Generating with Confidence: Uncertainty Quantification for Black-box Large Language Models
Authors: Zhen Lin
Shubhendu Trivedi
Jimeng Sun
Venue: Accepted by TMLR ()

Abstract: Large language models (LLMs) specializing in natural language generation (NLG) have recently started exhibiting promising capabilities across a variety of domains. However, gauging the trustworthiness of responses generated by LLMs remains an open challenge, with limited research on uncertainty quantification (UQ) for NLG. Furthermore, existing literature typically assumes white-box access to language models, which is becoming unrealistic either due to the closed-source nature of the latest LLMs or computational constraints. In this work, we investigate UQ in NLG for *black-box* LLMs. We first differentiate *uncertainty* vs *confidence*: the former refers to the ``dispersion'' of the potential predictions for a fixed input, and the latter refers to the confidence on a particular prediction/generation. We then propose and compare several confidence/uncertainty measures, applying them to *selective NLG* where unreliable results could either be ignored or yielded for further assessment.  Experiments were carried out with several popular LLMs on question-answering datasets (for evaluation purposes). Results reveal that a simple measure for the semantic dispersion can be a reliable predictor of the quality of LLM responses, providing valuable insights for practitioners on uncertainty management when adopting LLMs.

Published in Transactions on Machine Learning Research (05/2024)
Generating with Confidence: Uncertainty Quantification for Black-box Large Language Models
Zhen Lin" shenLing@illsnots edu
Shubhendu Trivedi
shubhendu@csasl, l. mät. edu
Jimeng Sun 1.2
Universily of Illinois at Urbana- Chanapgggaaa
Corle's Mlinois College of Medicine, University of Illinois at Urbana- Channaan
Reviewed on OpenReview: https: // opwnrevee..e.ee.e.e.eee net/ forum? id- DWK.C
Abstract
Large language models (LLMs) specialixing in natural language generation (NLG) have recently started exhibiting promising capabilities across a variety of domains. However, gauging the trustworthiness of responses generated by LLMs remains an open challenge, with limited research on uncertainty quantification (UQ) for NLG. Furthermore, existing literature typically assumes white-box access to language models, which is becoming unreal- istic either due the closed-source nature of the latest LLMs computational constraints. In this work, we investigate UQ in NLG for black-bor LLMs. We first differentiate uncer- tainty vs confidence: the former refers to the "dispersion" " of the potential predictions for a fixed input, and the latter refers to the confidence on a particular prediction /gnnrirtionnnnntiion. We then propose and compare several confidence/ coufidence/uncerriint uncertainty measures, applying them selec- tive NLG where unreliable results could either be ignored or yielded for further assessment. Experiments were carried out with several popular LLMs question-ansswring datasets (for evaluation purposes). Results reveal that simple measure for the semantic dispersion can be reliable predictor of the quality of L.L responses, providing valuable insights for practitioners on uncertainty management when adopting LLMs. The code to replicateour experiments available at https://giithi: com/ wwt.//woomc////.ee. 2/1////nn/////
1 Introduction
Large language models (LLMs) have recently gained significant attention in natural language generation
(NLG) (Touvron et al. 2023a Katz et al. 2023 OpenAl 2023 Chowdhery et al. 2022). Trained on vast amounts of data, they exhibit impressive abilities in generating human-like responses. As such advances invariably lead to wider adoption LLM for language generation tasks, such as question-answering (QA), it is crucial to quantify their uncertainty.
Uncertainty quantification (UQ) is well-studied within machine learning. A reliable measure of uncertainty is important to decide when to trust model. When model shows high uncertainty returns low-confidence predictions the input should either be rejected or yielded to further evaluation (Gal & Ghahramani 2016
See Section 3.2 for discussion uncertainty confidence.

