Multi-LLM Adaptive Conformal Inference for Reliable LLM Response
Authors: Kangjun Noh
Seongchan Lee
Ilmun Kim
Kyungwoo Song
Venue: ICLR 2026 Poster (2026)
Keywords: LLM Response Factuality
Conformal Inference
Multi-LLM
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
TL;DR: Develop a new conformal inference method to guarantee the factuality of LLM responses and maximize its practicality by using multi-LLMs.

Abstract: Ensuring factuality is essential for the safe use of Large Language Models (LLMs) in high-stakes domains such as medicine and law. Conformal inference provides distribution-free guarantees, but existing approaches are either overly conservative, discarding many true-claims, or rely on adaptive error rates and simple linear models that fail to capture complex group structures. To address these challenges, we reformulate conformal inference in a multiplicative filtering setting, modeling factuality as a product of claim-level scores. Our method, Multi-LLM Adaptive Conformal Inference MACI, leverages ensembles to produce more accurate factuality scores, which in our experiments led to higher retention, while validity is preserved through group-conditional calibration. Experiments show that MACI consistently achieves user-specified coverage with substantially higher retention and lower time cost than baselines. Our anonymized repository is available at https://github.com/MLAI-Yonsei/MACI.git

Published as a conference paper at ICLR 2026
MULTI-LLM ADAPTIVE CONFORMAL INFERENCE FOR RELIABLE LLM RESPONSES
Kangjun Noh', Seongchan Lee², Ilmun Kim?, Kyungwoo Song'
1 *Department of Applied Statistics and Data Science, Yonsei University *Department of Mathematical Sciences, KAIST
ABSTRACT
Ensuring factuality is essential for the safe use of Large Language Models (LLMs) in high-stakes domains such as medicine and law. Conformal inference provides distribution-free guarantees, but existing approaches are either overly conserva- tive, discarding many true-claims, or rely on adaptive error rates and simple linear models that fail to capture complex group structures. To address these challenges, we reformulate conformal inference in multiplicative filtering set- ting, modeling factuality as product of claim-level scores. Our method, Multi- LLM Adaptive Conformal Inference (MACT), leverages ensembles to produce more accurate factuality-scores, which in our experiments led higher retention, while validity is preserved through group-conditional calibration. Experiments show that MACI consistently achieves user-specified coverage with substantially higher retention and lower time cost than baselines. Our repository is available at https: /ttp://ghtth//// com/MLAI Yonsei MatiseeiMMAII.. git.
INTRODUCTION
As the performance of Large Language Models (LLMs) continues to advance, attempts to directly utilize their responses in high-stakes domains such as medicine and law are increasing. However, studies continue to report that LLM responses may contain false information (Wang et al., 2024). Therefore, to use L.LMs reliably these critical fields, guaranteeing the factuality of their responses has emerged as an important challenge.
Various methods have been proposed to guarantee the factuality of LLMs, but some are difficult to apply to black-box models (Meng et al., 2022; Zhang et al., 2025; Chen et al., 2024a) quire access to large external databases or online databases (Chen et al., 2024b; Lee and Yu, 2025). campling-based methods (Manakul et al., 2023; Sawezyn et al., 2025) relatively free from the constraints, but the process of repeatedly checking for response consistency incurs considerable time and financial costs, and they face difficulties in rigorously providing a statistical guarantee at user-specified error rate.
Recently, studies applying Conformal Inference (CI) (Papadopoulos al., 2002; Vovk al., 2005; Lei et al., 2018; Angelopoulos and Bates, 2022) to guarantee the factuality of LLMs have been proposed. For instance, Mohri and Hashimoto (2024) apply the concept of CI to the existing frame- work of decomposing LLM responses into independent claims and assigning a factuality-score to each one. Their method proposes filtering out claims that do not pass a predetermined threshold. However, because this single, global threshold is applied uniformly to all data, it provides only marginal coverage and can be overly conservative, resulting in the removal of a of true infor- mation. To improve upon this, Cherian et al. (2024) introduce conditional conformal inference. Instead of a single static threshold for all data, this method employs a threshold function that allows it to change based the characteristics of given sample. But it relies on adaptive error rates that are unsuitable for high-stakes applications requiring a fixed guarantee. Its threshold function also struggles to capture the characteristics of the complex grouping criteria of LLM responses that are separated by their semantic properties. Fundamentally, they typically define conformity score
"Corresponding Author.

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Published as a conference paper at ICLR 2026
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CC1
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Target Coverage: High
Target Coverage: Not Enough
Target Coverage: High
Retention: Low
Retention: High
Retention: High
Comparison of Conformal Inference Methods. T (true) and F (false) denote ground- truth labels per claim. Basic Conformal Inference (Mohri and Hashimoto, 2024) attains coverage by aggressive filtering, yielding low retention. Conditional Conformal Inference (Cherian et al., 2024) proposes adaptive thresholds but relaxes guarantees; MACI achieves both high coverage and retention.
Figure 1:
based on a single worst-case score (e.g., the highest confidence score among false-claims), which ignores the collective confidence of other claims and renders the calibration process highly sensitive to estimation error of worst-case score, leading to filtering many true-claims out.
In this context, we propose a new methodology called Multi-LLM Adaptive Conformal Inference (MACI). The core objective of MACI is to preserve as much factual information as possible while strictly adhering to a low user-specified error rate. Assuming an ideal oracle factuality-score we cumulative product of probabilities that retains
derive an explicit filtering rule which is written the maximum number of claims subject to a target coverage constraint. Inspired by this finding, we design new adaptive CI framework that uses conformity score in the form of cumulative product of estimated factuality- scores. In contrast to previous methods that calculate conformity scores relying on the single worst-case score, the conformity score of MACI aggregates estimated factuality-scoree between claims, significantly improving the robustness of the calibration. We fur- ther theoretically prove that the quality of the factuality-score directly impacts the efficiency of MACI, which is quantified by retention ratio. Accordingly, we adapt our strategy to maximize factuality- -core quality through a multi-LLM ensemble. As result, MACI not only theoretically guarantees group-conditional coverage but empirically demonstrates robust group-conditional cov- erage across diverse datasets, all while showing substantially higher retention ratio than existing methodologies. Figure shows an actual example demonstrating MACI's superior coverage and retention ratio compared to existing conformal inference methods.
Our main contributions are:
We introduce a multiplicative filtering framework that models factuality as the product of claim- level scores (factuality-scom) while preserving finite-sample guarantees.
We provide, to our knowledge, the first retention theoretical analysis in conformal inference, linking
deviations to true-claim preservation and motivating ensemble design. We extend conformal inference with coup-condittonal calibration and - ensemble, ensuring group-conte conditional coverage and showing substantially higher retention than conformal baselines in high-stakes domains.
RELATED WORKS
Existing Basic Conformal Inference (BCI) (Mohri and Hashimoto, 2024) performs false-claim fil- tering by applying a factuality-
threshold to individual claims within LLM responses, providing a distribution-free guarantee of an emrr rate below over the entire data. However, because BCI provides this guarantee only over the entire sample distribution (marginal coverage), specific subgroups may experience group-specific under- or over-coverage, and biases may persist.
Much research in the CI field has been conducted to move beyond this marginal coverage and pro- vide group-conditional coverage guarantees (Vovk, 2012). Hohert-Johnson et al. (2018) propose
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enhanced predictive guarantees that simultaneously satisfy calibration across various computation- ally definable subgroups through multicalibration. Jung et al. (2023) propose the batch multivalid conformal prediction that introduces the concept of multivalid coverage where coverage is guar- anteed simultaneously across multiple groups and various threshold levels. This approach further strengthens simple group-conditional coverage. In practice, when such guarantees are applied, while coverage deviation across groups decreases, the prediction set tends to become larger and retention tends to be lower. Ding et al. (2023) and Gao et al. (2025) analyze the trade-off between group-based coverage and efficiency (set size) by clustering classes groups or by combining surrogate informa- coverage
tion, and they propose improvements. Liu and Wu (2025) apply these methodologies to ensure the factuality of LLM responses. They systematically evaluate factuality guarantee performance across demographic subgroups by applying multicalibration and multivalid CP to claim-level score calibration and document-level conformal prediction. Detommaso et al. (2024) enhance the reliability of LLM confidence itself through group-specific multicalibration us- ing embeddings and self-annotation. However, stronger croup-conditional multivalid guarantees typically require more conservative prediction sets, which can hurt efficiency (e.g.. larger sets or more abstentions) in standard conformal settings. When such methods are naively applied to LLM response filtering, this conservatism can translate into lower retention, raising concerns about prac- tical usability in high-throughput applications. Our research continues the context of guaranteeing group- condiiioniii coverage, but focuses on maintaining practically high retention ratio while en- suring group- conditional coverage under realistic group definitions based on specific features. Feng et al. (2025) propose a framework that ensures claims by applying RAG (Lewis et al., 2021) to the calibration and filtering. However, this approach essentially shifts the application of conformal inference from the LLM to external components like the embedding model and retriever.
conditional coverage while increasing retained
Cherian et al. (2024) share the same goal with MACI, which is retaining many true-claims as possible under group-conditional coverage. They extend conditional conformal methods (Gibbs et al., 2025) to learn sample-specinf thresholds from calibration data. They propose framework that achieves conditional guarantees based on groups defined by prompt and response features while actively improving true-claim retention through techniques like adaptive error rate and conditional boosting. This approach of simultaneously pursuing the two practical goals of group-conditional coverage and retention ratio presents new balance point between ensuring factuality in LLM re- sponses per group and preserving information. Meanwhile, limitations such as the practicality of adaptive a in high-risk environments and the limited expressiveness of the threshold function still remain.
BACKGROUND AND PRELIMINARIES
Document structure and factuality-scoree. Let P denote the space of prompts and C the space of claims. Each document D = (P, C,Y) consists of prompt P E P, set of claims C = (cI..., C C. and labels Y E {0, 1 indicating which claims are factual (true-claims). We assume documents are drawn i.idd. from distribution P. which implies exchangeability of calibration and test data. A factuality-score function p P x C - [0, 1] assigns each (P, c) the probability of being factual, with oracle p" and estimator p.
Filtering operator. Given a score function p. threshold r E [o, 1]. and optional randomization U, the filtering operator F(p, r, U; P,C) S C returns the claims retained under т. We denote by F... the data-driven version of this operator, obtained by calibrating on held-out data to achieve target level a; this calibrated rule is the central object of our analysis.
Group-conditional coverage. Exact document-level coverage is infeasible in distribution-fiee setting (Vovk, 2012; Foygel Barber et al., 2021). Instead, we require validity within subgroups that capture meaningful distinctions such domains, topics, example is a full document Dz+1 = (Pn+1, Cn+1, 1n+1), drawn i..d. from the same distribution as the calibration data. Formally, let g: P C + {1,... {1...., K} be grouping function that assigns each document (P;, C2) to one of K groups. Each document consists of a prompt P; and a set of generated claims C (cr,1.-.., G.N. where N, Cel denotes its number of claims. Let A, = {Cr. E C yi. = }} denote the true-claim set of document D, We then require that, for every group [I....,K], K}.
populations. The held-out
PP (FFna(PPt1. (Pu+1, Cnt1) S A (g(ppti, Cn+1) k) 2 1 a.
(1)
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This mirrors the Mondrian conformal prediction framework (Vovk al., 2005), but is applied here to prompt-claim pairs rather than individual data points. In our experiments, we instantiate using high-level, dataset-specific categories (e.g., medical question types entity groups).
4 MACI: MULTI-LLM ADAPTIVE CONFORMAL INFERENCE
Building on Section 3, our goal is to design filtering rule Fn. that satisfies the group-conditioall coverage guarantee (1). The baseline BCI method applies one global threshold, which simple but ignores group heterogeneity and relies on single predictor. Inspired by adaptive conformal prediction (Romano et al., 2020), we propose MACI, which aggregates scores from multiple L.MSs and calibrates group-conditional thresholds. This approach improves retention while maintaining the required coverage guarantees.
