0.63, 0.06
0.81, 0.64, 0.05
GPT-4
0.92, 0.85, 0.02
0.94, 0.88, 0.01
0.96, 0.89, 0.01
Llama
0.8, 0.65, 0.07
0.81, 0.66, 0.05
0.83, 0.68, 0.04
Flan-ul2
0.76, 0.91, 0.07
0.78, 0.94, 0.06
0.8, 0.96, 0.06
SQuAD
Mistral
0.79, 0.90, 0.06
0.81, 0.93, 0.05
0.84, 0.96, 0.04
GPT-4
0.89, 0.81, 0.04
0.9, 0.82, 0.02
0.91, 0.83, 0.02
Llama
0.91, 0.70, 0.04
0.92, 0.71, 0.03
0.92, 0.71, 0.02
Flan-u12
0.86, 0.79, 0.05
0.87, 0.80, 0.04
0.87, 0.80, 0.03
CoQA
Mistral
0.80, 0.60, 0.07
0.81, 0.61, 0.06
0.81, 0.61, 0.05
GPT-4
0.93, 0.84, 0.04
0.94, 0.86, 0.03
0.95, 0.86, 0.02
Llama
0.81, 0.4, 0.06
0.82, 0.41, 0.05
0.85, 0.45, 0.04
Flan-ul2
0,86, 0.43, 0.04
0.87, 0.45, 0.02
0.93, 0.47, 0.02
NQ
Mistral
0.80, 0.37, 0.06
0.81, 0.39, 0.06
0.83, 0.42, 0.05
GPT-4
0.9, 0.77, 0.03
0-91, 0.78, 0.04
0.93, 0.79. 0.02
Table 9: Percentage of prompt perturbations entailed by the original prompt for the SQuAD dataset using deberta-large model. This dataset also has context unlike some of the other Q&A datasets and hence, is more challenging case of our features to maintain semantics. As can be seen perturbations produce the intended effect of maintaining the semantics of the original prompt in most cases.
Paraphrasing Sentence Permutation Entity Frequency Amplification Stopword Removal
99.81%
99.23%
99.66%
99.12%
20

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Published in Transactions on Machine Learning Research (06/2025)
Up to four important features (absolute coeflicient value > le=2) ranked based on our logistic regression model for the different dataset and LLM combinations. feature, while Rank 4 is the least important amongst the four.
Table 10:
Rank 1 indicates the most important
Dataset (LLM)
Rank
Rank
Rank
Rank
TriviaQAA (LlaiaQQ(Lllmaa)
SD lexical
SD semantic
SR lexical
PP lexical
similarity
set
similarity
similarity
(FiviaQA(Flan-u2))
SD lexical
SD semantic
PP semantic
PP lexical
similarity
similarity
IHiviaQA(Mistral)
SD lexical
PP lexical
SP semantic
SD semantic
similarity
similarity
set
set
(GPTT)QA(GPT-I)
ST) lexical
ST) semantic
PP lexical similarity
similarity
set.
SQuAD(Llama)
SP lexical
EFA semantic
similarity
set
SQuAD(Flan-ul))
SP lexical
similarity
(MusD(Misttal)
SP lexical
EFA semantic
similarity
SQuAD(GPT--)
SP lexical
EFA semantic
similarity
set
CoQA([lama)
SD lexical
EFA semantic
SD semantic
SR lexical similarity
similarity
set
set
CoQA(Flan-ul2)
SD lexical
FFAA semantic
SD semantic
SP lexical
similarity
set
set
similarity
CoQA(Mistral)
SD lexical
SD semantic
EFA semantic
EFA lexical
similarity
similarity
CoQA(GPT-1)
SD lexical
EFA semantic
SD semantic
SP lexical
similarity
set
set
similarity
NQ(Llama)
PP lexical
SD semantic
SD lexical
SP lexical
similarity
set
similarity
similarity
NQ(Flan-ul22
SR semantic
ST) lexical
SP lexical
PP lexical
set
similarity
similarity
similarity
NQ(Mistral)
PP lexical
semantic
SD lexical
SP lexical
similarity
set
similarity
similarity
NQ(GPT-4)
PP lexical
SD semantic
SD lexical
SP lexical
similarity
similarity
similarity
CNN(Pegasus)
SD lexical
EFA lexical
SR lexical
SP lexical
similarity
similarity
similarity
similarity
CNN(BART)
SR lexical
SP lexical
EFA lexical
SP semantic
similarity
similarity
similarity
set
XSUM(Pegasus)
SD lexical
EFA semantic
PP lexical
SD semantic
similarity
set
similarity
set
XSUM(BART)
SR lexical
SP lexical
EFA lexical
SP semantic
similarity
similarity
similarity
set
21

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Published in Transactions on Machine Learning Research (06/2025)
Table 11: AUROCs on four Q&A and two summarization datasets (CNN, XSUM) using a total of six LLMs (Llama, Flan-ul2, Mistral, GPT-4, Pegasus, BART), where the number of queries the LLMs is the for the baselines and our method. Higher values are better. Best results bolded.