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This is an area of active rescarch typically called selective classification (Chow 1970 Lin et al. 2022b; Geifman & El-Yaniv] 2017]. Similarly, in NL.G, one might refuse the generation by the LLM to the input with high uncertainty/lon /low confidence. Selective generation basing on uncertainty estimates could potentially improve the decision making process, espocially for high-stakes applications such as medical or legal question-answering.
While UQ has been an important topic for classical machine learning tasks like classification or regres-
sion [akshminarayanan et al. 2017 Gal & Ghahramani 2016 Hernández-Lobate & Adams 2015 Abdar
jet al.
2021a), it bears specific challenges for NLG and has attracted limited attention until recently (in
jet al.
2022a Kuhn et al. 2023: Malinin & Gales 2021). Numerous challenges hinder the direct application
of UQ methods from classification or regression. First, the output space has forbiddingly high dimensional- ity, rendering most measures, such as the exact entropy of predicted probabilities, unfeasible. Furthermore, distinct token sequences may convey identical meanings, nocessitating an effective uncertainty measure to operate in the semantic meaning space (Kuhn et al., 2023). Lastly, many existing LLMs are black- boxes served via APIs, implying that end-users lypically lack while-box access. Even when such access is available, many LLMs are too large to run for most. Note that such considerations are orthogonal to the problem of overconfidence: We need a ranking-wiss wise informative confidenco/ confidence/unccrtiinty uncertainty mcasure before the problem of overconfidence even appears (see discussion in Section
In this study, we explore the problem of uncertainty quantification in NLG with black- box LLMs. Unlike some prior research (Mielke et al. 2020; Lin et al.) 2022a Kadavath et al. 2022 Kuhn et al. 2023), we focus on the more realistic scenario where we only possess access to the generated text, rather than the numerical outputs such token-level logits. We first introduce set of measures designed to assess the uncertainty of the imput and the confidence of each generation The main idea entails estimating the uncertainty /confidence from multiple generations from the LLM. Then, we demonstrate the application of these uncertainty estimates in the context of free-form QA. QA datasets are used for the simplicity of evaluation, as completely open-ended NLG tasks require expensive human evaluation. The main contributions of this paper are summarized as follows:
We investigate uncertainty quantification for black-box LLMs, a previously under-explored lopic, and assess its value in the downstream task of selective natural language generation.
We put forward several simple yet effective tochniques for estimating uncortainty associated with input
data and determining the confidence level of each individual generated response.
Through extensive experiments several popular LLMs and question-answering datasets, observe that proposed measures demonstrate high level of effectiveness in pinpointing challenging (uncertain) questions and predicting the quality of their corresponding answers.
2
Related Works
The quantification of uncertainty has emerged as significant area of research across various machine learning domains, including natural language processing (NTP). However, previous NLP studies have predominantly addressed the associated UQ challenges similarly to classification or regression methodologies (Desai &
Durrett 2020 Jiang et al., 2021 Kamath et al. 2020 Wang et al. 2022 Xiong et al., 2023). For instance, task as a multiple-choice problem, reducing it to a de facto classification task rather than directly engaging with froe-form generation. As rocently argued in Kuhn et al. (2023), such approaches enable the application of UQ measures akin to those employed in more extensively researched classification regression contexts, but overlook the generative aspects and distinct challenges of NLG.
Kamath et al.
2020) examines the selective question--nsseriin
Recently, some research started to study uncertainty quantification for NLG. One line of research involves asking the LLM itself for its confidence, with or without additional fine-tuning (Kadavath et al. 2022 Lin let al. 2022a Mielke et al., 2020 Chen Mueller] 2023). Apart from being expensive, such approaches can be hard to generalize due to opaque training details or differences between LLMs (Kuhn et al., (023). The work most relevant to ours is Kuhn et al. (2023). which proposes to compute the "semantic entropy" by
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considering the equivalence relationships amongst generated answers, and requires training. Nonetheless, it still requires access to the token-level numerical output of the LLM,, which is not always available.
As discussed, one of the most pertinent applications of uncertainty quantification in NLG involves the de- velopment of methods for selective NLG (or, NLG with rejection). This emerging field has limited research to date, but shares close ties with classification with rejection. Both tasks can be viewed as determining when to trust a model, whether is classifier or an LLM. Numerous classification with rejection methods emphasize the identification of reliable confidence score (some of which are jointly trained with the classi- fier) (Corbière et al,, 2019: Fumera et al., 2000 Geifman & El- LLaiv 2017 Jiang et al. 2018 which is often not only dependent on the input but also on the prediction. As existing uncertainty quantification rescarch for NLG primarily focuses on input uncertainty (Kuhn et al. 2023 Malinin & Gales 2021), it overlooks the crucial aspect of confidence, which is essential in deciding when to trust an LLM's response (see Section 3.2 for more discussion). Recent works have explored selective classification 2022a b). However, the distinct generative nature of NLG precludes the direct adaptation of confidence measures from the classification with rejection literature. This paper serves as step to bridge this gap and enhance the effectiveness of uncertainty quantification in NLG.
NLP tasks (Varshney et al.
It is worth noting that the issue of LLMs being overconfident Mielke et al. 2022: Si et al. 2022 Xiong
et al. 2023) is orthogonal to our work, as we evaluate measures basing how they rank different samples uncertainty quantification methods like in [Schus-
such measures may then be calibrated by distribution- ter et al. 22022 Giulianelli et al. (2023) also provides an interesting exploration of the inherent uncertainty in human responses for many NLG tasks, in a black-box manner. Finally, carefully designed prompts have been proposed to improve the quality of the generated responses in general (Zhou et al.] 2023 Si et al.
2023 Wei et al. 2022 Orthogonal to UQ but related to selective NLG. Varshney & Baral (2023) focuses
reattempting rejected samples, with the help of an auxiliary model trained on an additional dataset that pre- dicts the correctness of the generation. This paper focuses on providing quantitative sucertainty/confideen measures, which can be used to identify high-quality gencrations.
3
Background
In this section, we describe the specific type of uncertainty under examination in the context of Natural Language Generation (NLG), while introducing terminologics used in the rest of the paper.
3.1 Predictive Uncertainty in NLG
Predictive uncertainty is prevalent subject in probabilistic modeling, especially Bayesian literature MMalinin & Gales 2018 Gal & Ghahramani] 2016). It quantifies the degree of dispersion in the predicted distribution of Y. conditioned on the input X x=a x. As it is generally characteristic of the predicted distribution, we denote it as Uz), dependent only on x. For example, when YJX = x adheres to a Gaussian distribution, the variance serves as an indicator of the uncertainty.
NLG can be viewed as a classification problem characterized by an exceedingly high dimension. In classifica- tion, the uncertainty is frequently measured by the entropy of the prediction (e.g.[Adda Abdar et al. (2021b): Kuhn et al. 2023); Sun et al. (2019); Wellmann & Regenauer-Lieb (2012 ). The predictive entropy is formally defined HYY
p(y x log (p(vlz))dy, which becomes the following uncertainty score in NLG:
U(x) = H()))) = p(s()) log (p(ss)).
(1)
Here, x represents the input, and the summation is taken over all potential sequences (responses).
Predictive uncertainty is occasionally characterized as total uncertainty, encompassing both epistemic and aleatoric uncertainty. Epistemic uncertainty (model uncertainty) can potentially be reduced with additional information, such as the use of better model and/or additional training data (Hällermeier Waegeman,
See Appendix for results on calibrated confidence measures.
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Lahlou et al. 2023 For example, an enhanced LLM trained more math problems could po- tentially generate better proofs with lower epistemic uncertainty. In contrast, aleatoric uncertainty (data uncertainty) pertains to the irreducible component of uncertainty inherently associated with the data gener- ation process (Senge et al., 2014). In a sense, this is related to the "open-endedness" for the question "when did the Philadelphia Eagles win their latest. Super Bowl" (as of 2023), the answer could be either 2017 or 2018, as the game took place in February 2018 but belongs the 2017 season. Some questions (x) intrinsically allow for markedly different answers (s). Decomposing aleatoric vs epis- temic uncertainty is, however, often complex and typically not required for learning algorithms in real-world predictive applications HHällemeiier & Waegeman 2021
2021
NLG. For instance,
Like most existing literature, we focus on quantifying the total uncertainty We would like to emphasize again that like Kuhn et al. (2023). we adopt QA datasets due to the simplicity evaluation. Methods proposed this paper could still potentially be applied more open-ended tasks with no reference answers, but the question then becomes: First, can we evaluate the quality of the UQ in a scalable fashion, given that extensive human evaluation might be necessitated? Second, how do we formulate the task and what does uncertainty entail in practice, when the questions are intrinsically open-ended? These are interesting but arguably much harder and less well-defined questions that could be explored in future work.
3.2 Uncertainty vs. Confidence
Uncertainty and confidence are sometimes deemed antonyms. However, confidence scores typically bear a slightly different meaning, especially outside the Bayesian literature. Specifically, while U, as discussed in Section 3.1] only depends on and is a property of the predicted distribution, the confidence scores are generally associated with both the input and the prediction and can be expressed C(x. y) (Chow) 1970;
Corbière et al. 2019 Jiang et al. 2018 Lin et al. 2022b). As concrete example, for P(Y[z) (N, the variance 2² is an uncertainty measure. For a particular prediction Y = y, the negative z-score Wー could be a confidence measure. Notice the use of lower-case y in the notation, instead of a upper-case Y that represents random variable. In the context of classification, one of the simplest and most used confidence measures is just the predicted probability p(Y = y|ı) (Geifman & El- Elniv 2017 Hendrycks & Gimpel 2017). The corresponding confidence score in NLG is the joint probability:
C(r, ((x,s)=f($1 [(s|z) = II ((s1lSe0,)).
2
Obviously, Eq. 2 requires access to the original LLM In Section we will elaborate some alternatives that do not require such white-box access.
Existing literature sometimes uses uncertainty estimate U(x) to predict the correctness of particular response (Kuhn et al. 2023 Malinin & Gales 2021), ignoring the distinction between uncertainty and confidence. Section shows that this is problematic, and confidence is a more reliable indicator of the correctness of a given response.
4 Quantifying the Uncertainty for NLG
In this section, we discuss several uncertainty quantification methods that can be applied to black-box LLMs. Some of these methods are sourced from the existing literature, while the majority are simple proposals of our own The discussed methods can be structured as taking the following steps:
1. For a given input x, generate m response samples $1...., Sm-
2. Calculate the pairwise similarity scores a(s)) $y2) for these m responses.
3. Compute an uncertainty estimate (xx or a confidence score C(a,sy) using the similarity values.
PIn fact, when the level of "open-endodness" of input distribution is similar, the ranking of total uncertainty should still be indicative of that of the epistemic uncertainty.
Here, we assume auto-regressive LLM. For other models, computing such quantify might require different approach, passible.
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4.1 Measuring Response Similarities
We mainly focus on two ways to compare the similarity between pair of responses.
Jaccard Similarity: The Jaccard similarity is a widely employed metric for determining the similarity between two sets. It is calculated by dividing the cardinality of the intersection of the two sets by the cardinality of their union. extensively utilized in Natural Language Processing (NLP) tasks (Cronin al., 2017 Pilchvar et al. 2013 Qurashi et al. 2020), where sentences or documents are treated as sets of words. Specifically, the Jaccard similarity between two responses 8/1, and Sj (considered as sets of words) where jı.j2 E [mn] is computed as:
rule-based metric that is easy to implement, the Jaccard index has been
aJoccara(S], S/,) = S3 Os/ul/ls, US/22 E []
(3)
Despile the computation efficiency, Jaccard similarity has certain limitations, including the lack of consid- eration for word order and the inability to capture crucial expressions such as negation.
Natural Language Inference (NLI): As noted above, rule-based similarity may not effectively capture the nuances present in generated responses. potential alternative approach involves utilizing a Natural Language Inference (NLI) classifier for this task. Numerous NLI datasets are available for training such classifiers (Williams et al. 2018 Bowman et al.] 2015 Poliak 2020 In Section 5 we will adopt the methodology outlined by Kulm et al. (2023) and employ an olf-the-shelf DeBERTa-large model (He et al., 2021) as the classifier. A NLJ classifier typically predicts scores (logits) for three distinct classes: entailment, neutral, and contradiction. We can use the predicted probabilities as the similarity, denoted as aNLI(S;,, 8j2). To obtain a continuous value ranging from O to 1, we apply the softmax function to the predicted logits, resulting in Poontra (8j1: (0j:88p 812) and Pentait (S) 812) (both depend on x7). We then define the following:
aN Lenttad (S), Sjz) = (e. Sja)
Loutrr (Sj,, = Pcontra S8,).
(4)
It should be emphasized that obtaining Pentail and Peontra is not in conflict with our primary objective of quantifying the uncertainty of black-box LLM, for two key reasons. Firstly, the NLI model can be (and is) substantially smaller than the LLM, because NLI is a considerably simpler task, and the NLI model is not required to have the same "knowledge" as the LLM. Secondly, the LLM's function in NLG is to generate responses (sequences of tokens); thus, any additional information, such as token-level logits or embeddings. is not part of the standard output and may not be accessible to users. In contrast, the NLI model's outputs are the probabilities we utilize.
4.2 Estimating Uncertainty and Confidence from Similarities
In this section, we aim to convert similarities from Section 4.1 into mncertainty/connidee measures.
Number of Semantic Sets was first proposed in Kuhn et al. (2023). The original paper proposed to use a NLI classifier to group responses into several "semantic equivalence" subsets (which form a partition of all responses). They use such "semantic equivalence" classes as well as the numerical output of the base LLM to compute the "semantic entropy While such method cannot be applied to a black-box LLM, in their experiments they also used the number of "semantic sets" (equivalence classes), which is an uncertainty measure applicable to black-box LLM:F] We denote this uncertainty measure as Ununsed | For example, for the question "What city was Zeus the patron god of?", the three responses "Olympia", "Zeus was the patron god of Olympia, Greece", and "Corinth" form two semantic sets (with the first two responses in one set). Intuitively, in the m responses, the L.L generates more semantically different answers, then the total uncertainty is high.
TThis is an improved version of Eq. response, p(s(z) is replaced with p(c x > sac. p(c[2) correspondingl..
where Σ. is replaced with Σ. where denotes semantic concept instead of
/tttps://github http::///sshubb.cmmooreeree com//oorenzzubb/eemmntt. uncertainty
Te tollow the bi-directional cutailment algorithm in Kuhn all 2023 to construct such semantic sets using LNLLeentod and aNLi contra and assuming transitivity semantic equivalence. Specifically, we iterate over j1 < j2. and merge 822 into the same semantic set of 85, if fentaa (8), , 852) > Peontra (8). 852) and Pentai((/22 85,)) > Pcontra Pcontra(8j2:88)) (8)))
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Sum of Eigenvalues of the Graph Laplacian In reality, whether two responses share the same meaning is not black-and-whice In the example of Zeus above, potential responses "Olympia" and "Greece" are neither exactly the same nor completely different. Moreover, there is no guarantee that the semantic equivalence judged by the NLI model (or any other measure) transitive. As a result, a more nuanced and "contimuous" way to measure the number meanings is preferable.
Since we only know the pairwise similarities aj between response S/2 and S/2, but not the embeddings of the generated responses, natural choice for the clustering responses is spectral clustering. Fixing imput 2, first treat each generated response node and define the symmetric weighted adjacency matrix as W = ie where Wj. (Ajizz b + @ja.i)//2. 1/2. The symmetric normalized graph Laplacian is then given by
I D D WDW
(5)
{ Exelm Whis'
(i = j2) (1 # j2)
Dji.2
(6)
A continuous version of UnunSat could be defined with A ... < Xms the eigenvalues of L:
UEigy max(0, -xx).
(7)
To see the connection, between Usigy and Unussat, we recall the classical theorem:
Theorem 1. Voo Luaburg 2007) The mulliplicity of the eigenvalue O of L is equal to the number of connected components in the graph represented by W.
In other words, with binary W (two responses are either connected not at all), the multiplicity of the zero cigenvalue coincides with the number of semantic sets (UxunSet). With a weighted W whose entries are continuous, there is typically only one connected component. However, in spectral clustering, the distribution of the eigenvalues is typically used to determine the number clusters (Von Luxburg 2007 An illustration is provided in Fig. [ with URigy roughly corresponding to the "number of semantic meanings". In Eq. 7 we ignore eigenvalues larger than as only the smallest few eigenvalues carry important information about the clusters (Von Luxburg 2007
Unigu 2.139
Floyd
Unigy 5.984
Pink Floyd in Edinburgh
Low Blood Preseure Carvical cancer Cancer
1.0
Pink Floyd
Shanbles
0.5
Pink Floyd
0.5
Breast cancer
Floyd.
Tuberculoeis
Pink Floyd
Cancer
Pink Floyd
Pink Floyd
Cancer (in 1953 41)
2
6
8
10
Pink Floyd
10
Throat cancer
eigenvalue
eigenvalue
The distribution of the eigenvalues of the graph Laplacian generated by @NI.Lentail, for two questions from trivia, as well as the 10 generated responses. On the left, the question and reference answer are "Q: Dave Gilmore and Roger Waters were in which rock group? A: Pink Floyd". We have somewhere between two and three meanings, and Usigy is slightly above The question on the right is "Q: What claimed the life of singer Kathleen Ferrier? A: Cancer", and observe more diverse responses. As result, higher Usigy (almost 6).
Figure 1:
The Degree Matrix The previous two methods helped us define the uncertainty (xx) but cannot assign confidence score to cach generated response. We utilize the degree matrix D in Eq. 0 to define both metrics, as D already contains relevant. information: A node with high degree is well-connected to other nodes, suggesting that it lies in confident region of the LLM. We thus define uncertainty estimate (z)g(z)
In particular, "gap" between small eigenvalue a large one A441 indicates that there k clusters.