4.1 ORACLE FACTUALITY
Given the definition of factuality-score function in Section first consider an idealized regime where the factuality-score coincides with the true conditional probability. In this oracle setting, the model has complete distributional knowledge of claim correctness, so that for any prompt-clan claim pair (P, c) it can evaluate Pyy = 1 P. c) exactly.
Definition 1 (Oracle Filtering Rule). For any prompt-claim pair (P, e) with binary factuality label y {0. 1}, the oracle factuality-scae -core defined as p²(,,c) == Ply D, (P,, C₁ Yi) with claim set C, (c,,1...,, the oracle score for each claim is
c) For a document
(c = p" (P), PYY,, P.,
Let [VV.] = {1. 2........ N.) and n. : [N.] + [N,]] be a permutation that orders the claims by de- 2 (C,,,(()))) with ties broken arbitrarily. Define Il = I P. (c),)))) with the convention Io = 1 and IIN = 0. For a threshold T E 1]. define the cutoff index and filtered set
creasing oracle scores, pl (1))
& (T) == max {R EE [N:: II ()))))))
=CP :J JSS.S k()))
with the convention max 0 = O. Thus F* ensures coverage at level 7 and is monotone in t (7 S 72 F= PA,), But conservative since coverage typically exceeds т. To obtain exact -
coverage, we randomize at the boundary index (²) Define ()(F) = U E [0. (with (T) = 0 if the denominator vanishes). With U, ~ Unif 1), the randomized oracle rule is
(Cam.1)) :j < Kf(r)};
)>>((T);
F(w", T, U1, Pú, C.)
(o.,).)) i K)(T) 1}, U.S((T)
This randomization balances inclusion and exclusion the boundary, achieving exact coverage at level 7 while maximizing expected retention ratio.
4.2 ADAPTIVE CONFORMAL INFERENCE FOR FALSE-CLAIM FILTERING
Before presenting our adaptive conformal inference (ACT) framework, we clarify the perspective taken in this section. Conformal inference aims to guarantee validity by designing filtering rule that achieves a target coverage level. In principle, one could satisfy this requirement by discarding most claims, but such strategy would be overly conservative. Thus, after ensuring validity, examine how efficiency is affected when using an estimated factuality score rather than the oracle score, and how this behavior compares the oracle benchmark under simple idealized assumptions.
Based on this distinction, the remainder of this section first explains how our method secures valid- ity, and then analyzes its efficiency in producing informative filtering sets.
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Validity. As discussed in Section 4.1, the oracle filtering rule requires access to the oracle factuality-score p" and therefore serves only as theoretical benchmark, In practice, p" is unknown and should be replaced with estimated score obtained from black-box verifier (e.g., LLM). The resulting filtering operator F(ý. r; U: P, C inherits the oracle structure but depends critically on how the threshold is chosen. The objective is calibrate in data-driven manner that, with high probability, all retained claims are factual, thereby achieving valid finite-sample coverage. We estimate using conformal quantiles computed on held-out calibration data. This calibration step guarantees coverage even when ý is only an imperfect approximation of the oracle score p'.
To carry out this calibration, we require a conformity score that captures the document-level filtering event in scalar form. Recall that for each document (Ps, C, Yi), A, = {cL, E C2 : Yi.j 1] denote the set of true-claims, and let U; ~ (n. 1) be the randomization variable. We define the conformity score F; inf E [0.1] F(2. T, U, P., C2) S A,), which represents the smallest threshold at which all retained claims are true-claims. Each E, compresses the leeument-level filtering requirement into single scalar quantity, making it directly suitable for conformal quantile calibration (Lemma in Appendix A). Applying the standard conformal quantile argument then yields the following finit-ssaple guarantee.
Theorem (Marginal Coverage Guarantee). If the samples (Pi, C₁ for exchangeable, ACI (Algorithm 1) satisfies
n n +1, are
P(Fn.a(Pae Cu+1) S An+1) 21-a
Furthermore, if the scores E, are almost surely distinct, the marginal coverage is nearly tight:
TP( P(Fn,, (Pn+1, C.+1) S. Ant1) S -a n+1
Marginal validity ensures that the overall error rate is controlled on average, but this may hide differences between groups, with some subpopulations receiving weaker guarantees. To address this, we extend adaptive conformal inference to group-conditional setting, so that validity is enforced separately for each group. This extension follows the Mondrian conformal framework of Vovk et al. (2005). Instead of pooling all scores, calibration is restricted to examples from the same group the test instance. For group & with calibration set Ig {i g(P,, C.) = k), the threshold is Q14) Quantilo (E; i Ix), a). Given test instance in group k the filter is F(g (Pr+11 Cn+1) Flø, ((h) Un+1; Pa+1, Cx+1)
Theorem 2 (Group-conditional Coverage Guarantee). If the samples (P,,C,, Y) are changeable, the
conformal inference rule satisfies
P(P (Pn-1, Cn-1) S An+1 g( (n.11)
2 1-a,
for all k (1,...,K) with P(g Cn+1) k) >0. 0.
A key implication of Theorem 2 is that ensures finite-sample, distribution validity within each group, in contrast to the marginal guarantee of Theorem 1, which holds only in aggregate. Each group achieves a level 1 - a based on its own calibration size ni,, ensuring that even small groups covered, albeit with more conservative thresholds and reduced retention.
Efficiency. Group-conditional validity holds for any estimated score p. but validity alone does not determine how many claims are retained. A filtering rule may satisfy the coverage requirement while admitting only small set of claims, yielding outcomes that are valid yet not practically useful. This motivates analyzing efficiency alongside validity.
To characterize the efficiency attainable under the validity constraint, we consider the oracle regime p = p" and assume that, given (Pr, C.), the labels yi.j are independent Bernoulli with mean p (Ci.g). This idealized assumption is used only to obtain tractable benchmark: (I) it factorizes the condi- tional joint distribution across claims, (2) turns the coverage constraint into a product of claim- Bise probabilities, and thus (3) yields an optimal valid rule that reduces claim- wise thresholding of the oracle score. Under this simplified structure, the oracle filter has a closed-form expression maximiz- ing retention subject to coverage and serves as the efficiency benchmark for our framework.
In this oracle setting, the resulting conformity scores become exactly uniform on Jo. 1] (Lemma 2 in Appendix A). Uniformity ensures that Theorem 2 achieves maximal retention efficiency: coverage
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is attained precisely at the target level. without conservatism, true claims are unnecessarily discarded. To formalize this notion efficiency, we introduce the retention ratio, which measures the proportion of claims retained under given factuality-score function and threshold. Formally, let (P, C. Y) ~ P be a random document (cf. Section 3). For a factuality-score function and threshold define the retention ratio
R(p,7) == P(cc E (r
Theorem (Retention gap with MSE). Let p" denote the oracle factuality-score and p an estimated score. Fix a threshold [o. 1] and let A == R(p, R(p", T) Suppose the oracle factuality- scores satisfy a margin condition around T: there exist constants e > and 3 2 0 such that
((Ip -| s ee
VE O.
Then, for any 6. > O,
Bi(β p-')
A S
Cc..
2²
Optimizing the right-hand side over e yields
A < C (B( - pz) )1) 242
where e' depends only on (c, B).
The assumption in Theorem 3 mirrors the classical margin assumption in statistical learning (Au- dibert and Tsybakov, 2007, Eq. (1.7)). The exponent 3 quantifies how sharply the oracle factuality- score separates
around the threshold. When 3 >> 0, the condition reflects genuine margin property, whereas 3 = 0 corresponds to the trivial no-margin case that is included only to unify notation. Under this assumption, Theorem shows that the retention gap decreases mial rate in the estimation error [ - p²) }²] The constant C influences only the overall scale of the bound. As long as is finite, the essential conclusion remains the same because smaller estimation error still guarantees smaller retention gap.
polyno-
Margin-type conditions are widely used across statistical learning theory, particularly in binary clas- sification and empirical risk minimization. Such assumptions play a central role in deriving mean- ingful statistical guarantees and are therefore commonly adopted in the literature.
4.3 MULTI-LLM ENSEMBLE
From a statistical perspective, ensembling multiple predictors reduces variance in the biri-variance tradeoff and lowers the MSE, bringing the estimator closer to the oracle benchmark. Yet directly minimizing MSE is not practical because the oracle score is unobservable and binary labels drive predictors toward overconfident outputs, which makes ensembles prone to overfitting. We therefore surrogate objective based the retention decomposition. By keeping recall above tolerance and reducing the FPR, we directly improve retention while avoiding overcontidence. This surrogate remains aligned with the oracle goal and, our Figure 3 shows, also reduces MSE in practice.
Let = Pyy = 1) denote the marginal probability that a claim is true. The retention ratio can be decomposed as follows.
R(P,T) = TPRR(, 7) + (1 p) FPR(P, r),
where
TPR(p, P(CE F. 1)
FPR(P. P(CE (p:
1 1-P
Maximizing R(p,7) T) therefore amounts to increasing TPR while decreasing FPR, but the two cannot be optimized simultaneously. To prevent trivial solutions that sacrifice recall, we require the true positive rate to remain above a fixed tolerance, TPR(p. r) 2 1- 5 for ó E (0, 1). With Tp.5 denoting the d-quantile of factuality- among true-claims, we thus focus on minimizing the FPR subject to this constraint:
p' = arg min E FPR(p, Tp. (p))))
(4)

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Since direct fine-tuning toward the oracle p" is impractical for black-box LLMs, we instead target the surrogate optimum p" in (4) by adopting a multi- ensemble strategy. Given base factuality- scorers {pm JM and weights (W), (.1...., Was). the ensemble predictor
Peas (P,cw)
Σ um Pm (P,c).
with w optimized minimize the empirical FPR under the tolerance constraint (see Appendix B.1 for details). Algorithm 2 summarizes the MACI framework, which combines group-con coup-conditional conformal inference with the ensemble to maximize retention while preserving exact coverage.
EMPIRICAL RESULTS
Dataset Structure and Evaluation Protocol. We empirically validate the superiority of MACI using three datasets with distinct characteristics, MedLFQA (Jcong et al., 2024; Cherian et al., 2024), WikiBio (Min et al., 2023; Cherian et al., 2024), and ExperIQA (Malaviya et al., 2024). Following prior work on false-claim filtering (Mohri and Hashimoto, 2024; Cherian et al., 2024), each example represented as a quadruple { (Prompt, Response, Claim Set, Ground Truth)) where the response is decomposed into atomic claims and each claim is annotated with a binary factuality label. For these datasets, we define a representative grouping criterion for each benchmark and general false-claim risk grouping. In all experiments, the underlying responses are fixed to those released with each dataset (mainly generated by GPT-4 and GPT-3.5-turbo), and every method, including CCI, BCI, and MACI, is applied to exactly the same pool of responses.
To estimate factuality-scorc we query an ensemble of M LLMs with verification instruction that asks them to output a verbalized factuality-score in [0. 1] for each [(Prompt, Claim)) } pair. In our main experiments, we use M models, Llama-3.3-70B 3.00---3--7--Insttrrutt Qwen-2.5-728 -.sttuutt and DeepSeek-V3. V3. These factuality- -ccoress serve as base uncertainty sig- nals. MACI aggregates the M scores via the optimization- baaed ensemble described Algorithm to obtain single factuality-scon scores, it can be used as a plug-and-play filter for arbitrary generators beyond the LLMs that pro- duced the benehmark responses.
Estimating Factuality-Score
for each claim. Since MACI only needs access to per-claim scalar
A detailed explanation the datasets, transformation to Log space, Fact-checking prompts (for ver- balized factuality- sccore)) group criteria, selecting LLMs, and evaluation metrics is in Appendix D. Additionally, Appendix contains extensive additional experimental results, including comparisons with Multi Valid Conformal Inference and Group Clustering methodologies, Conformity score vari- ants, Joint probability Modeling, and discussions on MACT's operation under Covariate Shift.
5.1 OVERALL PERFORMANCE
Table compares the group-conditional coverage and retention ratio for three datasets with dis- tinct characteristics against two prominent baselines in the false-claim filtering field: BCI and CCI. MACI demonstrates robust performance in settings where the other two baselines falter, consistently achieving the target coverage across most groups while maintaining the highest retention ratio.
