Trained on Tokens, Calibrated on Concepts: The Emergence of Semantic Calibration in LLMs
Authors: Preetum Nakkiran
Arwen Bradley
Adam Golinski
Eugene Ndiaye
Michael Kirchhof
Sinead Williamson
Venue: ICLR 2026 Poster (2026)
Keywords: calibration
LLM
semantic
uncertainty
theory
Primary area: foundation or frontier models, including LLMs
TL;DR: We show that LLMs can be semantically calibrated, and we develop theory for when and why.

Abstract: Large Language Models (LLMs) often lack meaningful confidence estimates for the semantic content of their outputs. While base LLMs are known to exhibit next-token calibration, it remains unclear whether they can assess confidence in the actual meaning of their responses beyond the token level. We find that, when using a certain sampling-based notion of semantic calibration, base LLMs are remarkably well-calibrated: they can meaningfully assess confidence in various open-ended question-answering tasks, despite training only on next-token prediction. To formalize this phenomenon, we introduce "$B$-calibration," a notion of calibration parameterized by the choice of equivalence classes. Our main theoretical contribution establishes a mechanism for why semantic calibration emerges in base LLMs, leveraging a recent connection between calibration and local loss optimality. This theoretical mechanism leads to a testable prediction: base LLMs will be semantically calibrated when they can easily predict their own distribution over semantic answer classes before generating a response. We state three implications of this prediction, which we validate through experiments: (1) Base LLMs are semantically calibrated across question-answering tasks, (2) instruction-tuning procedures systematically break this calibration, and (3) chain-of-thought reasoning breaks calibration (intuitively because models cannot predict their final answers before completing their generation). To our knowledge, our work provides the first principled explanation of when and why semantic calibration emerges in LLMs.

Published as a conference paper at ICLR 2026
TRAINED ON TOKENS, CALIBRATED ON CONCEPTS: THE EMERGENCE OF SEMANTIC CALIBRATION IN LLMS
Preetum Nakkiran Apple
Arwen Bradley Apple
Adam Golinski Apple
Eugene Ndiaye Apple
Michael Kirchhof Apple
Sinead Williamson Apple
ABSTRACT
Large Language Models (LLMs) often lack meaningful confidence estimates for their outputs. While base LLMs are known to exhibit next-token calibration, it re- mains unclear whether they can assess confidence in the actual meaning of their re- sponses beyond the token level. We find that, when using a certain sampling based notion of semantic calibration, base LLMs are remarkably well-calibrated: they can meaningfully assess confidence in various open-ended question-answering tasks, despite being trained on only next-token prediction. To formalize this phe- nomenon, introduce " "B-calibration," notion of calibration parameterized by the choice of equivalence classes, Our main theoretical contribution establishes mechanism for why semantic calibration emerges in base LLMs, leveraging a recent connection between calibration and local loss optimality. This theoretical mechanism leads to testable prediction: base LLMs will be semantically cali- brated when they can easily predict their own distribution over semantic answer classes before generating response. We state three implications of this predie- tion, which we validate through experiments: (1) Base LLMs are semantically calibrated across question-ansswrring tasks, (2) instruction-tuni ceannig procedures sys- tematically break this calibration, and (3) of-thoughtt reasoning breaks cali- bration (intuitively because models cannot predict their final answers before com- pleting their generation). To our knowledge, our work provides the first principled explanation of when and why semantic calibration emerges LLMs.
INTRODUCTION
As Large Language Models (LLMs) become increasingly capable. it is important to understand the nature and extent their uncertainty. Addressing this is an active research question: can we extract a meaningful notion of confidence in an LLM's answers? This question is scientifically interesting even aside from applications:
a way of asking, informally, do LLMs "know what they don't
know"? (Kadavath al. 2022)
In the classification literature, well-understood criterion for uncertainty quantification is calibra- tion: do the predicted probabilities reflect empirical frequencies? For example, if an image classifier set of inputs, then it should be correct 80% of those predictions. To apply
80% confident this definition to LLMs, one approach is to treat the LLM as a classifier that predicts the next-token, given all previous tokens. There strong empirical and theoretical evidence that base LLMs,, which are only pre-trained with the maximum likelihood loss, are typically mext-token-calibratte (OpenAl 2023 Zhang et al.) 2024 Desai & Durrett 2020). Next-token calibration is meaningful notion of calibration in certain settings like True/False or multiple choice questions, where single token en- capsulates the entire response (Kadavath et al. 2022, Plaut al., 2025). However, when the model produces long-form answers to open-ended questions, we desire a notion of uncertainty with respect to the semantic meaning of the response, which next-token calibration does not directly capture.