# of
Lexical
(LLM)
EigenValue
Eccentricity
Degree
AVC
Ours
S$
Similarity
TriviaQA(Llamn
0.74
0.76
0.76
0.77
0.77
0.76
0.79
0.88
(Flan-u2(
0.82
0.81
0.87
0.86
0.86
0.85
0.81
0.95
(Mista)A(Mistral)
0.65
0.72
0.76
0.75
0.75
0.68
0.73
0.81
[GPV)a
0.89
0.91
0.91
0.92
0.91
0.92
0.94
0.96
(LaaA()llma))
0.65
0.72
0.74
0.58
0.72
0.61
0.61
0.83
SQuAD(Flan-ul2)
0.6
0.7
0.67
0.65
0.67
0.63
0.66
0.8
(Mssta)isstal))
0.59
0.7
0.67
0.65
0.67
0.62
0.64
0.84
SQPAD(GPT-1)
0.79
0.82
0.84
0.79
0.83
0.81
0.86
0.91
CoQA(Llama))
0.61
0.74
0.76
0.76
0.77
0.64
0.78
0.92
0.61
0.76
0.78
0.78
0.79
0.63
0.76
0.87
(Mitrriss
0.56
0.74
0.79
0.77
0.79
0.59
0.75
0.81
CGPA())T
0.81
0.86
0.88
0.87
0,88
0.89
0.91
0.95
NQ(Llama)
0.65
0.75
0.75
0.73
0.74
0.68
0.74
0.85
NQ(Flan-ul2)
0.76
0.76
0.86
0.86
0.86
0.81
0.84
0.93
NQ(Mistral)
0.66
0.73
0.77
0.77
0.78
0.68
0.75
0.83
NQ(GPT-4)
0.81
0.85
0.85
0.85
0.88
0.89
0.9
0.93
CNN (Pegasus)
0.51
0..67
0.73
0.72
0.72
0.55
0.73
0.77
CNN (BART)
0.51
0.59
0.52
0.48
0.54
0.53
0.5
0.57
XSUM (Pegasus)
0.51
0.58
0.69
0.70
0.71
0.54
0.71
0.73
XSUM (BART)
0.51
0.59
0.54
0.52
0.52
0.52
0.53
0.57
Table 12: AUARCs on four Q&&A and two summarization datasets (CNN, XSUM) using a total of six LLMs (Tlama, Flan-ul2, Mistral, Pegasus, BART), where the number of queries to the I.I.Ms the same for the baselines and our method. Higher values are better. Best results bolded.
# of
Lexical Similarity
(LLM)
EigenValue
Eccentricity
Degree
SE
AVC
Ours
SS
TriviaQ TriviaQA(Llama)
0.76
0.8
0.81
0.79
0.8
0.83
0.7
0.72
0.73
0.73
0.73
0.71
0.72
0.74
(Mistal))Mistral)
0.55
0.63
0.64
0.64
0.64
0.58
0.63
0.64
(GPV))
0.8
0.84
0.84
0.84
0.82
0.84
0.85
0.89
(LaaA()lama)
0.3
0.36
0.37
0.28
0.36
0.36
0.31
0.68
SQuAD(Flan-ull)
0.73
0.95
0.83
0.82
0.83
0.78
0.83
0.96
(MuAD(Mistral)
0.72
0.93
0.82
0.82
0.76
0.83
0.96
SQPAD(GPT--)
0.7
0.72
0.72
0.63
0.66
0.69
0.71
0.83
CoQAA(llama)
0.56
0.67
0.67
0.67
0.67
0.61
0.66
0.71
0.7
0.79
0.79
0.79
0.73
0.77
0.8
CoQAA(Mistral
0.46
0..62
0.64
0.63
0.64
0.51
0.62
0.61
CGPA()PT-4)
0.68
0.73
0.72
0.73
0.74
0.72
0.76
0.86
NQ(Llama)
0.37
0.41
0.42
0.41
0.41
0.39
0.42
0.45
NQ(Flan-ul2
0.41
0.44
0.47
0.46
0.45
0.44
0.45
0.47
NQ(Mistral)
0.32
0.38
0.40
0.40
0.39
0.36
0.39
0.42
(Q(GPT-1)
0.69
0.73
0.74
0.74
0.74
0.73
0.72
0.79
CNN (Pegasus)
0.45
0.51
0.53
0.43
0.52
0.18
0.47
0.74
CNN (BART)
0.21
0.22
0.21
0.21
0.21
0.23
0.23
0.34
XSUM (Pegasus)
0.16
0.17
0.19
0.17
0.17
0.21
0.19
0.27
XSUM (BART)
0.21
0.22
0.20
0.21
0.22
0.23
0.22
0.35
22

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Published in Transactions on Machine Learning Research (06/2025)
Table 13: AUROCS two summarization dalasets (CNN, XSUM) with GPT-4 judge. Higher values
better. Best results bolded.