Comparison with BCL. BCI only guarantees marginal coverage via single threshold, which leads to alternating overcoverage and undercoverage depending on the difficulty differences be- tween groups. This phenomenon particularly evident in the results for the False-Claim Risk groups across all three datasets and the View Count groups in WikiBio. Moreover, BCI uses the conformity score of a document using a single extremal claim. This design discards potentially useful factuality information from the remaining claims and makes the document-level conformity score overly sensitive to estimation error in that single claim. Since conformal calibration will meet the target coverage, such sensitivity tends to induce conservative thresholds, which in turn filter out many true-claims. In contrast, MAČI uses a multiplicative, cumulative conformity score that more directly reflects the plausibility that the entire retained set jointly factual. By aggregating informa- tion across multiple retained claims rather than relying on a single extremal claim, MACI provides a finer-grained conformity, enabling higher retention while maintaining stable coverage.
Comparison with CCL CCI alleviates the overly conservative thresholding of BCI by employing an adaptive threshold function Jocr (Theorem 3.1 of Cherian et al. (2024)). It adjusts the threshold

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Table I: Group-conditional and marginal coverage, retention ratio for three datasets with distinct characteristics. The marginal results are in the row corresponding to the dataset name, followed by two rows showing the results for two representative grouping criteria for that dataset. Coverage within 1 1 are marked with green dot while values that fall outside this range (indicating either over or undercoverage), are marked with a red arrow 41. Compared to the two conformal inference baselines, MACI consistently achieves the target coverage in most cases, regardless of the group. Furthermore, its retention ratio is the highest across almost all groups. Cov. denotes coverage. Ret. denotes the retention ratio. The result with the highest retention ratio, achieved without under-coverage, is marked in bold. All reported values are the mean over 30 repeated trials. The performance of CCI is result of fixing the target coverage (1 a).
Targel Coverage: 80% (a = 0.2)
Targel Coverage: 90% (a = 0.1)
Targel Coverage: 95% (a = 0.05)
BCI
CC1
MACI
BCI
CCI
MACI
BC1
CC1
MACI
Group
Cov.
Rel.
Cov.
Rel.
Cov,
Ret.
Cov.
Ret.
Cov.
Rel.
Cos.
Ret.
Cov.
Ret.
Cov.
Rel.
Cov.
Rel.
MedLFQA 0.80.
0.06
0.81
0.56 | 0.80+
0.71
0.90
0.02
0.90.
0.31
0.90+
0.50
0.95.
0.01
0.95.
0.18
0.95.
0.30
Medical Content
Info
0.81.
0.06
0.761
0.54
0.80. 0.70
0.91
0.02
0.86
0.30
0.90.
0.96.
0.01
0.93
0.18
0.95. 0.30
Interpret
0.80+
0.07
0.847
0.58
0.79.
0.69
0.89.
0.03
0,931
0.33
0.90.
0.47
0.94.
0.01
0.96.
0.21
0.96.
0.26
Action
0.79e
0.06
0.851
0.49
0.80+ 0.73
0,90+
0.02
0.927
0.27
0.90. 0.53
0.96.
0.01
0.96.
0.16
0.95. 0.33
False-Claim Risk
Low
0,847
0.831
0.68
0.79.
0.78
0,947
0.03
0.91+
0.89
0.971
0.95.
0.28
0.95.
0.37
Medium
0.831
0.06
0.81 0.66
0.79.
0.70
0.89.
0.03
0.90.
0.39
0.91
0.46
0.94.
0.01
0.95.
0.25
0.95.
0.31
High
0.73,
0.06
0.781
0.43
0.80.
0.64
0.88.)
0.01
0.89.
0.89
0.94.
0.94.
0.12
0.95.
0.26
WikiBio
0.81
0.02
0.79.
0.19 | 0.81
0.43
0.90
0.01
0.89.
0.11
0.90.
0.25
0.95
0.01
0.93
0.06 | 0.95.
0.13
View Count
0.74|.
0.79.
0.18
0.81 0.36
0.87L
0.01
0.88
0.91.
0.94.
0.92]
0.06
0,96.
0.11
Medium
0.84
0.02
0.78.
0.19
0.81.
0.46
0.91.
0.01
0.88.
0.11
0.91.
0.24
0.95.
0.01
0.92.
0.06
0.95.
0.12
High
0.85
0.02
0.81
0.20
0.81
0.51
0.91
0.01
0.921
0.12
0.91+
0.24
0.95.
0.01
0.95.
0.07
0.96.
0.12
False-Claim Risk
Low
0.81
0.03
0.80.
0.21
0.821
0.40
0.90.
0.01
0.90.
0.11
0.90.
0.23
0.95.
0.01
0.93]
0.07
0,94+
0.17
Medium
0.81
0.02
0.78
0.19
0.81+
0.42
0.91
0.01
0.89.
0.11
0.90.
0.25
0.95.
0.01
0.93.
0.06
0.95.
0.12
High
0.81.
0.02
0.79.
0.18
0.81
0.45
0.89.
0.01
0.881
0.11
0.90.
0.28
0.94.
0.01
0.921
0.06
0.96.
0.09
ExpertQA 0.911 0.13
0,857
0.18 | 0.80
0.45
0.91
0.13
0.85
0.17
0.90+
0.15
0.911
0.13
0.855
0.17 | 0.95.
0.10
Question Domain
Bio/Med
0.14
0.861
0.18
0.821
0.47
0.921
0.14
0.86.
0,921
0.921
0.13
0.86
0.18
0,97*
0.10
Tech/Sci 0
0.14
0.867
0.17
0.81
0.44
0.90.
0.13
0.85.
0.16
0.89.
0.21
0.91
0.13
0.85.
0.16
0,94.
0.10
Common
0.907
0.13
0.84
0.18
0.781
0.43
0.89
0.13
0.85
0.17
0.89
0.891
0.14
0.841
0.17
0.95.
0.09
False-Claim Risk
Law
0.95*
0.13
0.851 0.31
0.81 0.57
0,941 0.13
0.843
0.31
0.89
0.35
0.95.
0.13
0.84]
0.31
0.96.
0.16
Medium
0,917
0.13
0.871
0.18
0.81
0.42
0.91
0.13
0.86.
0,18
0.89
0.23
0.911
0.13
0.86.
0.18
0,96.
0.11
High
0.87
0.13
0.851
0.12
0.79.
0.37
0.871
0.13
0.85
0.12
0.90.
0.87]
0.13
0.85]
0.12
0.95.
0.07
at the sample level, yielding a higher retention ratio. However, its conformity score for D still relies on single extremal claim. As a result, even with well-optimized SeCI. this maggle-laaim-bsee conformity score remains sensitive to estimation error in that single extremal claim and may lead to conservative thresholds, thereby limiting retention gains. Moreover, gCcr operates within linear feature-space framework, which imposes additional limitations when applying it to some grouping scenarios. The grouping criteria for each dataset, such Medical Content False- -llse- Risk, are complex semantic functions implemented based on prompt and claim parsing. It is therefore difficult to capture such criteria using the simple linear functions and features proposed by CCI, leading to undercoverage or overcoverage. In contrast to ICCI, our grouping function is an arbitrary measurable function that partitions the space into a finite number of groups, therefore unaffected by the complexity of grouping criteria in threshold calculations. The constraints inherent in JCcI are also reflected in its retention ratio. JccI risks calculating an overly conservative threshold depending on how well it captures the grouping criteria. This leads to a lower retention ratio compared to MACI, which calculates group-conditional thresholds directly. CCI proposes improving retention by applying per-sample adaptive error rates that reflect each sample's characteristics. The method learns a as a function and lowers a for each sample instead of merely exceeding a minimum retention target. However, this adaptive a differs from our objective. Our goal is to design filtering rules that guarantee, with high probability, that the filtered set contains no false-claims, ensuring applicability in real high-stakes domains. Adapting a raise retention produces filtering rules that are difficult
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Published as a conference paper at ICLR 2026
to deploy in such settings. Figure 2 compares CCI with adaptive and MACI with 0.1 on WikiBio. The upper plot sets CCT's target retention to MACT's average retention and outputs a per-sample adaptive a, showing that CCT's values generally higher than MACI's fixed small a. The lower table reports actual coverage and retention for both methods. CCI raises retention to nearly match MACI by increasing a overall, but the actual coverage is lower than MACI because the target a is larger.
D4
CO te Iskuthed
CCI (a=adap.)
MACI (a=0.1)
0.3
Group
Cov.
Ret.
Cov.
Ret.
WikiBio
0.72
0.26
0.90.
0.28
View Count
0.1
Low
0.71
0.24
0.91.
0.29
Medium
0.73
0.26
0.90.
0.27
0.0
High 0.74
0.28
0.90.
0.31
View Count (Sortod)
Figure 2: Performance comparison of CCI (adaptive a:) and MACT measured by View Count on the WikiBio dataset. The horizontal axis of the left graph the sample index sorted by View Count, and the vertical axis is a. The left graph shows the variation in a when CCI (adaptive a) sets its target retention ratio MACT's average retention ratio. CCI (adaptive a') trades off higher a achieve higher retention ratio, and the table below shows the resulting decrease in coverage.
5.2 MULTI-LLM ENSEMBLE
In Sections 4.1, 4.2, and 4.3, we have discussed the importance of the factuality p. Conse- quently, first verify to what extent our proposed multi-LLM ensemble and optimization method (Section 4.3) improve the performance of ý compared to a single-LLM, and how the retention ratio correspondingly improved. We first find that models exhibit significant disagreements in false- claim detection. Figure 3 (a) shows the high Jaccard distance (Jaccard, 1901) between the sets of claims that different LLMs classify as false. The analysis performed exclusively on the subset of MedLFQA claims with false ground truth. This high distance implies that the models have different patterns for detecting false-claims, suggesting significant potential for performance enhancement through an ensemble. Figure 3 (b) shows that the FPR and MSE are sequentially improved from the single-LLM to the arithmetic mean ensemble, and finally to MACI. Figure (b) also shows that an improvement in FPR is consistently accompanied by an improvement in MSE, and demonstrates that MACI's ý superior estimator factuality-score Figure (c) demonstrates that the cor- responding sequential increase in the retention ratio aligns with objective of maximizing it by enhancing the quality of p.
5.3 MACI UNDER COVARIATE SHIFT
In deployed systems, the calibration queries used to fit MACI's thresholds and the test-time queries need not follow the same covariate distribution. To study this covariate-shift setting, we construct an explicit shift on MedLFQA by ranking responses with a SelfCheck- factuality score and as- signing more factual easy queries to the calibration pool and more ballucination- -proon hard queries to the test pool. This induces a clear mismatch between the calibration and test distributions while keeping the underlying annotation procedure fixed. Following the density-ratio correction idea of Tibshirani et al. (2019), we consider a variant, MACI-DRE, that estimates the density ratio r(x) = 23 (e))pp(() (z) between test and calibration covariates using a lightweight classifier on deployment-time features (summary statistics of SelfCheck scores and prompt/response lengths). We then resample the calibration set according to the estimated ratios and run the original MACI pipeline on this resampled set, without changing any internal components of MACI. Table 2 sum- marizes the results on MedLFQA under the covariate shift. MACI exhibits under- or over-coverage in some false-claim risk groups, whereas MACI-DRE moves group-wise coverage closer to the tar- get level while maintaining comparable retention. This shows that MACI can be combined with lightweight density rattoo estimation to mitigate covariate shift in practice. Detailed explanation of MACI-DRE is described in Appendix E.

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Published as a conference paper at ICLR 2026
Deppndbr3
0.000
0.002
0.727
0.602
0.000
0.685
0.727
(a) Jaccard distance
(b) Marginal FPR. and MSE
(c) Marginal Retention Ratio
between LLMs
Figure 3: (a) shows the high Jaccard distance between different LLMs' predictions claims known to be false in MedLFQA, indicating diverse false-claim detection patterns that support using an ensemble. (b) demonstrates the sequential improvement in FPR from single-LLM and simple arithmetic mean ensemble to our proposed MACI. also demonstrates that as the FPR improves, the MSE also improves in practice; (c) demonstrates that the FPR and MSE improve, the retention ratio also increases. A more detailed analysis is in Figure 7.