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and confidence score CDeg (2, sy) as
Upag(0 (x)) trace(ml - Dm/m²
Cbag (2,, 8s) Di j../m.
Here, we assume Winda E [0, 1]. Upeg can also be interpreted as the average pairwise distance.
Eccentricity Recall that one challenge from earlier is that we only have the similarity (or distance) between different responses, but do not know their actual embedding space. The graph Laplacian, however, can provide us with coordinates for the responses. Denote u...... Up I as the smallest & eigenvectors of L, then an informative embedding of s, is simply v, [ui.j.... uk (Ng et al.] 2001 Von Luxburg 2007) As a result, we could use the average distance from center as the uncertainly measure, and each response S distance from the center as the (negative) confidence. Formally, the "eccentricity" estimates are:
Use (z) = lIvv ................................ Ym 1l22
CEoo sy) ~1liíl1
where v3g = وド== Vy' represents the offset from the average embedding. Ren et al. (2023) uses a similar idea for OOD detection for LLMs, which however requires white-box access to the original language model. Fig. 2 illustrates a sample embedding (from coqa).
Remark: The confidence scores Cece and Coeg in the current form are hardly interpretable. For example, Cnee could have a value of 10 or 0.4. In practice, they could casily be calibrated match the probability of whether the answer is correct. In Appendix C.55 we show that with simple calibration these confidence scores could faithfully reflect the probability of correct answer.
5 Experiments
In this section, we evaluate the quality of uncer- tainty and confidence mcasures proposed in Sec- tion
Dominique Ansel Bakery, next to the Broadway-Lafayette subway stop
Dominique Ansel
5.1 Setup for experiments
Dominiqu
Datasets Following Kuhn et al. (023), we use the open-book conversational question answering dataset, CoQA (coqa) (Reddy al. 2019), and the closed-book QA dataset, TriviaQA (trivia) (Joshi et al. 2017). In addition,
Ansel
Bakery
Globaly
also use the more chal- lenging closed-book QA datasct, Natural Questions (nq) (Kwiatkowski et al. 2019). We use the devel- opment split of coqa with 7,983 questions, the vali- dation split of nq with 3,610 questions, and the val- idation split of the re. nocontett subset of trivia with 9,960 (de-duplicated) questions. We repeat all experiments 10 times, each time with random subset of 1,000 questions as the calibration set for hyper- parameters of U and C' measures, and test the performance on the remaining data. We report the mean and standard deviation of all evaluation metries (see Section 5.2
Figure 2: An example of the 2D UMAP (McInnes et al. 2018) projection of the embeddings used in Ecc for 20 responses. The question and answer are "Q: What is the bakery's s name? A:the Dominique Ausel Bakery". Similar answers tend to live closcly together, justifying the of distance-based uncertainly and confidence measures (Uzee and CRee).
LLMs We followed Kuhn al. (0023) and include OPT (hang al. 2022) in our experiments. We also test three more recent models that have demonstrated superior performance: LLaMA (Tourron et al. 2023a LLaMA2 (Touvron et al.] 2023b) and the black-box -ot-.b served by OpenAl via an API For both LLaMA and OPT, we use the 13B versions. We use the default generation configs for all models.
Baselines We compare all uncertainty and confidence measures listed Section 4 including NumSet, Deg, Ecc and EigV. Deg, Ecc and EigV are constructed with three versions using a a.Jccerdd LNLneattll, L..oora
All gpt-3. S-turbo used in this paper are the 0301 version.

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(with suffix J/E/C, respectively). We also include "lexical similarity" (UteziBin) from Kuhn al. (2023) which measures the average rougel, between responses. Note that only UKnnSet and ULaxiSin have appeared in literature before (Kuhn et al. 2023). To benchmark these methods with the existing literature, we include two white-boz basclincs for non-GPT LLMs:
Semantic Entropy (SE) (Kuhn et al. 2023 This uncertainty estimate (Usz) groups answers like NumSet, and then computes the entropy over the aggregated semantic sets. This requires access to the token-level logits from the base LLM.
P(true) Kadavath et al. 2022 : This confidence measure Cpcarue) estimates the probability that a We follow the prompts provided in Kuhn et al. (2023) Kadavath al. (2022 and convert Cp(trua) an uncertainty estimate UFctrus) by taking the average over all responses.
model's gencration correct by asking the model itself of
5.2 Evaluation
Effective uncertainty measures must reflect the reliability of LLM responses, with higher uncertainty and lower confidence more likely leading to incorrect generations. Following prior works (Kuhn et al. 2023 Band et al. 2022), we evaluate the quality of the proposed UQ measures by using them to predict whether a generation is correct or not, and compute the Area Under Receiver Operating Characteristic (AUROC) for such prediction. Specifically, if we denote acC.,j = 1 (st, correctly answers x;): compute average AUROC using C(z., to predict acc.g for j E [m]. Unless otherwise noted, m = 20.
Evaluation Metrics:
AUROC however bears two limitations: it can only be applied on binary labels, and its value is hard to interpret. Thus, we use Area Under Accuracy-Rejectin Curve (AUARC) (Nadeem et al. 2009), as an alternative evaluation metrid"] Illustrated in Fig. 3 Accuracy-Rejection Curve (ARC) is computed as the average target (i.e. accuracy) when we reject a subset of the samples basing on predictor (i.e. U or C). As we reject more high-unccetainty samples, the remaining samples should have higher accuracy. The "Oracle" (max) AUARC is achieved by directly using the target as the predictor, while the AUARC of random predictor equals to the base accuracy without rejection.
Correctness of Generations: We assess the correctness of the generated responses automatically using 3pt-3.5-turbo
5-turbo from the OpenAl API. This model is provided with the question, reference answer, and L.M-generated response, and it assigns a correct score between 0 and 1. Responses with scores above 0.7 are doemed correct. Like Kuhn al.] (2023) (which used rougeL. score as a heuristic to evaluate the correctness of generated responses), we also perform human verification the correctness of the auto- generated judgment by gpt-3. 5-turbo and found that the accuracy is about 0.95 (see the Appendix for more details).
5.3 Results
Uncertainty Measures: We first evaluate the uncertainty measures by using them as the predictor to predict expected accuracy (shorthandod as U+EA), estimated with E[acc;] m Σド= acc;,j (m=20). The AUARCs are shown in Table (AUROC is skipped expected accuracy not binary). For readability, we include only aNLLenntait which empirically performs the best, with full experiment results in the Appendix. We hypothesize that Lntailt outperforms aNnta because even if generations do not contradict each other, they could be still mostly meaningless, whereas generations that actually align could indicate low uncertainty. A such example could be found in Fig. 5 (see more discussion in Appendix B.6j. Deg, Ecc, and EigV perform similarly as an uncertainty measure. In some cases, such as trivia the AUARC (for Deg) is close to the upper-bound (Oracle). In general, we found that the choice of similarity (E-entailment, C-contra, J-Jaccard) matters more than the choice of Deg, Ecc, EigV.
Note that Ptrru) may not require white-box access if one is willing sample lot of respouses from the LLM, which is however computationally expensive.
11 While originally AUARC was defined with binary accuracy lahel, one could generalize any contimuous label, such the expected accuracy (using Monte Carlo estimate).
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ROC
ARC
NurnSet
True Postiive Rate
0.8
Average Accuracy
0.9
EigV(E)
Ecc(E)
DeglE)
SE
0.4
P(true]
0.7
Oracle
Base Accuracy
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
False Positive Rate
Rejection Rate
The ROC (left)) computed using the accuracy of the 1st generated response and the corresponding confidence O (when available, otherwise U), for LLaMA on trivia. ARC (right) is computed using U and expected accuracy. (For ARC, average accuracy is noisy at high rejection rate due to small sample size.) The suffix (E) denotes for @NLi.entail Different ways to construct U or from aNLL,entod make a relatively small difference, but are all noticeably better than the baselines (including the white-box baselines). The ARC suggests that if we select only the top 50% samples using, for example, Deg (E), we could improve the accuracy from 62% to around 90%.
Figure 3:
0.90
AMMBE (Covtounce)
0.85
085
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0.0
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Figure 4: AUARC for confidence (left) and uncertainty (right), with different number of sampled responses. A small m like already greatly improves the accuracy in selective generation, compared with no rejection or rejection based on random noise (Base Accuracy). In general, confidence-based measures (Deg and Ecc) achieve better performance the C+TA setting, confirming the intuition that response-dependent confidence measures predict the quality of individual responses better.
Confidence Measures: To evaluate confidence measures, we compute AUROC and AUARC, and take the average across all m generations. We refer to this as the 'C+IA" (Confidence + Individual Accuracy) setting. The results are reported in Table (with AUROCs deferred the Appendix). Note that Ecc and Deg actual confidence measures and EigV is uncertainty measure (not response-specific). As result. able to further distinguish the quality of each response, Deg and Ecc typically observe higher performance in Table 2 compared with Table [ while the uncertainty-only measures like EigV stay the same, echoing the discussion in Section
Varying the Number of Generations: Fig. we report results when we vary the number of generations m, for trivia and LLaMA. It is observed that even with m 3, the confidence or uncertainty measures already achieve good performance. As m increases, the quality of C' and U typically increases. Note that for AUARC(C+IA), only the confidence mcasure (Creg and CBcC) improves, and the other mcasures, being response-aguostic uncertainty measures, stay the same. NumSet seemingly deteriorates plot, because it seems to predict the quality of the first few responses (for LLaMAttrivis only) better. As result, as we take the average of more samples, the higher performance flattens out. The trend is different for other settings, and flat in general when considering all data and LLMs. full presentation of the results varying the number of generations could be found Appendix C.1 Tn practice, one could choose the quality-computation trade-off by picking number of generations that appropriate.
increases in this
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Q: why was the plague that struck athens so devastating
A: close quarters and poor hygiene
Generations:
There's s no cure
large flock birds
because the plague was unknown at the time
The plague was devastating then because the diseases were unknown and a mystery to the \
ancient world and the there where numerous diseases spread
was so short and caused so much quaratic
Athens was located within a part of Greece that was full of fleas
The plague of Athens was a pandemic that hit the Greek city of Athens and surrounding
areas in 429-428 BC
Soldiers returning from battle because they were because it happened just as farming season began
Figure 5: An example where contradiction- UEagy indicates low uncertainty, yet the generated responses are mostly low-quality, Entailment- similarity suggests high uncertainty.
Table 1: AUARC when using (z) to predict expected accuracy. The best black- methods are in bold and the best overall is underscored. In general, our proposed uncertainty mcasures perform significantly better than the baselines, sometimes purperforinng the white-box methods.
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Table 2: AUARC when using C(r, s) to predict individual accuracy. The best black-box methods are in the AUARC for confidence measures (Deg
bold and the best overall is underscored. Compared with Table ! and Ecc) generally improve, as they could discriminate the quality of each response.
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6 Conclusion and Discussion
In this paper, we studied the problem of uncertainty quantification in black-box LLMs, with an emphasis on asscssing the quality of generated responses to a diverse set of questions. We developed and testod a range of easily implementable uncertainty and confidence measures. Our results demonstrated that using similarity as determined by an NLI model, along with simple measures that measure dispersion based on these similarities, can effectively identify difficult questions and confident answers, often outperforming existing white-box benchmarks. The objective of this paper is to provide practitioners with simple and effective methods to manage uncertainty, reduce incorrect answers (possibly by excluding them), and apply LLMs with confidence.
To conclude, we also note some limitations of our work and challenges remaining to be addressed. Currently, the evaluation of uncertainty ccontaiinty/oonideenee measures is restricted to question-answering tasks, because it is
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Published in Transactions on Machine Learning Research (05/2024)
generally difficult to acquire labels on whether response is "reliable" or not for open-ended conversations. Even for the datasets used in our paper, while our use of GPT as the judge improves upon previous heuristics (of using rougeL), the evaluation can be improved if human labels are used. Moreover, we currently evaluate uncertainty and confidence separately. Methods that consider both have the potential to predict the quality of generations better. Finally, the uncertainty and confidence measures this work (and those mentioned in Section reflect those in the "posterior" represented by the LLM. This has two implications: When the sampling temperature is 0 (i.e. greedy decoding), all methods will give degenerate results, and one might resort to white-box methods external information to quantify the uncertainty. Also, methods may not identify factual errors propagated from the training corpus or identify when the LLM is being overconfident (which is related to calibration, an orthogonal research topic). We hope this work could serve as a foundation for future research in these directions.
Acknowledgements
This work was supported by NSF award SCH-2205289, SCH-2014438, IIS-1838022, NIH award R01 1R01NS107291-01.
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Susan Zhang, Stephen Roller, Naman Goyal, Mikel Artetxe, Moya Chen, Shuohui Chen, Christopher Dewan, Mona Diab, Xian Li, Xi Victoria Lin, et al. Opt: Open pre-trained transformer language models. arXiv preprint vrXic:2205.00066 2022.
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Published in Transactions on Machine Learning Research (05/2024)
A Proof for Theorem
Theorem the same as Proposition 4 in Von Luxburg (2007) (Note the L defined in Theorem is equivalent
L Lay
Von Luxburg [2007]) We briefly recover the proof below, beginning with proposition:
Proposition 1.
Von Luzburg (2007) For every f E R
f
1'
m
Wii
ft dd
(10)
dg
Proposition ! can be verified by simple algebra. Now, suppose the graph has connected components. We first find & orthonormal eigenvectors. This can be done by letting v1...., Uk be k vectors such that
4
j St
m(()) =
Σ
Mes
(11)
& St
It is easy to verify that 47 1,...,,, h; | 1. For 4 + 12, (1), mz2) = 0 because S₁₂ and S are disjoint. Finally, the i-th entry of v /yyes, dy Luz is 0 if i & Si, and
Vd. VT,
Σ
Wij Vd
Vd. =0
(12)
j
if E St (because L D- - D
Now, we argue that cannot find another vector that zero eigenvector. By Proposition 1 must. be be scaled version of component S,. w.l.o.g,, assume v(j) * with j Sp. Then, by Proposition [ 3e such that Vj' E Si,((")) evr(j).. This means (v, vz) f O. Thus, cannot find another zero eigenvector.
B Additional Experiment Details and Ablations
B.1 Prompts for Response Generation
TriviaQA: We use the exact prompt in Touvron et al. 2023a for TriviaQA, which is reproduced below:
quastions:
Q: Scotland bothy/bothie is
A:
House
[Provided quaation]
A:
Natural Questions is much harder dataset. than TriviaQA, use the same 5-shot prompt version of
the prompt in Touvron et al. (2023a) (with 5 questions randomly picked from the training set).
CoQA: For CoQA, we use the code provided by Kuhn et al. (2023) prepare the data prompts below for convenience:
12 We provide the
[The provided paragraph] (additional question-answer paire]
Q: [Provided quastion]
A:
where additional question-answer pairs preceding turns of the conversation about the paragraph consist- ing of questions and reference answers.
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Published in Transactions on Machine Learning Research (05/2024)
B.2 Prompts for NLI similarity
Similar to Kuhn et al. 2023), we feed both the question and answer to DeBERTa-largd - with the following prompl for any pair of answers to the same question in order get the entailmenn//onntradiicci scores:
[question]
[anewer_1]
(SEP)
[question]
[anexer_2]
B.3 Automatic accuracy evaluation
We use 1pt-3.5-turble to evaluate the similarity between each response and the reference answer. The prompts for coqa, trivia and are shown below. The few-shot examples are chosen from the training split.
CoQA:
Rate the level of consietency between the ansver to the question and the reference anewer, from to 100.
Question: When the Vat fornally opened? Reference: It was forually established in 1475
Ansver:
Rating: 100.
Question: what 1s the library for?
Reference: research
Ansver: tourisn
Rating: 0.
Question: [question]
Reference: [reference answer]
Ansver: (generated response] Rating:
TriviaQA:
consistency betxeen ansser question ansxer, 100.
Question: In Scotland bathy/hothic is a?
Reference:
House
Ansver: Bouse
Rating: 100.
Question: Where in England vas Dane Judi Dench born?
Reference:
Ansver: London
Rating:
Question: [question]
Reference: [reference anever]
Ansver: (generated respense] Rating:
Natural Questions:
of consistancy
question
ansver, 100.
Question: who nakes up the state council in ruseia Reference: governore presidents
Ansver: governors and presidents
Rating:
Question: when does real time with bill neher come back
Reference: November
Ansver: September 8, 2000
Bating:
Question: [question]
Reference: [reference ansver]
Ansver: [generated response]
Rating:
99.34% of the judgements by GPT can be parsed integer between and 100 in the first attempt with simple str. split in Python. We skipped the remaining samples for the experiments in this paper, but if necessary one could devise a better parsing algorithm as well as improved prompts to clicit judgement with higher coverage.
18