of SS
Lexical Similarity
Dataset(LLM)
EigenValue
Eccentricity
Degree
SE
AVC
Ours
CNN (Pegasus)
0.54
0.65
0.76
0.77
0.75
0.61
0.75
0.81
CNN (BART)
0.55
0.64
0.55
0.52
0.58
0.56
0.54
0.64
XSUM (Pegasus)
0.56
0.62
0.72
0.74
0.73
0.6
0.75
0.79
XSUM (BART)
0.55
0.63
0.56
0.54
0.55
0.56
0.59
0.61
Table 14: AUARCs two summarization datasets (CNN, XSUM) with GPT-4 as judge. Higher values are
better. Best results bolded.
# of
Lexical Similarity
(LI.))
FigenValue
Fecentricity
Degree
SE
AVC
Ours
SS
CNN (Pegasus)
0.49
0.55
0.58
0.49
0.57
0.52
0.53
0.77
CNN (BART)
0.25
0.26
0.27
0.26
0.26
0.27
0.29
0.35
XSUM (Pegasus)
0.19
0.22
0.23
0.2
0.21
0.23
0.21
0.29
XSUM (BART)
0.26
0.26
0.25
0.27
0.27
0.27
0.26
0.37
Table 15: ECEs two summarization datasets (CNN, XSUM) with GPT-4 as a judge. Lower values are better. Best results bolded.
# of
Lexical Similarity
Dataset(LLM)
EigenValue
Eccentricity
Degree
SE
AVC
Ours
SS
CNN (Pegasus)
0.18
0.14
0.11
0.1
0.09
0.15
0.07
0.05
CNN (BART)
0.48
0.17
0.24
0.25
0.22
0.22
0.22
0.14
XSUM (Pegasus)
0.18
0.18
0.13
0.11
0.09
0.17
0.1
0.06
XSUM (BART)
0.23
0.19
0.21
0.23
0.23
0.22
0.2
0.16
23


Reviews:
[{"review_id": "812w3vwQFK", "reviewer": "Reviewer_3vgB", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- **(S1)**: The black-box nature of the method makes it broadly applicable across different LLMs.\n- **(S2)**: Empirical results are thorough, covering multiple datasets and models, which are strengthening the paper.\n- **(S3)**: The idea of model transferability is intriguing and potentially valuable in practice.\n\nWeaknesses:\n- **(W1)**: **(Solved)** I believe, important aspects such as how the semantic sets are computed (e.g., thresholds, NLI model configuration) and how lexical similarity metrics are aggregated are under-specified. Generally, the work lacks many small details, that could be helpful for reproducibility.\n- **(W2)**: **(Solved)** While logistic regression is interpretable, the paper does not fully utilize this advantage. It only reports top features but does not test their statistical significance (e.g., via t-tests or permutation importance), which could highlight robustness or redundancy. The feature importance rankings are based on absolute coefficients, as I understood, which can be misleading without statistical tests or confidence intervals.\n- **(W3)**: **(Solved)** The paper can be difficult to follow. The definitions and motivations for individual features need clearer exposition and mathematical clarity. Also the figures are low resolution, and tables could be improved. \n- **(W4)**: Many LLMs used (e.g., Flan-ul2, LLaMA 2) are now relatively old by current standards. This raises concerns about the generalizability of the findings to more recent, more capable models.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a black-box confidence estimation framework for large language models (LLMs) using prompt perturbation, derived features and logistic regression model. The authors introduce a set of prompt transformation strategies to extract semantic and lexical features from the outputs. These features are then used to train a logistic regression classifier that estimates the confidence of the LLM’s response. The method is evaluated on four QA datasets and two summarization tasks across six different LLMs.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- Provide precise details for semantic set creation and lexical feature aggregation, and other missing details\n- Clearly describe baseline implementations and ensure experimental parity.\n- Add statistical tests (e.g., permutation importance or t-tests) to back up claims of interpretability.\n- Discuss the limitations of the selected LLMs and address whether the results are likely to hold for newer architectures.\n- Improve the quality of presentation, i.e. structure, tables and figures resolution", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "rqIy6tgHJw", "reviewer": "Reviewer_2YJT", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength\n* The task definition is clear.\n* There are sufficient experimental results to support the claims.\n\nWeakness:\n* Figure 1 is low resolution. Therefore, it is better to use a high resolution for Figure 1.\n* Table 2 should be in the middle for better reading.\n* This work uses predefined features, so it is not clear whether these features still work well on a large domain.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This work builds a method to evaluate the LLM confidence under the black-box. To be specific, the author uses different predefined features to train a model to estimate the confidence. The extensive experiment results have supported the effectiveness of their methods.