Table 2: Comparison between MACI and MACI-DRE. MACI-DRE reduces miscoverages in the scenarios where under- or over-coverage occurs by utilizing resampled calibration thresholds
Targel Coverage:
% (a. = 0.2)
Target Coverage: 90% (a = 0.1)
Target Coverage: 95% (a 0.05)
MACI
MACI-DRE
MACI
MACI-DRE
MACI
MACI-DRE
Group
Coy.
Rel.
Cov.
Ret.
Cov.
Ret.
Cov.
Ret.
Cov.
Ret.
Cov.
Ret.
MedLFQA:
False-Claim Risk
Medium
0.65
0,77
0.84
0.56
0.82
0.55
0.94
0.27
0,87
0.42
0.96
0.19
High
0.77
0.62
0.75
0.67
0.88
0.39
0.89
0.38
0.92
0.27
0.93
0.28
5.4 TIME COST
Time efficiency is critical for real-time filtering. We compare MACI with baselines across two phases: Factuality-Score Generation, where factuality-scores are created, and Calibration, where parameters and thresholds
optimized and calculated. We exclude the negligible filtering time. Table 3 reports the costs on the WikiBio dataset. The Factuality- -ccore Generation phase employs Llama-3.3-70B via the OpenRouter. The results indicate that sampling based methods suffer from high latency due to repeated response generation (SelfCheck) or knowledge graph construction (FSC-KG). In contrast, MACI achieves the lowest total wall-clock time by utilizing a single-pass scoring and streamlined calibration process that avoids the complex parameter search of CCI.
Table 3: Time-Cost Comparison. Wall-Clock time is calculated as: calibration time + (score gen- eration time x # test samples). Values represent the end-to-end time for 500 WikiBio test samples. Note that CCI's score generation time is identical to SelfCheck as it relies on the same sampling process, and sampling- methods do not require separate calibration phase.
Phase
SelfCheck
FSC-KG
CC1
MACI
Factuality-Seen Score (s)
3.25 + 0.43
19.30 + 2.81
3.25 0.43
1.20 + 0.13
Calibration
10.33 1.18
3.24 + 0.65
Wall-Clock Time (s)
1643.91
598.98
CONCLUSIONS
We reformulate conformal inference through a multiplicative filtering structure, providing a frame- work for false-claim filtering with finite-sample, distribution-free guarantees. Our analysis reveals how deviations from the oracle factuality-score impact retention, motivating the use of ensemble methods narrow this gap. Building on these insights, we develop MACI, which uses ensemble- based factuality-score and group-conditional calibration to provide conp-condiiiooal coverage guarantees. Experiments demonstrate that MACI achieves user-specified coverage while substan- tially improving factual claim retention and running more efficiently than existing methods, offering practical solution for deploying L.LMs in high-stakes applications.
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ACKNOWLEDGMENTS
The authors gratefully acknowledges support from the National Research Foundation of Ko- rea(NRF) grant funded by the Korea SgvermmccnMMMSTTRRS--04400002222 RS-2023-00211073),
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APPENDIX
Overview of Appendices. Appendix A contains the proofs of the main theoretical results that were omitted from the paper. Appendix B provides additional methodological details, including precise definitions of the ensemble objective and empirical quantities used in our framework. Appendix C reviews background material on conformal inference and its adaptation to false-claim filtering. Ap- pendix D reports implementation details, datasets, and evaluation metrics for our numerical ex- periments, together with supplementary results. Appendix E contains the additional experimental results. Appendix F reports the use of Large I Language Model for our research.
A PROOFS OF MAIN RESULTS
Lemma 1. For each i E {I...., nJ. each threshold E [0, 1]. and each auxiliary randomization U, Unif(O, 1), we have
[EST) = (F(A r, P,P,C S A.).
Proof. Fix í E (1....,n). threshold r E [0, 1}, and a randomization variable U2 ~ Unif(O, 1). By the definition of Es
E, =n E [0, F( T. P;P C;) S A.).
(=) Suppose E. S r. Then, by the definition of the infimum, there exists S r such that
F "00,T", UU; Pu, Ci) S A..
Since the retained set F F(p, U, C.) is monotone non-increasing in т, we have FÚ C.) A..
(+) Conversely, suppose that
F(A,T,U; P. Ci) S A.
Then belongs to the set
P, C.) C A.). E E F(i T, TU; U.,
Hence, by the definition of the infimum, obtain T. Combining the two directions establishes the desired equivalence. That is,
= (F( C A.).
A.1 PROOF OF THEOREM 1
The proof follows the standard argument for marginal coverage in conformal prediction and is re- stated here in our setting.
Lower bound. By Algorithm I and Lemma 1, the event that all retained claims are factual is written
[Fa, (P Cn+1) S An+1) {En+1 Q(1))
Since the samples (P,, C₁,, Y2) are exchangeable and the randomizations U2 are i.i.d., the conformity scores EEI...., En+1} are themselves exchangcable.
P(En+1 S Qiia) 2 1-a. a.
Using the equivalence above, we obtain
((Fn,a(Pat Ca+1) S An+1) 2 a,
which proves the marginal coverage lower bound.
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Upper bound. Assume the conformity scores {= are distinct with probability one, eliminating the possibility of ties. Denote their order statistics by E(i):...; B(m+1). 1), which under this condition
form a strictly increasing sequence almost surely. Let k = [(1 an(n + +1] By construction,
{ (Pn+1 Cn+1) S Ant1) - (En+1 I Egg)-
Since the conformity scores exchangeable and distinct, the rank of En+1 is uniformly distributed on (I,..., + I]. It follows that
k
[(1 - ()) +11
P(En+1 S E(K))
n n+1
n+1
Finally, since
[11 (1-a) (I= 1)]
< - a
n2 11
n2 n+1
we conclude that
P(F (Pa+1, Cn+1) S An-1) S
which establishes the upper bound.
A.2 PROOF OF THEOREM 2 Fix (1,....
.ll...,KKK... K} with P(g(Pn+1s Ca+1) > By Algorithm and Lemma 1,
CR+1) C An+1] > (E s 21 Herz),
where I = {i [m] ((P,, Cj) = k}.
Condition on s(Pnt1, Cu+1) k. Then the conformity scores {E(1) i Inj U (E are
exchangeable. Let m Ixl and set [[1 - am(m 1)7. If E(h) S ... & E are the order
statistics, then
(1 S 2122 ((E())))
> (E [
By exchangeability, En41 is equally likely to occupy any of the m + 1 ranks. If ties occur the cutoff, the event (F < F(()) only becomes more likely. Therefore,
(k) P(E I L (() (( k) 2 m +1 2-a
This completes the proof.
Lemma 2 (Uniformity under oracle factua)ity-scorr), If = p", then conditionally on (P,, C.) the conformity score E, is uniformly distributed [0, 1].
Proof. Recall from Section 4.1 that Foracle (P,, C.) denotes the set of retained claims at threshold т. The conformity score is defined as the smallest threshold at which the retained set entirely factual:
E == inf e j0, 1] F. Foracle (P:, C.) S A.).
By construction of the randomized oracle filter, the retention rule is calibrated to satisfy
PP Foracle (Ps, C.) S A, P; Ci, This equality holds for every E [0, 1]. Consequently,
P(E < P., C.) =
Equivalently, the conditional distribution function of E, is
GE; T.
Thus, conditional (P,, Ct), have F. Un).(0,1).
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A.3 PROOF OF THEOREM 3
For notational simplicity, we suppress the dependence on (P, and write ÿ == (PPcc) and p" = p" PP, c) throughout the proof. We also emphasize that the argument below focuses simplified thresholding rule with a fixed threshold т. Rather than analyzing the full data-dependent and multiplicative filtering procedure used in MACI, this proof considers basic decision rule hp = [p 2 } to clarify how estimation error in p affects the retention ratio.
Recall from (2) that the retention ratio can be written as
R(p. r) = P(cc E (.(::
T) this simplifies to R(p,T) r) = =E(hp] with hp = (p 2 r). Therefore, the retention gap
In the thresholding case where (p: C) {c:
[R 7) - R(p", , T)
= |Bhhg] E(hp-]]
= |bhp hpp11 S Blhh - hg [],
where we used the inequality [||] S 21].
Since he, hp- E {0, 1]. their absolute difference equals 1 precisely when the two thresholding decisions disagree. Hence
A s P(h + hp:)
= =P((8- - (p - T) < 0).
Fix e > O. If he * hp. then one score is above T and the other below. This can only happen in two
cases:
1. p" lies within e of r, i.e. Ip - S
2. p" is farther than from but P crosses the threshold, which forces - 1-p1 > c.
Therefore,
(he # S S| Taking probabilities and applying the union bound gives
((
+P
- c)
(L)
(L)
For (I), by Markov's inequality,
P(D >)]>)) El p²
For (II), assumption (margin condition) ensures
P(lp" - S( s EC
Hence for every > O,
E -2-
A <
+ Ce
2
Let V = E((p- - p') The inequality
V AS
+ 3
holds for any 0. Hence, we may minimize the right-hand side over Balancing the two contributions by setting = v1/(9+2) (up to constant factors) yields
A A
where e depends only (C, (8). This completes the proof.
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Algorithm Adaptive Conformal Inference (ACI)
1: Input: Calibration dataset Deal {(P, Ct Y)) Pe a black-box classifier P, and an error level a E 10,
) of size neals new instance (Pa+1, Cn+1),
2: Output: Filtered set F.. (P (Pa+1, Cn+1) that satisfies marginal coverage.
Calibration Phase
3: for I...., neal do
4: Sample U, Unif(0, 1).
5: Let A; = Ga E E; Di.j = } be the set of factual claims.
6:
Compute conformity score
E = inf E [0. F(ū,T,U; P. C.) S A.).
7: end for
8: Compute empirical quantile
Q1-a = inf { [0
1
Σ
I[ S 2 2-
Real 1m1
Filtering Phase
9: Sample Un+1 Unif 1).
10: Construct the conformal filter
Cn+1) = F(D, Qi-a, Un+1: Patti Cu+1).
11: Return (n,o(Pat1s C441)
B METHODOLOGICAL DETAILS
B. DETAILS OF MULTI-LLM ENSEMBLE OBJECTIVE
This appendix provides the formal definitions of the proxy objective, empirical quantities, and op- timization procedure underlying our multi-LLM, ensemble (MACI), complementing the description in Section 4.3.
Recall from (3) that
R(p r) p- TPR(p, (1- (1- -))PP)(p,7)
Because TPR and FPR cannot be optimized simultaneously, we enforce a tolerance & E (0, 1) such that TPR 7) 2 1 5. Let Tp.ó denote the d-quantile of factuality among true-claims. The population-level objective is then
p = arg min E FPR(P, Tp.s)].
where FPR the false positive rate at threshold Tp.5.
Since the distribution P is unknown, we approximate the objective using a hold-out set Dope {(Pe, Ce, 12) Let N1 Ers Hay E Ce ye, 1} be the total number of truc-claims. The empirical 8-quantile among true-claims is
Thep
T = inf
Σ Σ
S()
E cECr:y=1
For document (Pr, Cr, Yt), the empirical FPR is defined as
FPRe(p.7) Hc ivi(ce C 0}
where a V max(a, 6). The empirical optimization problem then
i argmin
FPR Fp.8).
Tapp E=1
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Algorithm 2 Multi-LLM Adaptive Conformal Inference (MACT)
1: Input: Data Papt ((P;, Cs 12 and Danl = {(Ps, Ci )= of sizes Nopt and naa respectively, new instance (P, (Pn+1: Co+1). collection of base classifiers (fm) b mm= grouping function g. an enror level (0, 1), and TPR tolerance ò E (0,1). 1).
2: Output: A filtered subset (( (Pn-1, Cn+1) that satisfies group-conditioaal coverage.
Optimization and Calibration Phase
3: for each group & {L..... K} do
4: Define the optimization indices Zopt,k = {ie E Dopt ((,,, C.) = }.