Prior works have proposed a variety of notions semantic confidence for long-form text, including verbalized measures and sampling-basee measures (e.g. semantic entropy of Farquhar et al. (224 See Vashurin et al. (2025) for comprehensive overview. However, from the empirical data it is unclear whether LLMs naturally calibrated with respect any of these notions of confidence,

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Sample answers
Collapee semantic closses
orgmax prediction
calibration
What is the oxptor of Pronce?
1.0
smODE::0
00,03
genantte Accurao
00.8
1 sample
The copitil in Rork
1. Pora (roN)
Somoriic Prodictior:
Paris
x
6.0
LLM
make
Rome (20%)
Dig ToOO
70%
6.4
Borlin (103)
Semandi Accurocy KNT
8.02
NMM Eado
0.00
0.2
0.4
0.6
0.8 1.0
Semantic Conficence
LLM-induced semontic cloesif
Semantic clase/feation
Semantic colibration
Figure 1: Semantic calibration refers to calibration of an LLM-induced semantic classifier (dashed box): the classifier induced by post postessocsssing LLM outputs with a given semantic collapsing func- tion. which we refer to B throughout. To measure semantic confidence calibration: for given question, sample multiple temperature T generations, and extract semantic answers by applying the collapsing function B (e.g- strong LLM prompted to extract one-word answers). This yields an empirical distribution over semantic classes (above: Paris, Rome, Berlin), which we treat as the classifier output. This classifier output defines a semantic prediction (=argmax probability) and semantic confidence (=max probability). Semantic confidence calibration means, over all questions, these predictions are sefidence-calibrate the standard elassification sense.
without being specifically trained for calibration (Kadvvath et al. 2022; Yin el al. 2023 Band et al, 2024 Kapoor et al.] 2024. Yoon et al., 2025 Mei al. (002)). Empirically, calibration may depend on many factors: the test distribution (math, trivia etc.), the post-training procedure (RLHF, DPO, RLVR, none, etc.), the inference-time procedure (few-shot examples, chain-of-thought (CoT), best- of-K, etc.), the model size, the model architecture, the sampling temperature, etc. All of these factors have been posited to affect calibration, for reasons that are not yet well understood (Kadavath et al.
2022 OpenAl 2023 Leng et al. 2025 Xiao et al. 2025 Zhang et al. 2024 Wang al. (025)).
A priori, there is reason expect emergence of any of these forms of semantic calibration as a product of standard pre-training with the maximum likelihood loss. In this work, we show both theoretically and empirically that a particular type of sampling-basse semantic calibration actually does emerge for a large class of LLMs. Our definition is closely related to semantic entropy (Far- definitions of confidence in Wang al. (023)) 3)
quhar et al. 2024), well as the sampling-basee
Wei et al. (2024), and Lamb et al. (2025). At a bigh level, our approach involves treating the LLM as standard multi-class classifier (by collapsing outputs with the same semantic meaning). and then applying recent theoretical results from the literature on classifier calibration (Gopalan et al. 2024 Blasiok et al. 2023] 2024). Fig. Iillustratss the overall setup, described detail the next section. work is the first to propose theoretically plausible mechanism for semantic
To our knowledge, calibration in LLMs, and we validate the predictions of this theory empirically.
Summary of Contributions. We empirically show that LLMs are semantically-caliee callibbred surpris- ingly often, for certain settings and types of questions. We offer candidate theoretical mechanism to explain how this calibration emerges from standard LLM training (that does not explicitly encour- age it), and discuss under which settings and for which questions we expect it. The basic prediction of our theory is that semantic calibration is likely to hold when (1) the model is a base LLM, and (2) the model is able to directly predict the probability that its answer will land a given semantic class, even before it has started to generate it. Intuitively, in order to be semantically calibrated, the model must "know" how likely it to generate a "Paris"-type answer, before has determined exactly how it will phrase its answer. This theoretical insight leads to number of practical predictions about which models and tasks should be semantically calibrated, which we then test experimentally.