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Published in Transactions on Machine Learning Research (05/2024)
Verifying the correctness of GPT evaluations We sample 33 samples per dataset and model (OPT, LLaMA, GPT) and perform a human evaluation of the quality of the GPT judgement. That is, compare the human judgement on whether a generated response is correct or not with the GPT's judgement, like in Kuhn et al. 2023 In Table B we show the breakdown of the accuracy by datasets.
Table 3: The accuracy of the GPT evaluation the correctness of the responses, measured by the authors. For example, for trivia, the human evaluations and the GPT evaluations are aligned on 97 out of the 99 question/answer pairs, leading to an accuracy of 98.0.
coqa
trivia
nq
Accuracy of GPT evaluation
90.9
98.0
95.9
We also include a few typical examples where the GPT evaluation is more reliable than the rougel. evaluation used in Kuhn et al. (2023)
"Q: Were they nice to the other critters? A: no". GPT correctly identifies that "They mcan to the other animals" is correct answer, but the rougel, metric does not caplure this logic.
"Q: Who created the immunity plan? A: the Gulf Cooperation Council". GPT correctly identities that "GCO" is a correct answer (an abbreviation of the Gulf Coorperation Councol", but rougel cannot identify this mechanically).
"Q: when was the last. bear killed in the uk A: c. 1000 AD". GPT classifies the response "The last bear in the UK was killed over thousand years ago, the 9th century, so there no specific date available" correct, but rougeL fails to do so.
The GPT evaluation also introduces its own problems, though. For example, when it used to evaluate own response the question "when did it become law to stand for the national anthem" with reference answer 16/22/1942"
rates "There is no federal law that mandates standing for the national anthem in the United States" as 100. However, we believe it is using its knowledge here and largely ignores the reference answer. Such problems could polentially be improved with better prompts.
B.4 Hyper-parameter parameters for Uncertainty/Confidence Measures
For all U and C measures involving @NIJtenteil and [Nottrontta we need to choose temperature for the NLI model. The temperature is chosen from 0.1, 0.25, 0.5, 1, 3, 5, and 7. For Usee and Ceee, also need to choose a cutoff for cigenvalues. For simplicity we use the same threshold for each experiment/datte datasett and the threshold is chosen from 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9.
B.5 Computational Requirements
We perform all experiments machine with 2x AMD EPYC Milan 7513 CPU, 512 GB RAM, 8x A6000 GPUs. Generating 20 responses with OPT/LLaMA for coqa, nq, trivia takes from 2.5 to 15 hours, or from 2 to 11 seconds per question. CoQA takes the longest due to the story input. Rumning the NLI model for 20 responses (assuming the worst case that no two responses are exactly the same, which means 380 comparisons) takes about 0.8 seconds.
B.6 Full Experiment Results
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Published in Transactions on Machine Learning Research (05/2024)
In the main text, we present only results for LNLi.ettal- We show results for &Jecardd and cNntra in Tables 4 to B We observe that with the default generation configs, the choice of similarity measure (CNtL, enttait aJaccara) seems to play a bigger role than the construction of U and C measures. LN..ttttail consistently performs the best, especially on nq. This is likely due to the nature of the questions. For example, for questions starting with "why", for a collection of low-quality random answers, LN does not consider them as high-uncertainty because they don't actually contradict each other, but ONLL,enter captures such high-uncertainty case, as illustrated by Fig. On the other hand, for the more factual questions like in Fig. [ both all similarity measures agree on high-mneeraiinty morameerrtinny cases, as there are few other ways to state the correct answer. It will be interesting to see more future research on the choice of the appropriate similarity for different tasks (which could involve different types of questions). Fig. also provides an example showing why semantic-bas based similarities are preferred over lexical-based ones.
Table 4: AUARC, U(x) + methods are in bold and the best overall is underscored.
Expected Accuracy, with m = 20 (similar to Table The best black box
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20

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Stacy Lattisaw Published in Transactions on Machine Learning Research (05/2024)
Q: Who the lead singer with Stylistics
A: Russell Thompkins
10 Cenerations:
David McKee
Darius Rucker
Bruce Watson
Billy Jer osteen
Paul Michael McMurray
Jermaine Rogers
Billy Davis
Steve Winwood
David Bennett yeah it 6,'s,
Figure 6: An example where contradiction-based Usigy indicates high uncertainty. this case, the entailment-based and Jaccard-based similarity also indicates high uncertainty.
Q: How was he traveling then?
A: he was walking
by foot
by foot
on foot
walking
walked
Walking
walking foot with his legs with own feet
Figure 7: An example where purely lexical similarity performs worse than semantic-based ones. Although the generations are phrased differently, they convey the exact same meaning.
21