5: For any candidate weights w, compute the empirical threshold
Peo( == inf {
E R
Σ
Σ
1f (P,,c; St) 2 8).
NIA
TE cEC
where N1. = EiEIon. Heis CC : Vi. = t3 is the number of true-claims group k.
6: Compute the optimal ensemble weights wi by solving
wi arg min
Σ
FPR (wen(()), (ecc))))))
Kopt,
IEIopa.
subject
Σ
(Peaa Pau(()))) 2
Kupt.k
/ETmp.
7: end for
8: for each group & {1,...: K} do
9:
Define the calibration indices Teal, (i Deal :g(P =
Define the group-conditional ensemble classifier pi(c) Pens (c; w for each i E & do
11:
Sample U, Unif(0, 1).
13:
Let A; = c.j E C; Vt.j = } be the set of factual claims.
14:
Compute the conformity score
F2 inf in [0, F( P,P.C. S A.).
15:
end for
16:
Compute the croupcconiiiional empirical quantile
Q inf {q [0, Leat, iEI.IA
Σ
11
17: end for
Filtering Phase
18: Determine the group of the new instance: Kistt g(Pn+1, Cn+1).
19: Retrieve the corresponding optimal weights w and threshold Q (Rrm))
20: Define the group-conditional ensemble classifier pi.m. (c) = Pens (c; W W.n)))
21: Sample Un+1 (0, 1).
22: Construct the adaptive conformal filter:
(kmu n,
(Pat1, Cn+1)= = F(it (hhcc)) Un-1: Pu+1. Cn+1).
23: Return (kwaa)))
(Pat1, Cat1).
Direct fine-tuning toward p" is infeasible in black-box LLMs. Instead, let {pm. 1 factuality-scores and (wi,..., WM) non-negative weight vector summing one. The en-
m==1 denote base
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semble predictor is
Pens (P,G
Σ Wm Pun (P, c)
and the weights optimized by
Topt
argmin
FPRe (3w), Pon(-sw)
Napl 1=1
C BACKGROUND
C.I. CONFORMAL INFERENCE
Conformal Inference (CT) (Papadopoulos et al., 2002; Vovk et al., 2005; Lei al., 2018; Angelopou- los and Bates, 2022) is a statistical framework that provides distribution-free uncertainty quantifi- cation for any machine learning model. Under the sole assumption that the data is exchangeable, a condition satisfied by iid. data, CI generates a prediction set C(X++1) for new test point a. This is
Xx+1 that contains the true label Yn+1 with a user-specified probability of least achieved through a calibration process using hold-out calibration dataset, Dealib- The core mecha- nism involves defining non-conformity score function, S(;'), which measures how poorly data point (X,, 4, Y,) conforms to model's predictions. For instance, a common score for probabilistic classifier with score function á is S(X,, Y.) = 1 - - X;), where a higher score indicates that the true label was assigned a lower probability. These scores computed for each sample in Dealib, and a threshold sition in the sorted list of scores. For new test point Xn+1, the prediction set is constructed by including all possible labels y E y whose non-conformity score does not exceed this threshold, i.e.,
determined by taking the value at the [([ + 1) - a po-
C(Xn+1) (y E y 5(X++1;)) S +). This construction provides the powerful finite-sample marginal coverage guarantee, P(Yn+1 E C(Xn+1)) 2 1 a. offering robust foundation for building reliable machine learning systems.
C.2 FALSE-CLAIM FILTERING WITH CONFORMAL INFERENCE
Mohri and Hashimoto (2024) adapt the CI framework to filter false-claims from Large Language Model (LLM) outputs, proposing a foundational method we refer to as Basic Conformal Infer- ence (BCI). The process begins with set of prompts, {P.} For each prompt P. an LLM generates a response R,, which is then segmented into a collection of independent claims, == = [c.1, , C.N, J. Each claim C.jj is associated with ground- truth binary label Vi.j {0, 1), where yi = denotes true-claim and yi.j = 0 denotes a false-claim. Thus, each data point is tuple D; (P,, C₁ Y.), and the dataset {D; 1n is assumed be drawn i.i.d. from an unknown joint distribution P. A score function, p. assigns a confidence-scon p(ct,) to each claim. This score function can be constructed various ways, such as by directly querying an LLM (Tian al., 2023; Guan et al., 2024) or by capturing frequency (Wang et al., 2023; Manakul et al., 2023). Their for- mal goal is to output a filtered set of claims, Fn.a (P: C.) S C., that contains no false-claim with user-specified error rate, i.e.,
P(Pn.o Cn+1) & An+1) S aa,
They define the filtered set as all claims whose scores exceed the calibrated global threshold f, that is. Fr (P., C,) { E C: p(P., c.,) J.. The threshold f is determined by the conformal procedure. Specifically, they define a non-conformity score for each sample (C,, Y.) as the lowest possible confidence-scom threshold 7 that ensures all retained claims true:
S(C,, Y,) == inf(r [0. (.,,, C₁) C AA)
This non-conformyty score is computed for all samples in the calibration set Dealib- The global threshold f is then set to the (1 a) quantile of these non-conformity scores, detailed in Sec- tion C. They show that if the data samples are exchangeable, this procedure satisfies the desired probability guarantee.
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Figure 4: An example of independently decomposed claims in MedL and the aggregated re- sults of four methods that filter the false-claims of those claims. BCI yields conservative results, while CCI and FSC-KG show high retention but fail to filter out all false-claims, whereas MACI successfully filters out all false-claims.
D EXPERIMENT DETAILS
Transformation to Log Space. Our implementation performs all multiplicative computations in log-space. Given a claim-level factuality-score (),,c) E [0, 1] and small c > 0, we apply the complement-basee transform
((P.c) == - log(l c),, +e).
Let denote the ordering induced by P (ascending, low-confidence claims are considered first), and define the complement-prodtee
k
II (P..)) = II (1 ((P.cz()) + +e).
Ø=1
Then we have the exact log- oog-proucct identity
k
log (P.CC = Σ ((PP.c())
Hence the budget rule used in our filter, Σラ=1 ((P,ern)) S r, is equivalent to [T(P.C) 2 e Because the mapping x + log æ is strictly monotone on (0, 1]. this implementation of log-space preserves the induced selection sets and conformal quantiles, while improving numerical stability by avoiding product underflow.
D.1 DATASETS
MedLFQA. For the medical auestion-nnswere task, Cherian et al. (2024) create an experimen- tal dataset using prompts from the MedL.FQA benchmark (Jeong et al., 2024). To generate the data, they first prompt GPT-3.5-Turbo to produce new responses, which are then parsed into atomic claims by GPT-4o. For the crucial step of ground-truth annotation, they employ an automated verification procedure. For each generated claim, they prompt GPT-3.5-Ture to verify whether it is substanti- ated by the reference answer provided in the original MedLFQA benchmark, effectively treating the reference answer as the ground-truth source text. From this dataset, we randomly extracted 2,000 samples, comprising 33,833 claims, for our experiments.
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WikiBio. Cherian et al. (2024) follow the principles of the FACTSCORE (Min et al., 2023) dataset to construct a new, large-scale benchmark for evaluating the factuality of LLM output. To gener- ate the data, they prompt GPT-3.5-Turbe Wikipedia. To circumvent the high cost of manual annotation, they then employ a variant of the FACTSCORE procedure for fact-checking. For each generated claim, they use the BM25 algo- rithm (Robertson and Zaragoza, 2009) to retrieve ground-truth passages from Wikipedia and subse- quently prompt GPT-3.5-Turie to verify whether the claim is supported by the retrieved text. This completed dataset is referred to as WikiBio in our paper for convenience. We randomly extracted 2,000 samples, comprising 33,804 claims, for our experiments.
write short biographies for 8,516 names sampled from
ExpertQA. Malaviya et al. (2024) construct the ExpertQA dataset, large-scale benchmark for evaluating the factuality and attribution of LLM output. To generate the data, they first asked 484 qualified experts across 32 fields to formulate challenging, professional lives. To ensure high-quality annotations, they then employed an expert-in sepert-in-tlee in-the-loop evaluation procedure where the same experts validated sentence-level claims in responses generated by six representative LLMs. Using the rich, human-annotated information provided in this dataset, we construct binary ground truth for each claim. Among the datasets in our study, ExpertQA was the most challenging and also the most rigorously labeled, due to its direct validation by domain experts. We randomly extracted 2,000 samples, comprising 11,538 claims, for our experiments.
citoommtiionsceeiing questions from their
Of the 2,000 samples, 1,500 were used for the calibration phase, and 500 were used for the filtering (test) phase. Figure 4 shows an actual format of the data sample we use.
D.2 GROUPING CRITERIA
We employ complex grouping criteria that are likely occur in reality, yet simultaneously require prompt and response parsing along with numeric values. We create grouping criterion applicable to all three datasets and three classification criterion reflecting the characteristics of each dataset. The criteria are as follows:
Common: False-Claim Risk. This is a composite risk index calculated by analyzing features of the prompt and response texts. The risk score increases with longer response lengths, higher frequency of lists or numbers, and the inclusion of absolute or definitive expressions like 'always', 'never', 'cure'. Conversely, the risk score decreases when expressions citing sources or evidence, such as 'according to or 'research shows', are present. This index estimates the potential risk of containing false information based solely on textual characteristics.
MedLFQA: Medical Content. Medical-related questions are classified into three groups based on the 'intent' of the user's prompt:
Information-Seekken SSekinn (Info): Cases that ask for factual information about specific disease or drug, using keywords like "what is," "symptom," " or "treatment."
AnteppeetationSeekiice (Interpret): Cases that request an interpretation of what a specific symptom or condition means, using phrases like "what does it mean" " or "should worry." Action-Seeking (Action): Cases that ask for specific guidance actions treatment, using phrases like "should I," "can I take," or "how to."
WikiBio: View Count. Groups divided based the cumulative number of page views for each person's Wikipedia page in the WikiBio dataset. This is used as an indicator of public interest in or awareness of the person.
ExpertQA: Question Domain. Questions (prompts) from the ExpertQA dataset are classified into three high-level domains based on the academic field specified in the official metadata: Biology/Medicine (Bio/Med): Life science and health-related fields such Healthcare, Medicine, Biology, Chemistry, and Psychology.
Technology/Science (Tech/Sci): Engineering and physics-related fields such as Engineering and Technology, Physics, and Astronomy.
Common: All other academic fields that do not fall into the two categories above.
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D.3 FACT-CHECKING PROMPT
The prompt below is an example of the prompt we input to M LLMs to obtain a verbalized factuality-score for each claim.
Fact-checking prompt
Instruction. You are an expert fact-checker and logician with access a vast knowledge base. Your task is to assess the factual accuracy of specific claim extracted from a model's 's response to given prompt. Follow these steps to produce a precise evaluation:
1. Contextual Analysis: Understand the claim within the context of the original prompt.
2. Knowledge Verification: Using your general world knowledge and the information provided, check the claim against established facts, scientific or medical consensus when applicable, and basic logical consistency.
3. Probability Estimation: Estimate the probability that the claim is factually true.
Scoring rubric (probability truthfulness). Assign continuous score between [0.0, 1.0] representing your confidence:
1.0 (Certain Truth):
The claim axiomatically true or verified by very strong evidence.
0.8-0.9 (High Confidence): The claim is widely accepted as true by experts; minor context
might be missing but the core correct.
0.4-0.6 Ambiguous/Uneertaing The claim is debated, only partially supported, there is insufficient information to verify
0.1-0.3 (Low Confidence): The claim conflicts with available evidence contains significant inaccuracies.
0.0 (False): The claim is demonstrably false, logically impossible, entirely hallucinated. Input. Original prompt: (prompt_text)
Claim to evaluate: fclaim_text,
Output format. Return a single JSON object with an array "evaluations";
" "evaluations":
"claim id"; 1,
"reasoning": "Step-by-step reasoning here...", "score": 0.85
D.4 SELECTING LLMs
Although our methodology is model-free and works with any large language model, we choose to use Llama-3.3-770--Insttte (Grattafiori et al., 2024), Dwven22.-72B-IInstmu (Qwen et al., 2025), and DeepSeck-V. V3 (DeepSeek-Al et al., 2025). We selected these three white-box models for their high transparency and reproducibility. Their public availability and stable serving allow us to openly share and control all settings, such decoding parameters, logs, and the calibration pipeline, mak- ing our work easily replicable. This open nature also reduces our dependence on unseen changes to policies filters that come with version updates, which is key advantage in fields where repro- ducibility and auditing are crucial.