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "N/A", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "ucw1KPIruI", "reviewer": "Reviewer_6eJV", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n\nThe proposed method is simple and seems like a natural approach. The design of the feature set is sound and well described. Considering the improved performance shown in the experimental results, this could be a useful method in application.\n\nTo the best of my knowledge, the baselines that the method is compared against are sufficient. The variety of LLMs used also strengthen the results. The use of both large black-box and white-box models gives confidence that the results generalize.\n\n**Weaknesses**\n\nThe contributions are purely empirical and lack theoretical motivation. The use of logistic regression for its simplicity makes sense from a practical perspective, but it would be helpful to have some theoretical motivation for its use. Have other models been explored and does this method work well with other models?\n\nThe presentation of the paper can be improved. The results are presented as an extensive set of tables, making it difficult for a reader to extract insights efficiently. \n\nThe result on transferability is particularly interesting, but it could be explored in more detail, as it is a strong claim. This claim could be better supported with additional evidence.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors propose a method to estimate the confidence of an LLM output using a logistic regression model. This logistic regression model uses features derived from a set of LLM outputs generated using perturbed prompts. These features capture the variability in responses. Experiments spanning six datasets and six LLMs suggest improved performance from relevant baseline methods, measured by AUROC, AUARC, and ECE. Additional results shed insights on feature importance, the performance of the method when reducing the number of samples, and an experiment whose results suggest the model may generalize between LLMs.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "**Critical**\n\nThe presentation of results should be made more reader friendly. I would suggest visualizing some of these tables and moving some of the contents to an appendix. For example, Table 6 could be visualized for 5 decodings, color coded by dataset, with a line style for LLMs, while the results for 3 decodings could be moved to the appendix (This example is just a suggestion; please use your discretion.)\n\nI would appreciate clarification on whether this method relies on logistic regression or if it performs well with other models. A brief experiment comparing alternative models would help assess its generalizability. Additionally, providing diagnostics for the logistic regression models used in the paper would strengthen the analysis. If logistic regression is indeed the most suitable choice, a clear justification for its selection would be beneficial.\n\n**Strengthen**\n\nIs it possible that the transferability result relies on the distribution of features to be similar between different LLMs for a given dataset?  It would be helpful to have a brief qualitative analysis or visualization to shed light on the mechanism of transferability.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "XjSmZi9eWk", "reviewer": "Reviewer_EyEU", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "# Strengths\n1. The problem studied in this paper is important. \n\n# Weaknesses\n1. This paper is poorly written. The extensive textual description in section 3 makes reading very difficult. There are no formulas, no algorithms, and no graphs.\n2. The contribution of this paper is very limited. The proposed disturbed methods in section 3.1 are not novel and can almost be regarded as a standard text perturbation approach. \n3. The only contribution is to featuring the above strategies and training a logistic regression confidence model. However, if so, the transferability is poor and the complexity is higher. The author often emphasizes interpretability, but there are no experiments to prove it.\n4. There are no other compared baselines (e.g. semantic uncertainty) in the experiments.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "Estimating the uncertainty of LLMs is one of the most pressing problems. Existing approaches (e.g., semantic uncertainty[1]) propose to estimate the semantic similarity of multiple outputs with the same input prompt, i.e., disturbing input prompt to obtain multiple outputs, clustering the same semantic set, and calculating the entropy. This paper enhances this by training a (interpretable) logistic regression model to estimate the uncertainty.  The experimental results show that the proposed approach achieves impressive performance.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "See the weaknesses.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)