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Published in Transactions on Machine Learning Research (05/2024)
C
Additional Results
C.1 Number of Generations
In the main text, we include results with only m = 20. Hcre, we show the full results with m = 3, 5, 10, in Tables 7 to 9 The resulls are very similar to when m = 20: In general, Lentool seems to provide the best results, and the choice of similarity seems more important than the choice of construction (Ecc,Deg,EigV) EigV). As m increases, the performance typically increases as well. With three generations (m = =3) we already see noticcable performance boost if we perform selective generation (AUARC).
Table 7: AUARC, (x)) + Expected Accuracy, with 3, 5, 10 (similar to Table 1 The best black-bar methods are in bold and the best overall is underscored.
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C.2 Uncertainty& + Individual Accuracy
We present the result of using uncertainty to predict individual accuracy in Table 10 This is similar to the practice Kuhn et al.] (2023); Malinin & Gales (2021). Formally, this setting is U(x) + Individual Accuracy: We use the uncertainty estimated on m samples for each of the m samples to prediet each answer's accuracy. For AUROC, this means
T AUROCU+IA = Σ AU -U(x)), [acor,,,............. al)
(13)
1=1
C.3 Effects of Sampling Temperature of LLM
In the main text, we use the default generation config from hugginface or OpenAl's API for the LLMs. In Tables 7 to 9 the temperature is 1 for all models except for LLaMA2 (which uses 0.6) and top_p is 1 for all models except for LLaMA2 (which uses 0.9). Being sampling-basee based uncertainty quantification methods, our proposed measures obviously are affected by the temperature of the base LLM, and as we noted in the main paper, when very low Lemperature we do not expect such sampling- based black-box methods to work at all. Thus, in Tables to 13 we lower the temperature to 0.5 and observe the effects.
Lower temperature leads to a less divergent posterior, which makes it more difficult for sampling based methods to get an estimate of uncertainty or confidence. As a result, compared with results at higher
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Published in Transactions on Machine Learning Research (05/2024)
Table 8: AUARC, C(a,s) s) + Individual Accuracy, with m = 3, 5, 10 (similar to Table 2 methods are in bold and the best overall is underscored.
The best black-bor box
Dupx Eec (B)
Wiittshos 2(orw)
Rasions
Ocade
LessStn
Eigh (C)
Ecc (0)
Peg (4)
Eige [8]
Peg [X]
Digh 6/
Zec (1)
Deg 21
00:01-0:07
01:254270
11.0348.17
64:0-00
53:00-00
6400000
62-2360 33
30
05.000055 50
11. 08.4350-13
10
92 23.01-00 23
12.75 12.76:5..28 25
93 122221011
50.2000011
91.8cak.22 27
42/0200
copllyy
23.200035 55
11.
T3
73 73.0900.5 54
13.06 0.36
73.01-0.21
T1/00033 56
73.6--131
(5.004 56
76.864011
85.0011.33
56.02.0.355
86.56.0.30
$6:010030
01:25-11-22
(eglarsd))
66.11:01:5 35
10.01:1.00
08:3213:00
00112:100
59:78:11.0
19:04:133
10:3220005
17:70:1150
13:0
12-4200
01:5503000
70.5410.04
70885:4773
70:90:10:0
11.13:9003
24
10:355033
19:005130
(
76.880615 15
06311220
01:3042:0
01:001.11
5.55
01.59.0.10
$151111
A7.0710.17
00.1700:5
51.000133
22-5488:31
72 12
25
C0u400 16
5330
5200.0.35
tenaaapp)
23
03-57-1134
25
12
12:001022 022
T2/06033
oydaace)
23.730933
50321530
15/000.55
0.36815
57.0033.33
rey (eglared)
(4.0000 66
01:01:100
rejoet)
15:5000
[e-13 13
07
01.068.00
01.366.:11
15.3000 66
32.3888055
10.200
14.00000
16.00117
57.30
12
06.222200
00/00011
92.7010.33
00.265511
12 12
00:2700.11
52
11.57
25/0600:1
015442.15
413111.0
53-0000333 23
35.2010.17
56-72.0
13.22:117
51/0
opllan))
60:4000 10
53 5.30003
(ppiland))
1x-0ee0 15
41.72-06111 11
00.77-0-26
54 4:11-13
coqalapt)
TtAL19133 33
45.3610.20
17.85217
oy (glaar))
45
00:25-1115
091550
30.7100.00
91.07.2.18
723911211
76.3780.4
74.15-9.55
742510.71
76.2918.22
14.40:9.76
73.9010.76
12:000179
71382915
71.00.1.54
Table 9: AUROC, Clz,s) + Individual Accuracy with m = 3, 5, 10 (similar to Table 9) The best black- box methods are in bold and the best overall is underscored.
Wattee P(tzae)
MumBet LeciBia
ELpY (C)
(C)
Dog (C)
Ragl
Ece (E3)
Dog (E)
Eigf (1)
Boc (1)
Dex: (T)
Liviiillon)
82.58a0
80.78 70
$6.44000
88 00:00:001
66.01=0.10
57.86=0
80.0150:11 11
85.77:0.45
Si 00:00
S4.4020033
86.1860.10
77.06:20.1 14
00:42:000
70.25-0011 (
62.52-40.40
83.0410.30
50.00:0..2
62.45-00115
1111660.1
50 00:100.31
90 00.4640.24
80.500 53.0240.22 38
65:0060011
76.34-000 09
13.50--0.1
00.00-0007 0707
83.0910.46
63.75-0.05
50 70:00.0
79.75-0 19
80.05 33.0550..07 07
57.01 0.07
41.1950.1
08.8789021 21
71,5880.30 30
TLP6 151
12.2421 35
72.5455556
18.04:0.35 92
72.0020.05
72.11
72 2222
70.50en.45
71. 23.2260.22
02.8720.11 12
73 73.5823.28
7000100000 41
783445 22
79, 20.1005.33 20
70 00.17:4020 28
80.0350.92 92
70.
20.20=0019 1.
T0 10
77.00
71 76.23=0024
D5 05.1200.0 15
20.3860 22
70.50 20
72.47+0.24 24
71.70e 73
72.11-035 38
72.76-0 40
72.68-0.32 52
71.50:5.10
70 00:22:2222
70.21-0:29
70.07500.21 21
72 10:0000.0 20
50 00:0750.0
6148860.1 18
73 77
70.30:0-41 41
74344-0 31
70.84:0.00 56
78.00-0036
77:1440011
73 737000111 13
Ta.77 49
71. 27.0660.10 10
72 0.
65.5780.15
cousigp
51.4620.10 10
04.5 64.37 16
07.1100.20
61 61.20-41 16
06
07.42:0050
65 00
66 66.18-0.12
64 00:50:00
61.4ee1 05
05. 03:.3.3333
neliumal
01 01.30330.3
70.8220400
08.00221.47
05,
770 0::0
75.94:0422
72 12:11:1
70.5721135
73.70:00.99
68 68.84:00444
nellamad)
07 07.7025.445
70.022035
7222520.53
72:4155.45
72:92:00.20
29
22.20220.29
72. 22.2025.33 36
T1 11:6
00.k2100
7035459.46
71 71.15:03
ne/spn))
0L1820..2 52
61.0923688
64:5-0071 73
05.5220.87
07.0025.:0
73:07:0.02 42
03.7::1455
71 21.1025.566
58. 20
06.55210033
05.6650.58 55
5100
47.40:5030
ociapt)
00000.0.52
68:7260.75
05.8740.40
64:03-0.78
66. 66.11.00.7
04 04.52-0.70
70
64.3425:00
64.50
02 00:77777
61.4450.47
62 12.1950.78
tiviiillmm?
01911000
77.094015
17:01:010
0000000:0
01000
07:77:0111
02.01-0.36
00:00.000
184-0007
T:: 10
70.080.11
71.2980.22
52.01:0000
85.6450.00
41.77a0.10
00.0000
74.1633.9
70.34-0.2
75.544111
15.00
To
coqelllmaa
02.606000 09
72-4263.15
76.0 97
17
02.99-0.33
62.32-0.14
76 20.1
78.77a0 14
coqa(looo)))
02.01:0:22 26
75.5960.20
16.31-1555 99
75.0000
75.00-0.35 13
76.85-0.55 50
70.05:0.44
18.7700:5 95
72.6250.4
43.0250.00
0.53
6385750.11
70.08--.16
15 15.99-00
0.0
81.911-.311
10
17.30-055 50
45.01000 06
cossippi)
SLS1250.0
64.8823.16
10.2310.24
08 21
05. 11
0.01:0044
07.0620000
07.0050..0
50 1022
07.012051
67. 07.0110111
nellimal
02.0120.177
72 72.3150.37
70047
300
74 743820000 o0
To. 0:200.7
T4 21:
77.0150.20
50 5000
07 03:35
ngflama))
0 001220.3
22.00:3:2
000 (0
58
73 0000
T4 72-0..0
74.1100.30
74302-045
72 72
12.3
7246459.47
71 11.0G0044
02 10l66a0.3
0268500.5 45
G4.6220336
0G 00
68 68
68 mhe
79 0734:00511
T4 7474
17, 12.0960.36
51 11.0=0020 82
47.7780061
Gl.020033 39
03 03.5030.52
07. 07.56::0.00
07. 35:0046
67. 07:18400:71
65
05.5030.00
64 15
02 02.75-000
02 33.8860.54
m w 10
81 15:610115
86 00000-00 13
81 00
53.00
87 0.08
16
50.85 0.05
55 00:1
89 00:00-111
90 00.00000066
3F
(
$5.1620011
62.7900.11
80.11:0114
88
03.04:0.11 13
10:0000
85.3021.51
85.0050011
04 00:0
77.00:0
70.00
11:0:0011
80
00:00000
57.80e00 26
05.04-0007
72
41.5480.11
Givielap4)
7360050
77.0003.8
0.00-000
78.0110337 57
T0.51
76
20
01:004012
71
06:01-0111
0.00
83.02--011
020770010
02.0320.00
71.5883.0
11
76.70mn.14
89.3010.35
59.16:0.11
79.5720035
81.5020.11
69.04:20114
02:2420.12
65.5983.22
11444-0.54
07.77:51:20 20
09.97
11.22-000
69.19.5.10
67 20
07.4120.57
05.1720.33 20
mellinal
GL4750..1
74.3683338
(8:220000
00.51-00:5
72.00-0.66
74.
80.06:2044
12-18-0
T6 20030
78.7420.59
01.5660.50 220
50.41:0035
mglloadd)
7064400.77
72.464-.30
76.01-0711
75.47:40.51
16.10-0.58
75.00:0000
76. 76.44:-0.49
76. 51
T3 08.00:0:55
72.50-0.41 43
72.01
51 11.0640022
02.1050.55
68.7760711
67 06
72.53-0.51 51
71.30-0:00
76 16:00:000
00.73:50:2
61.34:0.43
77 47:0033
77.50--099
91
(6.6:40.5
neiapi)
(2.26200.2
64.7720.71
08.95
08.97:0.00
08.05:2045
68. 08.04:0.01
08.1220.12
08.1020.26
67.165568
50 00:1177
01.2121092
05. 4850.75
temperature, we observe that in general, the white-box method (SE) performs better than our black-box methods when m is small. However, as m increases (2 10), our methods' performance picks up.
23