D.5 SAMPLING-BASED METHODS
SelfCheck. Manakul et al. (2023) proposes black-box, zero-resource method that detects hal- lucinations by sampling multiple responses from large language model for the same query and quantifying the content consistency between the original response and the samples. Specifically, the method generates multiple stochastic responses for single prompt and calculates response reliabil-
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ity by aggregating mutual consistency the sentence/passaae level, using metrics such semantic similarity, NLI-based contradiction signals, and use Llama-3.3-70B- ----t--nntttutt the model to generate multiple samples for the same prompt when applying the SelfCheck procedure.
agreement. In our experiments,
FactSelfCheek (FSC). Sawczyn al. (2025) proposes a method for detecting fact-level halluci- nations by extracting factual units from response and multiple samples to construct a fact graph, then aggregating supporting and contradictory signals for each fact across all samples. The pro- cedure involves extracting facts (e.g,, seiitt-relatio--enet -etnntttetttttttee triplets) from an initial response and multiple samples, calculating the degree of consensus among these facts aggregate them into fact, sentence, or passage-level scores, and finally performing threshold-based filtering. In our experi- ments, we also use Llama-3.3-7005 300m-33.-700--Insttut to generate the multiple samples required for the FSC procedure.
D.6 EVALUATION METRICS
To evaluate our proposed method, we assess two key aspects: the quality of our oracle- approximating factuality-score function and the performance of the final filtering procedure.
Coverage. Coverage is the primary metric for verifying the theoretical guarantee of our confor- mal inference procedure. A sample D; is considered "covered" if its filtered set F(C:) contains no hallucinatory claims. The empirical coverage is the fraction of samples in the test set that are successfully covered. For given error rate a, valid conformal procedure is expected to yield an empirical coverage rate approaching or exceeding 1 a.
Cov.
Σ [ Gi E F(CC), Us.) = =1]- DEDee
[Drest
Retention Ratio. While coverage measures the safety of the filter, retention ratios measure its utility. Retention measures the average fraction of total claims remaining after filtering, indicating how much of the original text volume preserved:
F(C,) 1C|
Ret.
[Drest
DEDiE
E ADDITIONAL RESULTS
Comparison with BCI, CCI, MACI, with Unified Factuality-Score. Theorem demonstrates that the quality of the factuality-scom directly impacts MACT's retention ratio. Therefore, improving the quality of the factuality-scor through Multi-LLM- contributes to the high retention ratio compared to MACI's conformal inference-based baseline. meaningful verify how much the adaptive conformity score structure, which mimics oracle factuality by excluding the Multi-LLM ensemble part in MACI, contributes to the retention ratio. Therefore, we fix MACT's factuality-scom to a frequency score, which BCI and CCI use. That is, compare the coverage and retention of MACI with BCI and CCI, excluding the Multi-LLM ensemble component. Table 4 shows that MACI achieves the highest retention even under the same unified factuality-scom scoor. This demonstrates that MACI's oracle-motivated adaptive conformal inference structure itself is superior to the baselines.
weighted optimization ensemble directly
Consequently,
Comparison with MultiValid Conformal Inference. Research in the Multivalid Conformal Pre- diction family presents a robust framework that simultaneously guarantees coverage for multiple subgroups and threshold intervals. Specifically, Jung al. (2023)'s Batch Multivalid Conformal Pre- diction defines the concept of multivalid coverage, which simultaneously satisfies group-conditional coverage for multiple groups and cresshll--evel-condiiee
coverage for various threshold levels under single threshold function. Formally, letting s(X, YY) denote a conformity score, I, (X) the prediction set induced by a threshold function T(-). and g. I denote family of groups and score intervals, multivalid coverage requires that for all G E G and I E I,
PPYY E (x) s(X,Y) E 2 1 - 1-a-5
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Table 4: Comparison with conformal baselines, with unified factuality-score Coverage within + 0.01 are marked with a green dot while values that fall outside this range are marked with red arrow 11
Target Coverage: 80% (a = 0.2)
Target Coverage: 90% (a = 0.1)
Target Coverage: 95% (a 0.05)
BCI
CCI
MACI
BCI
CCI
MAC1
BCI
CCI
MACI
Group
Cov.
Ret.
Cov.
Rel.
Cov.
Ret.
Cov.
Ret.
Cov.
Ret.
Cov.
Ret.
Cov.
Ret.
Cov.
Ret.
Cov.
Ret.
MedLFQA: False-Claim Risk
0.841
0.831 0.68
0.821
0.70
0,947
0.03
0.91
0.41
0.91
0.971
0.95.
0.28
0,96. 0.32
Mediom
0.831
0.06
0.81+ 0.66
0.81+
0.71
0.89.
0.03
0.90.
0.39
0.91e
0.46
0.94
0.01
0.95.
0.25
0,96.
0.30
High
0.73 0.06
0.78. 0.43
0.81 0.58
0.881
0.01
0.89.
0.22
0.90.
0.36
0.94. 0.01
0.94.
0.12
0.95. 0.24
for small tolerance E 2 0. This is a stronger requirement than standard group-condittonal coverage, which only conditions on group membership.
While this approach provides a strong form of validity, it can lead to a conservatively large predic- tion set due to the need satisfy many groups and threshold levels simultaneously. MACI targets a more specific application setting. MACI's focus is performing false-claim filtering for responses generated by LLMs, aiming to maintain group-conditional coverage for a small number of mean- ingful groups (e.g.. groups based on false-claim risk) while achieving level of retention suitable for practical systems. Concretely. if Grisk denotes a task-specific partition (c.g., low/medium/high false-claim risk), MACI enforces
P(Y
2 1-a,
vg Grisk
without additionally conditioning on score intervals. Therefore, MACI focuses balancing cover- age and efficiency for the specific task of false-claim filtering, rather than providing general multi- valid guarantees across diverse groups and threshold intervals.
To quantitatively assess this difference, we implemented Multi Conformal Inference (MVCI, modification of Jung et al. (2023)'s BatchMVP tailored for the false-claim filtering environment, and compared it with MACI. Experiments were conducted on the WikiBio dataset, with both meth- ods configured use the same group definitions (c.g.. group partitioning based on false-claim risk), the same calibration/test split, and the same target error rate Table 5 shows the results comparing the group-conditions coverage and retention of the two meth- ods. MVCI satisfies the target coverage level in each group but achieves retention at significantly lower rate by producing conservative thresholds to simultancously guarantee coverage across groups and threshold levels. In contrast, MACI achieves much higher retention ratio than MVCI while maintaining a similar level of group- condiconni coverage under the same group definitions and sig- nificance level. This reflects the different design philosophies: MVCI prioritizes strong multivalid coverage, whereas MACI emphasizes balancing group-conditional guarantees with practical reten- tion in false-claim filtering.
Table 5: Comparison with MultiValid Conformal Inference (MVCI). Coverage within a + 0.01 are marked with a green dot while values that fall outside this range are marked with red arrow 11.
Target Coverage: 80% (a:
0.2)
Target Coverage: 90% (a 0.1)
Target Coverage: 95% (a 0.05)
MVCI
MACI
MVCI
MACI
MVCI
MACI
Group
Ret.
Cov.
Cov.
Cov.
Cov.
Ret.
WikiBio: False-Claim Risk
Low
0.80.
0.11
0.821
0.40
0.89.
0.03
0.90+
0.23
0.95.
0.02
0.94.
0.17
Medium
0.81
0.08
0.81
0.42
0.91.
0.02
0.90.
0.25
0.96.
0.01
0.95.
0.12
High 0.81
0.03
0.81
0.45
0.90.
0.01
0.90.
0.28
0.95.
0.01
0.96.
0.09
Comparison with sampling- methods. While our primary focus is on CI-based baselines, it is also practical to compare against recent non-CI approaches. We compare MACT's group- conditional coverage, marginal coverage, and retention ratio with sampling-basee methods that ap- ply to blx LLMs and do not rely retrieval. Brief descriptions of these baselines appear in Section D.5. Unlike Cl-based methods, sampling- guarantees; instead, they compute factuality-score p E [0, 1]. enabling false-claim filtering via
approaches do not provide statistical
24

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Published as a conference paper at ICLR 2026
1.0
0.8
0.4
6.-
Coverage (MACL gropp Sateition (MACI group)
Coverage DMAO custer group!
group)
Group
(a) MACI
(b) MACI-Cluster
Figure 5: Coverage and retention in large number of groups, using MACI and the MACI-cluster method. The top shows results split into 24 groups. The average coverage per group more main- tained by MACI than MACI-cluster, but with greater variance. Using the group clustering method allows for practical results by sacrificing strict group-conditional coverage.
threshold (e.g.. 0.5). Table shows that sampling-basse based methods attain high retention but low cov- erage. This highlights their limitation in meeting the strict requirement that the filtered set contain no false-claims. Moreover, their target coverage is not user-specified and thus unpredictable. These points underscore the need for MACI in high-stakes settings that require a user-specified high a.
Table 6: Comparison with sampling-based methods, representative black-box and non-retrieval approach for false-claim filtering. Sampling-base based methods generally exhibit very low or unstable coverage and high retention ratio. This suggests they unsuitable for the strict target that all claims have to be factual (the definition of Cov.). In contrast, MACI (a = 0.1) demonstrates the ability reliably guarantee the user's desired coverage.
SelfCheck
FSC-text
FSC-KG
MACT
SelfCheck
FSC-text
FSC-KG
MACI
Group
Cov.
Rel.
Cov.
Ret.
Cov.
Ret.
Cov.
Rei.
Group
Cov.
Ret.
Cos.
Ret.
Cov.
Ret.
Cov.
Ret.
MedLFQA
0.56
0.97
0.63
0.85
0.64
0.88
0.90
0.50
WikiBiio 0.12
0.97
0.33
0.77
0.37
0.70
0.90
0.25
Medical Content
View Count
Info
0.98
0.69
Interpret
0.54
0.98
0.59
0.90
0.63
0.93
0.90
0.47
Medium 0.13
0.97
0.31
0.79
0,27
0.73
0.91
0.24
Action
0.95
0.79
0.83
0.90
0.53
High
0.82
0.39
False-Claim Risk
False-Cluim Risk
Low
0.64
0.98
0.70
0.86
0.71
0.89
0.89
0.52
Low 0.19
0.98
0.41
0.78
0.43
0.72
0.90
0.23
Medium
0.56
0.98
0.63
0.88
0.60
0.92
0.91
0.46
Medium 0.08
0.96
0.28
0.74
0.32
0.68
0.90
0.25
High
0.97
High
0.78
0.35
Applying Group Clustering Method. MACI adopts a Mondrian-style group-conditional confor- mal scheme: for each pre-defined group g G. we calibrate a separate threshold from the calibration examples belonging to that group. Let Rg denote the nonconformity score distribution within group and let ii-a be the empirical (1 a -quantile of the calibration scores in g. Using this group- specific quantile the threshold yields exact finite-sample group-conditional guarantee
IP(Y E C,(x) 9) 1 Vg E G.
under group-conditional exchangeability between calibration and test examples. However, when the calibration sample size ny particular group small, the empirical quantile a1-a can have high variance, which in turn leads to unstable thresholds and increased variability in both coverage and retention across groups.
Gao et al. (2025) propose to mitigate this issue by clustering groups whose score distributions similar and then performing conformal calibration at the cluster level. Formally, let h 9 - K. map each group g to cluster & h(g), and let R4 denote the score distribution obtained by pooling all calibration scores from the groups assigned to cluster Mondrian conformal inference is then applied in the cluster space, producing threshold r that is shared by all groups g with h(g) k.