---

Published in Transactions on Machine Learning Research (05/2024)
Table 10: AUROC, Uzx) + Individual Accuracy. Compared with Table g the AUROC for confidence
measures (Ecc and Deg) are typically lower, which is consistent to the belief that confidence measures captures the quality of each responses.
Jumdet LeziBin
Eup (C)
(C)
(C)
Eage (E)
Rce (E)
Elat (3)
Rec (3)
Deg (3)
Pitroe)
mn-dd
bitsa/llats!
54 R2 12
56.64:00.12 12
50. 01.91:0111
54.5
14.85:11.35 25
BT 87.36m00.11 1
99. 20.66ci0.15 10
87. $7.54.50.3 ET
84.40c60.12
R2.5864122
84.3020 11
17. 77.3 14
tits//lluud))
00.01:0.16
25.52:0.45
50:00:0000 28
88.
08.56.00.00 39
89.45:00.0
8.7
R2 02:08-1.11
80.5 00.55:6..2 22
80 2..4650.2 21
80. 80.1840 .000
80.8430 21
64,5844.11
88.82.8.14
01.00:0.0
16
M1.89 11.1
851-80011110 11
82
90798100 06
40,2110019
tress/pps)
9:300
11.00:10000 1.88
75.04410020
TL.883932 25
coqul coqa/lamss
8.12
8.23
1000::00440 61
29..2.1.300 30
10124:00111 17
T9.37::0.22 28
70.8420 55
74.38240255
75:0520
75.224 24
92.7848015
coqu//
72:00-033
coquiops)
eoqaigne)
00:30.4
9.00
72 0.:00.00
02:00.00
70 0.00
13.02:00:20
71.80:0032
72.74:00
70.86:0000
09.10:00:0
70.7660
71.1660.00
11.8660.01
18:99.0
01:31:0.70
05
0.:55-0081
78.07:00-52
70.51:0033
00:00.70 70
06.53:0.05
63 11:00 75
81.0150.11
46.6640.74
eugge)
65.37 00.80
05.23 0.882
05.36 10.77
68.82:10.70
08.67 70
07:00:10:3 73
00:95.10.0
07. 07.113077
77
5
titsa/llana!
$2.94
17 88
52.03:00:10 10
22:00:18
50. 1::00:11
H0 77:00.12
02.22:1133 12
85.06280.00 06
84.51=005
83-56220 05
77.5448 to
65.144000
tissa/(landd)
58 12::0:18
50 005.0.11 16
80.39:00.10 10
35
80.53:00.20 20
Dic 22
84.50-0011
84.18:00 15
11-10:5010 17
G4.7450.12
nisleelog4)
Th 0:1
71 1:33::055
10.01:0.10
10
60.33
64:10:0000 07
83:67:011 11
51:30:50 07
40.33440 10
triss/ppp)
84.11 20.00:...22
90
74.32:121
2.:00000
75.842224
02,29400116
0.300
72
72:712033
01.27-022
zoqaiope)
12.79
eoquiape)
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12:30.58
11:00:000
00:31:0000 70
12 0.
76:72:0::0
48
72.420035
71.1100
53.3630 50
02.04 0..6
66 70-00.00
060
75.73:0.04
75.0510.47
15.6100.0 85
81.2540:33
45.7610.57
neigpe)
01:30:0.33
0..90
67.56 00.00
07.33 0.800
67.52.0.70
63.7911088
05.01
61:19:088
07:67:0000
63-3210881
10
tibss/llana
51 11
TG.14 0..1
66.61:0.38 18
56 7.:1111000 06
56. 0..80:0011 18
95.07
05.07:0.000
96 00.71:0000 08
87.60-90 12
85.40:0.00 07
84.61-007
83-50460 07
27.37 to
04.5440005
nista/laand))
55 11
59 79 11
65.43:00114 1.4
05:00:11 14
0-300-3
08-31:000 52
13:00 15
10c0 19
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96.53.00
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00.00:100 59
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10:10 00
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nists(op4)
17 77
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13
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85
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78.00 00556
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10- 12
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72.5000
cxpa(os))
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nellamms?)
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65:13-0000
FR.71 10.31
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82 12.52
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08.81 10.56
66.00
08.11
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67.11.1030
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01.00..:0
61:3710777
m 30
erven/llmm?
75 25.022:2114
10 00:01:0111
15-47:11.34
HT. ST.0049.00 OZ
81.15231110 10
75.51211.0 to
nist//landd)
10.77:0:11
0. 10
80.24:-0.1
18:0011 15
80.47:0.10 1.
10
20
00:000000 03
67.25:003
01:0940.11
nicts/ret)
7.077-3.10
80111-00112 12
65
10
83.30-0111
00:31-1000 135
40.23--0.15
74 01008.800
1118:00
10.90: 5.50
733
81.76
(opiloms))
M6311..11 0.14
76674
113
79.99 11:00
71.01.1111
70-191912 13
71.32101 1032
51.20110.19
exp/Band))
70.88:00442 63
T1:01:0055 58
71
11
20
73.5620.22
T466620 35
73:00-0:31
51,00400300
72:-0.06
76:07:0-14
16:10:111
74:85:0000 06
73:25:0016
T4-12-0 14
71.6860.11
45.77:0.14
(opige)
71.09:0.05
T1
07
07
74.11:122
75.3220
03.7922
61.20:20.44
01002:0411
16.20:0.99
16.10:20.1
10.80:0.411
74.1421.34
21.4050.77
73.56:50
73.40:...
81.1250..5
0.00
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07.41=0.07
77.5520.45
77.50e0.01
78.50e0 36
79.1660:35
45.2280.00
nejupe))
11.51
89.99
89.75.10.551
69.86.1066
000550033
89.57.10.08
07. 03.43.000
06 00.0000
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61.0010.22
C.4 Generation without Rejection
Previous results indicate the practical performance of using the proposed U and C for selective generation, where some hard questions are ignored. In Table we simply pick the most confident response. Even without rejection, most confidence measures can pick better responses and significantly improve the accuracy.
C.5 Confidence vs Calibration
As suggested in Section 2 model calibration is an ort research direction. Once the confidence mea- sures are given, a calibration method could then be applied to calibrate the confidence scores into something close to the probability of correct answer. We applied the classical histogram binning method Zadrozny & Elkan (2001) on all methods, and compute the adaptive calibration error (ACE) Nixon et al. (001)). The number of bins is set to 15 following the standard practice Nixon et al. (2019), and the confidence measures are calibrated on the 1st generation of 1000 calibration samples and evaluated on the rest. Confidence
measures are estimated using 20 generations. The results are in Table 16 After calibration, the confidence scores can faithfully reflect the accuracy. For example, for Ecc (E) on trivia trivia(llama), (11ama), the gap between calibrated probability and the actual accuracy is only 0.026.
24

---

Published in Transactions on Machine Learning Research (05/2024)
Table 11: AUARC, U(x)+ + Expected Accuracy, with m 3,5, 50,1 10, 20 (similar to Table ! and the temperature of the LLM set to 0.5. The best black-box methods are in bold and the best overall is underscored.
Righ
18.222111
85.21-9922
36.17.0.332
29100.0
06.17.0.14 16
$2.5840.95
06.14:0.24
52.740.24
80.0001.3 24
27
73 23.823455
40
90.88-9.0 30
81 10.0000.0
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03.3 23.2010.55
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29 15
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0011100
14:1--13
06.3148.00
37. 02:3225
56.334017
20.351300
11315119
34.945111
183200144 42
m(ar))
11 22
01522002
TL.119956 96
71.02101
11 11:41:9.75
1087-816
25

---

Published in Transactions on Machine Learning Research (05/2024)
Table 12: AUARC, Clz,s) + Individual Accuracy, with m 3.5, 20 (similar to Table 2 and the temperature of the LLM set to 0.5. The best black-box methods are in bold and the best overall is underscored.
hos
99.81:9-32
45.30.33
02.4400.33
6820.20
69.81:00.2
15-750013
12
35
43.21 $9.30
10.15-6.41
162720139
185022055
51242315
05.52:9-35 9-38
56
47
M5.0250.22
cogellane))
16.3224.05 04
37 224223044
59. 27
07.00:0.15
47. S7.1730020 26
STAT7-9-23 23
08 21
Thulee 23
43:0-0333
35.4140.55
85:1466.55
8.00-043
77 7:540.55
19.51 9.71
50.63 0.53
56.10 56.10-070
5101-05 8:
(T-0040 C3
on (egan)
0334:1135
06.21:9.39 11.01:0
75
06:00:0:4
18.6420000
55:41:0.04
57.79:1.00
03:040331
6164103
00.1:0.:0
08.15
1.35
66
10:10:131
07:11-145
48.5210000
08:43
0.45
46-30-1100
0801:011
58
07.40-0.70
68.03:0.24
7125-009
0535-003
20 20:552.00
00:0T-9.25
10.7410.16
00.71-0.36
00.770011 11
00.1-0011
90.000.11
012310.0
06 00.0-0:1
00.31.6.10
lamdd
01:070032
0L02-0034
50300011
67710011
3335000
130
03.07-6115
eppiaaoo)
TEOL
coqelcp))
41315527
53
cogelan))
0.155515 11
13202111 132
lyglavdd)
5001125
41.00u0.25 50
11:0000000
51-77-155
32001100 100
80 80771300
30.9911.01
36:0ad 50
oraar)
021050135
013224034
68.7211777
me-10 10
12220-33
U1
13:00033
02.000.37
02.07
77 73.32012
02:14.33
07.01033
07
06:338251
21222101
22
copdEased)
67.24001
41351015
66110011 13
062004317
esplig))
02:61005
1640000
02 22300500
13:5--21
myllam))
51325004
S25121.0
83 11331011
10 12:01-1
132211 100
10:0-155
01.033100
15.2201 43
1.1 181
00.311.00
23
(ente(aam))
544800000
95 414100005
MEAtE 13
10.75:0000
10
55 50 05
91.115116
10
11
33330035
1230:410
10
31555n01
02.3215.00
copslased
76274033
05001002
N5.2140.10
05-844312
copillane))
csplast)
61351022
8281-9911
07.391366
02400031
07.3510322
07:3740.10
oydanad
517316711
01
56.7348.71
56.231017
46.0001114
35 1120177
20.3011.0
5139-0077
33.0 14
02:9945-72
70
04.265135
18.71111 43
mrar))
22
01812072
01.9153 20
T1.11-9.66
71 11.3810773
T1.66:9.44
73 71
T1 71.29-0.76
71.00:881
18100913 31
0913135.70
26

---

Published in Transactions on Machine Learning Research (05/2024)
Table 13: AUROC, C(a,,) + Individual Accuracy, with m 3,5, 10, 20 (similar to Table g and the temperature of the LLM set to 0.5, The best blxck-box methods are in bold and the best overall is
underscored.
Baadina
Ours
Whiteloo
HasBet
LexuBio
Right (C)
Boe [0)
Dog (C)
Bigh (E)
Boc (E))
Dog (E)
Eigh L3)
Bus U7
Deg (3)
88
Pilxue)
10
tivin/Bamaa
78:71an115
20
0.200000117
02.12:0011
P1.42:0
78:62
S1.77=0.40
61.47=0.55
64-220011
28:21 122010111 13
70.10
N0.24 0.31
N0.95
00.36
81. K1.05..0 00.33
80.90
10.32
80.81 10.35
78 14-701008
18.3% 10:31
05 0.00
55 10112
15.0110.17
56.96.1013 13
06.25 10.27
88:00 11 .8
86.15
03.522 03.522113
06.
66.
10.43
66.17 0
66. 38 00.36
03.95.10.36
06.
06.16.10
10.28 26
60.68:41 19
74.0200.58
75.00 73.0660.3
74.08 74.06:40.25 22
74.00+0.00 98
74.54:0.00 50
7A 7A47:59.00 2.50
71. 71.7249123
706446000 61
T1 TL.0:0055
54
00:00:00
48.16 20
36 03.52:n.10 19
70.000..10 10
11.38611.3
71 01.8660.77 70
72. 07.07
79:15:0030
71 11:00:1100
71 11.ffe0.4 0.44
68 08006-00.00 23
05 00.43:601 18
70.26=00 50
51.1240 25
copstope)
07 02.34-00:13 15
08
50
09 00:99::1100 10
70 01.50 54
70 71--0
71.40 11.00-00-42
71.24:0.71
To 1200--0.00 46
08 22.40.000 75
08:77 08.77:1053 53
08 01 50
70 703315511
46. 16.02--0.0 0.333333
coquiase)
04.57:00:44
62.99 62.99200.47
64.15::0.00 30
64.95:20.75 75
68.37:1.16
23.01211.00
02.0120.44
nellaans)
05, 3..32291.44
00.74:0.30
72: 12.16
73
13.42:40.4 48
12:11:5145
70 70.2
07 05
12. 00:55
T1.00 110060036
58,50220.01
07:25.11:33
71.301.0:11
72.08
0.34
17-07 1i
70.16 111.8.0 49
10-97 1151
52:71 10:32
eqjope)
63-6211143
72:9110.000
10:3
T4-56- 1627
71:37111.58
05:37:0000 70
77 10.35
4637.10.11
meigee)
38.18111222
06.0911143
00:46:10099
80.06-1..0
00.06:1
00145:110.00 45
50 10.50-00.55
50.a0 0.60
38:97311477
06.11:103
87.31:00117 17
0:15
85.44:0022
K5.77 0.55
A5.79
10.30
15.67-00.68
k5.51 0.14
53 53.530014 15
83:33-0
13:33--1166 1.16
11.11.0.11
Lisis/spp)
Livis(app)
Livis/llmaa))
42.0025111 13
80.40en 12
G2.1Den 17
81.2501.18
68.21.0.33 83
15.03.0.16 16
06
00.00:0047 47
03.0000.00
50.01:0.00
55 1..00:00.30 30
05 08:61:0000
26
84 04:0000000
85. 00.633:0.21
05 0.:00:0000
S4 04:00:0114
55.00 55.00-0033
0.33
51.229107
55.002 19
81.2520.84
07.50 30
St SL.00000
07 0738-000
64.28as111
46.17
coqulllmm))
01.8224113
76.31 11:33
72.74.1122
coqufltma2))
01.88241199
11
22.2433225
122 TTAS0ET
51.1220025
copstop))
64.5340017 17
09.76-0.0 m0
T2.92 72:72-0-71
23. 77:00-077 76
7088750.22
70:51:033 30
Tt T1
copige)
(ellaan))
mylloass)
51:15:10:09
10:3110335
0811-:0.2
63:00003
70.0000.00
70.53 01:35
06.12 10.94
T399 0000
75:020024
01:11-1:2
210010-3
21251 1201101
T463=0.07 221000011
06.39100
11
72-33 19:99
74:09:0.49
71 71-011211
72 7.18
30.310331
61:22-0-1 -2
71.75-0-1
7125510-11
71 11
71 11
52.51=0.31
retope)
70:50:0.25
03:1:00:6 06
7000=007
742550010
70. 70.05+0.40
45
76 00:00:0.544
73.02:0.22
73.54:0.0 45
75 1100000000 d5
naiipe)
02.02-0.87
02.12:001
61:0220.47
10
litis/[lmaa
848225.14
8L17
S7 31.40:0
SS 02150.15
56
08:90.20
87 53
86. 20150.15
St. 67.84:0.17
55 14
85.120072
55.10
(itet((lwmaa))
65. 65.1510.12
81.51 0.00
10000 0.20
0.36
05 10
10.26
66:36 10.1
w0.11.01 0.11
15
0012
(ntex//ps)
72.15.0.15
85.76 0.22
51 18
10.13
85:17 1..3
6237
Lisiklau))
6444550.1 15
206528888 83
T1. 0454
210 129
71
71.82+0033
39
7310::00
7005250 21
20,0000 43
02:01:00000 200
02:55:40:20 30
22.52=0.5 59
76 00:000000 06
78-22--0-40
74 7455640226 26
74.00:033 33
02:00-4030 30
71.7000 m0
10
20
24
72.73-0-7 79
coqulope)
28.42211:23
22:32:122 22
12:00
40.704200255
coqp(ase)
12.87:91112
04:0200.24
05.45:0.75
08.05+0.00 S7
06 0630021:25 26
05.24:0544
77.00
10:13
22:0100:2
72:46:0-41
18.00-0.33 20000000
matope)
20
01.000000
10 08
31:31
7333--037
16.0620.41
13.66000
46.0250.65
musgee)
58 38.2450.35 35
00.5200.50
08.70-0.02
0.30
00.30-0
50
= 20
Litsse/lama)
45.82251.12
58 31
59 00.07:0..2
59 00:5520.14
85 85.0024330
99
86.44:51:2 36.3623109
50 50.00:10110
10
56.25620.15
86 #6.20e11.1 12
60.3700111 11
18:20.00 36
90 51
26
20.04:0 00
00
20:00
0.21
Trae0 25
57
22.2 05
60.36220 21
50 00.00-00000
64.4225011 15
Lisin/spi)
81.5850011 12
20.35:0 18
18
00:00 10
90 09:00 16
56 94 12
23
22.9722000 09
63,55200 14
6463-0008
46.1110117
tress gp1)
08:48111:1
16.4
T1-28-11225
coquillmms)
0220020124
24
33.3620-31
70-35:0-2
55
31.50-0.34
rogsioge)
$1.70en.10
64.122224
09:68:0.84
68.1811.88
00. 29
66.5621122
05.67:0:51
67.11:0.5
70.0650.45
71.4400.0
10:01:1:00
76.1855033
17
75.47-1.82
72.16:0
11:21-0.2
78.90
58.78e80.00
08.00:0.55
71002004
10.44:00.0
71.7059.39
76. 08.50:0.4
41
76,07:0.00
70.66:10355
74.01-0.45
71.22-0.2
18.6-0042 62
58.07
eqjope)
05. 05.8210177
M012:11
16.02:0434
77.251:0.05
25.02231:25
76.3821134
Tu.00.0.3
75.45201:35
17.35-022
78.1720045
46.54130..55
nelipe)
$9.282m1-45
05. 11.17
05:26-0.62
03.02:0.30 98
642253155
64:022204
52,0221722
0.77231222
10.20:00000
Table 14: Accuracy achieved by picking the most confident answer among m = 20 generations.
Base Accuracy
Ecc (C)
Deg (C)
Ecc (E)
Deg (E)
Ecc (J)
Deg (J)
P(true)
trivia(llama)
61.18+0.07
0..27:
71.78=0.11
74.04=0.32
75.75+40.15
04.25+10.06
10
09.1
trivia(llama2)
76.24+0.11 11
78 72.22:00.42
78.85:10.14
78. 08.68:0.20
78.96=0.23
78.4140.32
78.81+0.16
03.36+20.13
trivia(opt)
25.75:0..12
01.86:+0.8
31.91=0.15
39.10:0.288
00.25+t0.38
39.98:10.22
41.63:t0.17
17.89:1..13
trivia(gpt)
87.1210.08 08
87.7411.16
87.91=0.20
88, 88.08:0.11
88.36=0.11
87.9610.12
88.1510.11
coqa (laaaa
62.46:10111 11
63.02:10.25
73.18
73.96=0.68
76.33+1.33
71.73:10.17
77.4110.12
56.97.10.2
coga (llama2)
08.
79.64=10.00
80.83-10.15
80.30=0.200 20
80.84=0.18
79.91=40.57
80.86+10.16
05.36:t0.15
coqa(opt)
53.81=00118
52.01+00.65
63.86=0.30
67. 0..39
69.37+00.0
69.01-10.188
T2.64+00.18
36.87+00.22 .2
coga(gpt)
79.16=0.11
09.5
80.13:10.17 00.22:10.17
79.70+40.09
79.58=10.17
80.27=00.14
ng (llama)
23.63=10.36
26:00:20.62
20.93=0.55
36,32 46
38.59+1.07
35.8110.58
10.6610.50
25.16:0..0 0.40
ng (llama2)
44.13+4.668
45.18
46 66.69-0.83
47.90-0.81
46.62-40.48
47.78++0.84
41. 41.449-0.74
pu(pp)
8.60=10.18
0.35
8.03+10.29
14. 14.26:0.26
17.25++0.33
16. 16.26:10.61
18.33:10.40
5. 5.62:1:029
nq(spt)
62.72++0.39
63.10:+0.69
63.43=0.55
64.07=0.69
66.05
63.70:+0.56
63.90=0.56
27