25

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Published as a conference paper at ICLR 2026
Source (calib)
Target (test)
Resampled
Density
Mean Frequency Score
Logistic Classifier: AUC-0.943 ACC-0.894 F1-0.890
Figure 6: Covariate shift and resampling distribution in the MedLFQA dataset. The blue and red graphs show the distributions of calibration and test, respectively, based on the covariate shift vari- able mean frequency score. The green graph shows the distribution resampled using binary classi- fier.
This increases the effective calibration sample size for each threshold and reduces variance. At the same time, the group- condiconnaiional guarantee is relaxed: if the score distribution of each group g close to that of its cluster h(g) in the sense that
sup (t) - Fx(t) S Eg.
EER
where Fg and F5 are the CDFs of Rg and Rg, then the group-conditional coverage degrades from the exact a level to
P(Y E Cx(X) 2- = g} 2 a Eg
for all E G, as shown in Gao et al. (2025). In other words, group clustering trades strict finite- sample group-conditional coverage for lower variance and more stable thresholds.
We construct a variant, MACI-Cluster, by combining Gao et al.s group clustering mechanism with MACI's factuality-score and thresholding scheme. We first define fine-grained groups using the false-claim risk score, apply group clustering in the space of conformity score distributions, and then learn cluster-level thresholds that are shared by the groups within each cluster. We then com- pare MACI-Cluster with the original MACI in terms of empirical croup-conditional coverage and retention across many groups. Concretely, MACI-Cluster uses K 8 clusters obtained by k-means on vectorized conformity histograms, and applies single conformal threshold per cluster.
Figure shows the coverage and retention per group for MACI and MACI-Cluster under 24 group partitions. Over 100 repeated runs, MACI's group-wil -iie coverage is on average close to the user- specified level but exhibits large variance, making the realized coverage unstable. MACI-Cluster yields group-wise coverage that does not exactly converge to the target level due to distributional dif- ferences between clusters and evaluation groups, but its variance smaller, leading to substantially more stable coverage and retention when the number of groups is large and each group contains few samples. Consequently, the group clustering method is good approach that provides stable results when the number of groups increases and the sample size is small.
MACI under Covariate Shift. The calibration data used to conformal thresholds and the queries encountered at test time do not necessarily follow the same covariate distribution in deployed systems. This setting is typically formalized as covariate shift: calibration data {(x,, Y) (.ddd
p(s) xx and the test point (Xx+1, Y,-1) p(() Xy satisfy ((s) p(!)) Y|x while their marginal covari- ate distributions p!* x and ((t) may differ. Let p and (t) denote the densities of p(*) and p(t) respectively (with respect to common dominating measure). Tibshirani al. (2019) address this
26

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Published as a conference paper at ICLR 2026
mismatch by reweighting calibration examples using the density ratio
PX (z)
rax)
Let (X;;, Y.) denote nonconformity scores the source calibration set. A weighted empirical quantile for a target miscoverage level a is then defined by
E 11=1 r(X.) o S a).
aa infoa
Di=1((X)
which replaces the usual unweighted empirical tail probability by density-ratio weighted version that targets P. instead of ...
We adopt this density-ratio correction principle for the MACI pipeline and refer the resulting variant as MACI-DRE. Rather than modifying the quantile computation inside MACI, we first use an estimate F(z) construct renssty-raio--eiihhee calibration set by importance resampling, and then run the original MACI algorithm on this resampled set. Concretely, given calibration samples {(X,, 1:) from P, and estimated ratios F(X;), we draw indices
T(X,) Σ/=1 f(X))
I In Multinomial(n; TI, T mn)s
T
and form a resampled calibration set {(X,, YI, ME-1 For any bounded measurable function f.
a Σ (X Yin)
n Σ n: Tx,(), R BxxYY (((X))) if F(x)
(t (x(
Thus, when the density ratio is well estimated, running MACI on the resampled calibration set is equivalent, in expectation, running MACI calibration set drawn directly from the target dis- tribution Pt. while keeping all internal components of MACI (ensemble weight learning, subgroup- wise threshold estimation) unchanged.
For our covariate-shift situation on MedL.FQA, we construct an explicit covariate shift hetween the calibration and test distributions. For each data sample {(P, R, C)), obtain claim-level factuality-scores via SelfCheck (Manakul et al., 2023) method, and define a scalar feature
m(x) (
s(x)
(BeRCheck(I,5))
m(x)
where m(x)) is the number of atomic claims in the response. We sort all samples by s(x) and designate the upper region (larger s(x), average more factual responses) as the source test pool Pa. and the lower region (smaller szx), more hallucination-promm responses) as the target calibration pool Pt. This produces a clear marginal covariate shift in terms of the distribution of s(x), while keeping the underlying annotation procedure fixed.
The density ratio is estimated from features that are observable at deployment time and do not require ground-truth labels. For each sample æ, we build feature vector
σ(x) (s(z), (s) len(prompt), len(response). 1).
where ((x) and sz))) denote the mean and standard deviation of the claim-level SelfCheck scores, and the remaining components encode simple prompt and response length statistics. We then train binary classifier on Φ(x) to distinguish source from target samples, with label for P. and 1 for P.. Using logistic regression with approximately balanced priors, the estimated density ratio is obtained as
p(Y 1 x(x))
f(z)
p(Y = (xx))
which is then used for the importance resampling step described above.
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Published as a conference paper at ICLR 2026
Comparison with Conformity Score Variants. MACT's multiplicative score is motivated by the oracle formulation. Under an ideal oracle that assigns joint factuality-score to each claim, combin- ing the scores of multiple verifiers by multiplication provides simple approximation to this joint score. For numerical stability, we implement this the following log-product form:
N2
Smule (c) = = log P:)).
Conformal calibration depends on the ranking of conformity scores rather than their absolute values, so the product and log-product forms are related by monotone transformation and yield the same coverage guarantees.
We compare this multiplicative score against two alternative aggregation rules. The first is log-sum form,
Stog-sum (c) = log
pi(c))
The second is power-mean form,
1 N Σ P((())) 1/A
(Tm(c;
which allows us to explore different aggregation behaviors by varying the exponent A.
Table 7 reports the coverage and retention of these two variants and MACI. Overall, all three meth- ods stay close to the target coverage, but power-mean consistently achieves the lowest retention across settings. The log-sum form attains retention comparable to MACI in the low- and medium- risk groups, but its retention noticeably drops in the high-risk group and for smaller values of a. MACI's fog-product score has a clear motivation from the oracle formulation, is compatible with the conformal calibration framework, and empirically performs best overall in our experiments, so we use it as the default choice the main text. Sum-based (log-sum) aggregation remains a promis- ing alternative that could be explored further with more refined design in future work.
Table 7: Comparison with Conformity Score Variants (Power-Mean (A = 2), Sogg-Sum,aa and MACI's log-product) on MedLFQA, across false-claim risk groups
Target Coverage: 80% (a
0.2)
Target Coverage: 90% (o
0.1)
Target Coverage: 95% (o 0.05)
Power Mean
Log Sum
MACI
Power-Meun
Log Sum
MACI
Power Meun
Log Sum
MACI
Group
Cov.
Ret.
Cov.
Ret.
Cov.
Ret.
Cov.
Ret.
Cov.
Ret.
Cov.
Ret.
Cov.
Ret.
Cov.
Ret.
Cov.
Ret.
MedLFQA: 0.81
False-Claim Risk
0.81
0.75
0.78
0.91
0.90
0.52
0.96
0.33
Medium
0.80
0.66
0,82
0.70
0.79
0.70
0.91
0.45
0.90
0.46
0.91
0.46
0.95
0.31
0.95
0.31
0.95
0.31
High
0.81
0.54
0.80
0.60
0.80
0.64
0.91
0.35
0.91
0.36
0.89
0.41
0.96
0.23
0.95
0.24
0.95
0.26
Modeling Joint Probability. Our primary experimental setting decomposes each model response into independent atomic claims and then estimates factuality-score for each claim using only the (Prompt, <;) pair. This design follows the oracle formulation and enables conformal analysis at the claim level, implicitly adopting an independence assumption between claims. In realistic LLM outputs, however, claims may exhibit correlations, redundancy, or logical dependencies, so this assumption does not perfectly match the underlying generation process.
In our framework, factuality checking is delegated to M LLMs used as verifiers, rather than being resolved within single decoding of the generative model. Under this perspective, an important con- sideration is whether claim-wis modeling strategy remains effective compared approach that explicitly exposes the full Claim Set. To investigate this, we complement the per-claim setting with joint modeling experiment in which the verifier receives the entire Claim Set once and outputs conditional scores {p(c, C) Concretely, provide (Prompt, (c1...., CN, } as input so that the verifier can, in principle, exploit interactions among claims when assigning factuality-scores.
joint modeling
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Published as a conference paper at ICLR 2026
Drepoodb
0,000
0,632
0.727
1J000
00:70
0.213
1,000
0,30
EL345
Mbenaet 1988
Unome di 5,30
Unooe 2888
0,000
0.445
LJN30
11:923
0.300
04
Qpeers E1-70h
Desperok.y
Lieen
Deepeen-115
06
Jaccaad Distance Between LLMs
Q-statistic
(c) MCC LLMs
$.500
0.53
0.422
010 CJ43
0.522
1.11
6.39
0.2
0.2
NACI
Marginal
Te) Marginal Retention
(1) Marginal Retention by Model Increasing
Figure 7: (a)-(c) display diversity diagnostics including Jaccard distance, Q-statistic, and MCC; (d)-(e) show the performance metrics MACI; and (f) illustrates the retention trend by ensemble size. (a), (b), and (c) reveal high disagreement and low correlation between LLMs' predictions on false-claims, indicating diverse detection patterns that strongly support the necessity of using an ensemble. (d) demonstrates the sequential improvement in FPR and MSE from LingleLLLM and arithmetic mean ensemble to our proposed MACI, while (e) shows that these error rates improve, the retention ratio also increases. (f) demonstrates that as the number of models in the ensemble increases (k: = 1, 2) with the optimal combination selected, the retention ratio consistently im- proves, validating the efficiency of scaling the ensemble.
Table 8 shows that the per-claim setting and the joint setting yield similar coverage and retention profiles. This suggests that, even when the full Claim Set available, the verifier LLM can effec- tively treat each claim comparatively independent decision unit, and that claim-wise modeling a practically viable choice from the verifier's perspective. other words, the independence as- sumption used for oracle-basee analysis serves both convenient theoretical simplification and as an approximation that does not substantially degrade performance when compared to a more joint modeling strategy.
Table 8: Comparison between MACI and MACI-Joint. MACI-Joint scores claims jointly given the whole claim set, while MACI scores each claim independently.
Target Coverage: 80%
Target Coverage: 90
Target Coverage: 95%
0.05)
MACI
MACI
MACI
MACI-Joint
MACI
MACI-
Group
Ret.
Cov,
Ret.
Cov.
Ret.
Cov.
Ret.
Cov.
Ret.
Cov.
Ret.
MedLFQA: False-Claim Risk
Low
0.79
0.78
0.80
0.75
0.89
0.52
0.91
0.52
0.95
0.37
0.95
0.36
Medium
0.90
0.95
High
0.80
0.64
0.80
0.60
0.89
0.41
0.90
0.38
0.95
0.26
0.95
0.25
F STATEMENT ON THE USE OF LARGE LANGUAGE MODELS
We used an LLM for minor editing and scripting automation only; core idcas, experiments, and analyses were conducted by the authors.