---

Published in Transactions on Machine Learning Research (05/2024)
Table 15: Accuracy achieved by picking the most confident answer among perature of the LLM set to 0.5.
20 generations, with tem-
Base Accuracy
Ecc (C)
Deg (C)
Ecc (E)
Deg (B)
Ecc (J)
Deg (J)
P(true)
trivia(llama)
74.43++0.11 10
76.54+00.43
77.44-10.20
77.44: 0.30
77.50-10.26
76.96
77.48+0.11
70.13++0.18
(Iaama
75.92-40.1
77.90+0..25
78.58=0.22
78.60-4:0.1
78.91:40.28
78.44440.20
78.80-00.15
70.9440.
(rpvia(upt)
40.65:10.15 13
44.17:10.51
45.30=0.15
45.58=00 0.26
45.10=0.17
44.684:0.15
45.12=0.18
34.69:10.16
trivia(gpt)
87.79440.0 10
87.99=0.08
87.91=0.10
87.82:10.10
88.12:10.14
87.9640.11
88.00=0.10
coga (llama)
76.27:40.08
77.23+40.61
79.08=0.27
79.26++0.09
09.35-00.10
72.01+00115
coga [laama)
78.36:10.11
09.04
80.6320.28
79.711:0.43
81.00:10.22
80.19:10447
80.5620.13
73.20:10.17
cogu (opt)
70.40:+0.15
12.31+0.16
73.78=0.21
72.97400.13
74.08=00199
72.93++0.58
71.19=:0.16
62.99++0.21
(gpt)
80.0240.15
80.02:0.117
80.67:00.18
80.02:00.19
79.69=0.20
80.05+0.20
80.03=0.14
nq [lamaa
40.72++0.56
42.24440.65
44.34=0..9
46.20-+0.63
46.73++0.86
44.21++1.24
05.34=0.71
38.46-+0.74
ug (gllamaa))
44.01:10.64
46.05=10.75
46.7620.60
46.89:1:0.3
47.93:10.63
46.64:10.61
47.76:1..75
41.48:10.58
nq(opt)
18. 18.33777.37
19. 10:204422
20.66 15
21.93=0.422
22.25:10.73
22.00 22.06:20.45
22.28:10.39
15. 16.1611.31
nq(gpt)
62.90-10.45
63.74+00.2
63.00=0.50
63. 63.35
65.26:1044
63.77+00.50
62.93++0.49
Table 16: Adaptive Calibration Error after applying Histogram Binning Zadrozny & Elkan belter.
2001
Lower is
Buscinn
Outs
White- box
ACI (In 10
BuaSet
LexiBin
EigV (C)
Boc (C)
Dog (C)
Eigy (E)
Scc (E)
Dog (E)
Eigh (J)
Ecc (J)
Dog (F)
3B
Pitzue)
3.6820.54
3.88
2.5410.63 0.2477067
2.49
1.000.552
2.0010.51
2.37.0.60
3.49
2.501...6
4.5881100
4.300:1001
trivia/llmm2))
4.63100771
triets/opt)
3.2020.70 70
3.31::0.5
101100022
2.3810.30 38
2.84
3.1 01150
2.1510.00 0.00
2.0010551
2.83 8.64
2.621:0.0
23010.53
3.1620.62
4,19100799
trivislge))
3.22+0.54
2.03-0.54
2.766-0.94
2 2668+0.51
2.83-0.62
2.54=0.54
2.50+00.2
2.82+0.5
2.50-0.52
2.47+00711
27540020
mop((lamme)
47881.01
4.90=00.6
3.788-0.96
3 1.92+0..3 54
3.54=0.63
4.08c=1.11
$.0+
3.33=0.67
3.70=0..50 0.50
9.52:+-.77
32770075
3.85=00.0
4,90 1.57
coqo(llmaz)
3.88.0.00
3.871.0.38
3.87.10..3
3.32.0044 4.
3.62:0080
3.38100.0
3.07.00.44
3.38 0.39
3.08.100.8
5.2
3.62.10.54
0,161.0.73
coplege)
4.59:0.22
mus(ges)
1.32+0.91
3.06=0.02
9.50++0-43
3.73=00.90
3.28=0.75
3.50=0.74
3.80=0.00
3.90=0.54
3.71++0.55
mylllams)
4.71:50.06 96
3.54=0.72
8.07:00.02
4.51=1106
4.23:0.78
4.20:0..6
9.624:0.44
3.8620.07
4.12:0.75
3.504-0.57
5.57.0.86
4.45=11.03
4,06+ 1.6
mq(llamz))
5.0250.72 72
4.68:=0.00
5.012:1.44
5.5221.00 28
5,23=1.03
1.99:1.24
4.001:0000
4.87=1.10
5.172:0.22
1.22261.11
2.202:1.40
5,38:20.6
1,891:1166
ngfopt)
3.20.10.77
3.322.033
3.011.0.31
2.8710.46
3.12-00.1
2.6710.677
2.641:022
2.401.0.73
2.82-066
2.57.10035
2:5410.56
2.96220892
4.09110.54
nolape)
5.09.10:54
5.8410099
5.0.1.9
4.71-0.38
5.721073
5.67.11.06
5.0011.112
4.71.1222
28


Reviews:
[{"review_id": "1MM0Lw0Dcr", "reviewer": "Reviewer_GpdJ", "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": "- Measuring prediction uncertainty is an important problem in NLG and can lay the foundation for many NLG applications. Measuring the uncertainty on black-box models is even more critical and challenging, given that only limited information is available.\n- The algorithmic approach based on spectral clustering is very interesting; it is great to see a classical algorithm can still be useful in solving some new problems.\n\nHowever, I do have some concerns about the paper, specifically.\n\n- It’s unclear to me why semantic similarity was only measured using Jaccard similarity and NLI, is there a particular reason of focusing on these two methods? Given the algorithm design, it seems okay to use an independent model for similarity measurement, then how about using BERT scores or something like this?\n- Furthermore, is there a justification for why the performance of Deg, Ecc, and EigV are very similar (e.g., figure 3)?\n- In addition, I am not sure whether the distinction between “Uncertainty” and “Confidence” are necessary or even convincing. For example, a well-known issue of LLMs is giving over-confident predictions. In this case, defining “confidence score” using equation 2 seems misleading.", "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 paper presents a work on measuring prediction uncertainty in large language models without accessing the prediction probability or logits; therefore, it can be used on black-box LLMs. The proposed approach is built on semantic measurements (e.g., Jaccard Similarity or NLI with a independent model), then uses spectral clustering to identify semantic equivalence subsets, as a way to measure uncertainty.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please address the concerns raised in the previous section.", "broader_impact_concerns": "No concern", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "azu26OIc3X", "reviewer": "Reviewer_FXUK", "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\n- The methods are generally well-justified and are fairly simple to implement.\n\n- The experiments show that the proposed measures may be used to filter out responses when the model is uncertain or not confident (without access to its probabilities). This may help prevent misinformation when LLMs are deployed in third-party applications.\n\n**Weaknesses**\n\n- As the proposed methods require multiple samples for each question (and pairwise comparisons), the added computational cost may limit the adoption of these methods for use cases where latency or cost is critical.", "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": "In this submission, the authors propose strategies to quantify the uncertainty (distribution-level) and confidence (for each output $y$) of large language models in a black-box setting, i.e. without using the probabilities directly. They sample multiple completions, establish pairwise similarity (Jaccard or NLI-based) and then compute uncertainty/confidence estimates based on graph theory notions. The methods are evaluated on question answering tasks, for example by examining how accuracy changes when rejecting questions based on uncertainty or confidence.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Page 2: grammar \"requires\" -> \"require\"\n\nThis is more of a question than a requested change. You mention that the embeddings of the generated response are not available. Similarly to how you use a separate NLI model, could a standalone sentence embedding model be used to define other uncertainty/confidence measures?", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "VFVGhb4qFh", "reviewer": "Reviewer_PXYY", "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 approaches are novel and elegant.\n2. They are relatively simple to implement.\n3. The authors performed an extensive comparison with multiple LLMs and methods for weighting and clustering.\n4. The approaches significantly outperform existing SotA.\n5. The paper is well-written. The approaches are explained clearly, and intuitions for the approaches and for quality metrics are given.\n\n# Weaknesses\n I discuss the first two weaknesses in more detail in \"Requested Changes\"\n1. Arguably, the proposed weights $a_{Jaccord},a_{NLI,entail},a_{NLI,contral}$ are all representing the \"equivalence\" of answers suboptimally.\n2. The discussion understates the importance of uncertainty estimation in open-ended questions.\n3. The paper focuses on undirected graphs where the metrics that it uses are initially directed. In directed graphs, instead of measuring connected components we could try measuring strongly connected components, but I guess we don't yet have theory similar to spectral clustering for this case.\n\nOverall, I believe that the paper is very strong, and I only found minor weaknesses. I strongly endorse accepting this paper.", "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 family of novel uncertainty estimation measures for LLMs on closed-form question answering tasks. The approach only requires black-box access to the LLM, which allows it to be tested on models provided only through an API. The uncertainty measures are based on techniques of spectral clustering. This framework allows the authors to generalize the notion of the \"number of connected components\" to weighted undirected graphs, and allows to measure uncertainty in a more fine-grained manner compared to previous work.\n\nThe novel uncertainty estimation methods are tested on several QA datasets and on several LLMs, including the closed-source GPT-3.5. The new methods consistently outperform the existing state of the art in terms of predicting whether the output is incorrect. Both uncertainty (depending only on the question) and confidence (also depending on the answer) are studied. The authors also study different choices in the weighting scheme and different ways to generalize the \"number of connected components\" and show that the former has much more influence on the quality then the latter.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Based on the intuitions from Section 4.2, fundamentally we want the weights of the edges $s_i- s_j$ in the graph to represent whether the answers $s_i$ and $s_j$ are equivalent. If that were the case, the proposed uncertainty measures would tell the rough number of answers given. Equivalence means that $s_i$ entails $s_j$ _and_ $s_j$ entails $s_i$. If we have probabilities of entailment $p_{entail}(s_i,s_j)$ and $p_{entail}(s_j,s_i)$, then the probability of equivalence will be $p_{equivalent}(s_i,s_j)\\le\\min(p_{entail}(s_i,s_j), p_{entail}(s_j,s_i))$. Hence, in my view it would be more semantically meaningful to use the measure $\\min(\\hat{p}(s_i,s_j), \\hat{p}(s_j,s_i))$ based on estimated entailment rather than $(\\hat{p}(s_i,s_j)+\\hat{p}(s_j,s_i))/2 $ used in the paper. A more robust measure might also be a generalized mean with the power $\\le 0$. I do acknowledge that more semantically meaningful measures do not necessarily lead to better performance though. While it would be interesting to see the results for this measure, it is not required for acceptance in my view.\n2. At the end of Section 3.1 the authors ask:\n> <...> do we still care about the level of uncertainty when these questions are intrinsically open-ended?\n\nI think we do, maybe even more so when a model talks about nuanced issues. If the problem is harder to formulate, it does not mean that it should be discarded. The focus of this paper on closed-form QA is not a critical limitation in my view, but I would rather not have the authors discourage other researchers from pursuing the more challenging question too.\n\n## Typos\n1. In eq. (8) the authors write $m-D$, subtracting a matrix from a scalar. I suspect this means $mI-D$, where $I$ is the identity. Not sure if this is standard notation.\n2. Unclosed bracket in inline expression before eq. (1).\n3. \"Outperforms the best\" in Sec. 5.3. Maybe \"performs the best\" or \"outperforms others\"?\n4. Unclosed \"because\" at the end of page 9. Is there an intuition why NumSet deteriorates with increasing $m$?", "broader_impact_concerns": "In my opinion a method to quantify uncertainties of outputs of LLMs is a clear net positive.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The reviewers provided detailed feedback, and the authors addressed the concerns during the rebuttal period. The reviewers were generally positive about this paper. In particular, Reviewer PXYY championed it for acceptance. The method is technically solid and experiments are detailed. The experimental methodology is correct as well.
Decision: Accept (poster)