29


Reviews:
[{"review_id": "EfY9vbMqsy", "reviewer": "Reviewer_4aeR", "summary": "The paper proposes a subgroup-aware selective prediction procedure that learns a filter and threshold per subgroup to control FPR while maximizing retention and attains valid subgroup-conditional coverage. An ensemble over base filters stabilizes performance, and the method reports strong empirical FPR control with competitive efficiency. I appreciate the clear motivation and the practical design choices (e.g., monotonicity of FPR in the tuning parameter, which enables efficient search). My main concerns are (i) statistical stability in underrepresented groups when learning per-group thresholds/filters, (ii) the strength and usefulness of Assumptions 1–2 (Theorem 3 hinges on them), and (iii) limited discussion of PAC-style guarantees despite the monotone FPR structure. I also encourage a more robust experimental protocol (more than 30 trials). More discussion on retention-aware, adaptive-α strategy would be good.", "questions": "1. If test and calibration data distributions differ or if the filter is learned on the same data as thresholding without sample splitting, nominal control may not hold. It would be good to see clear explicit cross-fitting across subgroups (which could be unstable without clustering those with similar conformal scores as suggested) or an accounting for potential dataset shift, e.g., covariate shift with bounds on the density ratio.\n\n2. If C upper bounds either the change in FPR as a function of the parameter near the operating point or the inverse density, then taking it to be arbitrarily large makes the bound true but uninformative. Can C be tied to an observable quantity, e.g., plug-in estimator of f_S at the quantile with high-prob bounds? Or can it be acknowledged that small density at the threshold leads to large constants and loose rates; can this be characterized and shown more clearly? \n\n4. Are there ever settings where we must achieve a particular retention group-wise while enforcing FPR per group to be less than some threshold? If so, how might one go about doing this? \n\n5. For a group-conditional score, the efficient influence function for the \\tau quantile in the nonparametric model can yield a one-step estimator. Can the authors comment on this and its feasibility?\n\n6. Is there a retention vs FPR frontier per group? This could make the fairness-efficiency trade-off visually clear.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "* Clear problem framing at the intersection of fairness (group FPR control) and efficiency (retention), with a practical ensemble that helps smooth variance across base filters.\n\n* Good use of the monotonic relation between the parameter and FPR, which opens the door to simple line or binary searches and possible RCPS-style risk control.\n\n* Careful numerical studies and sensible metrics; the approach feels implementable.", "weaknesses": "* Learning a separate filter and threshold per subgroup can be statistically brittle when the sample size is small, such as in underrepresented subgroups; estimates of group-level score quantiles and FPR could swing widely, harming both fairness and efficiency. Consider clustering subgroups with similar score distributions (cf. Gao et al., 2025 or Ding et al, 2023) or hierarchical / shrinkage pooling of quantiles.\n\n* As written, C appears tunable or possibly loose enough that the assumption risks being vacuous, i.e., I could take C large enough to satisfy the assumption. More clarification on whether C is pop-determined, empirically verifiable, or enters the finite-sample bound in a way that tightness matters.\n\n* Assumption 1 and 2 seem potentially strong for minority groups; if score density at the operating quantile is small, variance explodes, and makes the bound useless. More discussion on failure modes when density at decision boundary is low is needed.\n\n* Because FPR is monotone in the parameter, RCPS techniques suggest that PAC coverage bounds could be obtained.\n\n* 30 trials used in Table 1 seems small. How sensitive are results to group size imbalance?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "mr3DHS2FbL", "reviewer": "Reviewer_eeqE", "summary": "The authors propose a conformal method (MACI) for filtering LLM outputs. MACI leads to higher claim retention that prior work while still maintaining marginal/group covergage.", "questions": "Questions (I apologize in advance if some of these questions were directly addressed in the main body):\n1. How exactly do you produce estimates $\\hat{p}$ / What is $\\hat{p}$ your experiments? It wasn't clear to me if you were using base uncertainty scores like in prior work (verbalized confidence, frequency scores, model probabilities, P(True), etc.) or if the \"black-box classifier\" is also something you train for each LLM.\n2. For the ensemble, do you mean that you are taking $\\hat{p}$ from each LLM: Llama-3.3-70B-Instruct, Qwen-2.5-72B-Instruct, DeepSeek-V3?\n3. Which model's output are you filtering? Or are you running these methods on outputs from all 3 and then taking an average in Table 1?\n\nSuggestions:\n1. I recommend citing a few other works related to group-conditional conformal prediction.\n- [1] Jung, Christopher, et al. \"Batch Multivalid Conformal Prediction.\" International Conference on Learning Representations.\n- [2] Liu, Terrance, and Steven Wu. \"Multi-group Uncertainty Quantification for Long-form Text Generation.\" The 41st Conference on Uncertainty in Artificial Intelligence.\n- [3] Hébert-Johnson, Ursula, et al. \"Multicalibration: Calibration for the (computationally-identifiable) masses.\" International Conference on Machine Learning. PMLR, 2018.\n- [4] Detommaso, Gianluca, et al. \"Multicalibration for Confidence Scoring in LLMs.\" International Conference on Machine Learning. PMLR, 2024.\n\n    In contrast to Cherian et al. 2024 / Gibbs et al., 2024, [1] and [2] frame group-conditional conformal prediction more similarly to works on multicalibration (like [3], [4]), and they also typically consider more groups than in your work or that of Cherian et al. 2024. Potentially, you could also compare to the methods evaluated in [2] (group-conditional versions split conformal and conformal quantile regression) to further bolster your experimental results since at the end of the day, the goal of those methods are also to achieve better group coverage and retention rates. However, I think just mentioning these lines of works would be sufficient.\n2. I don't see the value of comparing to sampling-based methods (SelfCheck, FSC) since as you stated, the goal of those methods are not to achieve some target coverage. I think the point you're making could also be made with any conformal method like just using BCI, so the comparison doesn't highlight to me anything about MACI. Hearing your thoughts on this would be helpful. In my opinion, you could remove these experiments without hurting the paper at all, and it would leave more room to explain other details in the main body.\n\n    Also, since these methods are in some sense, proposing better options for $\\hat{p}$, perhaps a more sensible comparison might have been to see what happens if you use these sampling methods as the base uncertainty score $\\hat{p}$ (instead of the \"frequency score\" that prior work often use) for BCI, CCI, multi-valid split conformal, etc. However, maybe that would make sense if BCI/CCI perform better using SelfCheck/FSC compared to frequency scores and that MACI still performs the best.\n3. Perhaps Figure 3b and c should be bar charts instead of line graphs.\n4. I think 5.2 and 5.3 could be moved to the appendix so that some other experimental details from the appendix can be put into the main body.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 4, "strengths": "The paper is well-motivated and the proposed method is novel. The the empirical results (Table 1) are strong and clearly communicated.", "weaknesses": "In general, I think this is a good paper. However, I think the empirical section does not clearly communicate some details. For example, I believe completing replicating the experiments from just reading the paper (without contacting the authors or looking through their code) would be challenging. While I understand that the authors are limited in space, I also did not find it easy to parse some of these details out of the appendix.  \n\nI am open to improving my score once some of my clarifying questions and suggestions are addressed, but currently, I am not comfortable giving a higher rating.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "SPEmp7GU9g", "reviewer": "Reviewer_p4aF", "summary": "The paper tackles reliable factuality filtering for LLM outputs using conformal inference (CI). Prior CI approaches either (i) apply a single global threshold (achieving marginal coverage but aggressively filtering, hence low retention) or (ii) learn sample-wise thresholds with adaptive error rates which improves retention but weakening fixed-level guarantees. The authors propose MACI, which (1) reformulates factuality filtering via a multiplicative conformity score (cumulative probability product across claims), (2) introduces group-conditional calibration to retain finite-sample, distribution-free guarantees within predefined subgroups, and (3) increases retention by improving factuality-score quality through a multi-LLM ensemble whose weights are optimized. They also give retention analysis in this CI setting, linking deviations from an oracle factuality-score to true-claim preservation, which motivates the ensemble design. Empirically, MACI maintains user-specified coverage while substantially improving retention and reducing time cost relative to BCI and CCI baselines; an anonymized repository is provided.", "questions": "1- How does MACI’s group-conditional coverage behave under covariate shift (prompt style, domain drift, temperature changes)? Could the authors report coverage/retention under controlled density-ratio bins or apply lightweight split-conformal covariate-shift corrections? \n\n2- What guidance can you offer for choosing group granularity 𝐾 given a fixed calibration budget? Any results on automatic grouping (e.g., clustering claim/prompt features) vs. fixed datasets’ categories? \n\n3- Please include diversity diagnostics (pairwise disagreement, Q-statistic) and an efficiency-quality frontier varying the number/type of models and weighting schemes. How sensitive is MACI to over-reliance on a single strong model? \n\n4- Did you compare the multiplicative conformity score to additive/log-sum or power-mean variants? A small ablation would help establish that the product is not only principled but also empirically dominant. \n\n5- Table 3 is helpful; could you add wall-clock numbers at larger evaluation scales and with streaming recalibration when groups shift or new domains appear?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1- Casting filtering as a cumulative probability product of claim-level factuality scores is natural yet non-obvious; it aligns the conformity score with joint “no-false-claims” events and helps reconcile coverage with retention. \n\n2- The paper provides a retention analysis linking oracle-estimator deviations to preserved true claims, thereby justifying the multi-LLM ensemble design rather than treating it as a heuristic. \n\n3- Group-conditional calibration preserves finite-sample guarantees at user-specified risk levels while avoiding the heavy conservatism of global thresholds; it also sidesteps adaptive-α issues in high-stakes settings. \n\n4- The method achieves higher retention at target coverage and shows favorable time-cost due to single-pass scoring and simpler calibration vs. conditional boosting/sampling methods.", "weaknesses": "1- The method relies on exchangeability/i.i.d. between calibration and test; real deployments may face distribution shift (domain, style, model temperature). The paper would benefit from experiments that explicitly stress test group-conditional coverage under covariate shift (e.g., density-ratio buckets, temporal drift). \n\n2- Guarantees are group-conditional, so the definition of groups meaningfully affects validity/retention. More analysis on group granularity vs. sample size trade-offs (and automatic group discovery) would strengthen the story. \n\n3- Retention gains hinge on factuality-score quality and thus on ensemble diversity and calibration. The paper could add ablations on ensemble composition, weighting stability, and computational budget vs. benefit. \n\n4- While the multiplicative score is well-motivated, exploring alternatives (e.g., log-sum, powered products) and reporting sensitivity would clarify robustness to mis-calibrated claim probabilities. (This is partly implied by the theory but not fully evaluated.) \n\n5- Many high-stakes domains require retrieval/attribution. It would help to discuss compatibility with retrieval-augmented pipelines and very long contexts where claim segmentation and score independence may be more brittle. (Background mentions database-heavy approaches; MACI’s stance could be clarified.)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "5RXgFVg5lY", "reviewer": "Reviewer_fF7o", "summary": "The paper proposes a conformal inference for the distribution-free uncertainty quantification. Overall, the paper is well-written and well-positioned. Two major contributions come to my attention: 1) the conformal inference is used to select the factual LLM-generated claims with group-conditional guarantees; 2) multi-LLM ensemble is used to get a preciser score estimator $\\hat{p}$ with higher retention rate. The proposed method is evaluated on three datasets for empirical benchmark.", "questions": "None", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Typically for conformal inference, the coverage guarantee is expected. The authors have shown the (marginal and group-conditional) coverage guarantees in Theorem 1 and Theorem 2. Also, authors provided Theorem 3, in which the improved accuracy in the estimated score $\\hat{p}$ will lead to more efficient retention. The motivation to multi-LLM ensemble is intuitive on top of Theorem 3. The flow of the paper is clear and reasonable. The empirical results align with the expectations.", "weaknesses": "A shortlist of weaknesses that might be worthy to address\n\n1. The review of the related works on conformal inference seems not sufficient in my view. The authors do not mention too many papers addressing the group-conditional coverage. A incomplete list is\n* Conditional validity of inductive conformal predictors\n* [Neurips 2024] Class-conditional conformal prediction with many classes\n* [ICML 2025] Bridging Fairness and Efficiency in Conformal Inference\n\n2. Second, I am concerned about the conditional independence assumption in Definition 1. Since the claims are generated by LLM from **the same one** prompt, it does not intuitive to assume the factuality of each claim is independent. Did authors try to model the probability of claims being true jointly given one prompt? I understand it can break the way to obtain the calibration threshold for the conformal inference. But it might be a good sensitivity check, that the modeling the joint probability of the claims being true is similar to modeling them independently.\n\n3. Third, I am confused by the dataset used in Multi-LLM ensemble. In specific, what is the M here. From my understanding, the prompt is repeated M times. But how can you make sure that the claim $c$ is the same in the repeated prompts to get the ensemble scores?\nDo you consider Some of the atomic claims are considered the same?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)