Organization. We start by formally defining the notions of calibration we consider in Sec. 2 In Sec. introduce our proposed theoretical mechanism for emergent calibration, and state
formal results. In Sec. 4 we apply the theory to make three concrete predictions about when LLMs semantically calibrated, and in Sec. we experimentally test these predictions.
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TriviQQA Mistrall
TrivisQAA Qwen?2 (senteecce procept)
SimpleQA LJama 70B (sentence procppt)
GSMSK Qwen2.5 (concise prompt) emtict: d,px?
GSMAX
OpenMatadnsttutt Qwen? Math 72B (concise prompt)
27t- pt (concise prompt) emtii:
(sentence prompt)
Semsnic Aconeyy
smbiiit: 0:335
amticc::
emtic: BuI32
entii::
0.3
Semontic Confidence
Semantic Confitenze
Semantic Confiiencee
Sensamic Exnfidence
Semantic Comfdencee
Semantic Confidence
Reliability diagrams demonstrating semantic confidence-caliration of base (pretrained- only) LLMs across various combinations of datasets, models, and prompts. Calibration error mea- sured with SmoothECE (smECE), average confidence and accuracy marked with a black cross, and density of semantic confidences shown in gray histogram; details in Appendix D.I
Figure 2:
2 SEMANTIC CALIBRATION AND B-CALIBRATION
We now informally describe our framework; formal definitions follow in Sec. 2.1 The core of our approach is collapsing function B which post-procesee the LLM's raw text outputs, mapping each generation to one of a finite set of classes. Of particular interest are semantic collapsing function which we focus on now As illustrated in Fig. semantic collapsing function implicitly transforms the LLM into an ILM-induced semantic classifier: For a given question, the classifier's output is distribution semantic classes, whose probabilities multiple generations from the LLM and applying B to each. From this distribution, we define the semantic confidence the probability of the most-likely semantic class, and the semantic accuracy as whether the most-likely semantic class matches the ground truth's semantic class. The LLM semantically confidence-calibrated if these confidences and accuracies dataset-e.g.. among questions with 70% semantic confidence, the average semantic accuracy is also 70%. This definition coincides with Lamb al. (2025)'s 's definition of "Empirical Semantic Confidence" when applied to the full distribution. For example, Fig. 2 measures calibration of several models using this approach (full experimental details in Sec.
be empirically estimated by sampling
calibrated across
2.1 NOTATION AND SETUP
We now establish the notation used throughout the paper. We assume that our semantic collapsing function outputs at most K E N classes, which we represent by the set of indices [K] = [I, (I...., K). We allow K to be arbitrarily large. We identify these classes with the set of standard basis vectors EK C R.K. The set of probability distributions over a finite set S is denoted A(S). For convenience, use the shorthand Ад [[[]] for the probability simplex over the K classes.
Language Model and Data. Let V be the model's vocabulary. We assume throughout that the evaluation data comes from a ground-truth distribution 'D over prompt-completion pairs (x. y) E V x V where N a maximum generation length. An LLM is function pe V A that maps a prompt æ to a distribution over output strings. We use conventional notation: Pa = ( x the entire distribution over sequences for a given prompt, so we can denote p=.(2) = (o x)) as the probability of a specific sequence 2. The conditional probability of the next token is denoted pa(2; x. Zgz). To distinguish model outputs from the dataset, we use 2 V for generated strings and y Y for ground-truth completions from D.
The core of our framework is the collapsing function B V' DN + [K] that classifies a given prompt-completion pair into one of K categories. In our theory, B is allowed to be arbitrary, but often will think of it "semantic collapsing" function, grouping many different strings into a single semantic class, as visualized in Fig. [ An example of such a function is described in App. D For convenience, write (z) B 2) emphasize its role classifier for outputs given fixed prompt æ.
Collapsing function.
'To implement this function, use strong auxiliary LLM prompted to extract canonical short answer from long-form string. Details in App.
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2.2 CONFIDENCE CALIBRATION
We first recall the relevant definitions of calibration in the multi-class setting (for unified treatment, see Gopalan et al. (2024, Section 2)). In the K K-class setting, classifiers ouiput values с E Ax and the true labels take values y EK (one-hot encodings). Calibration is a property defined for any joint distribution of prediction-label pairs (c,w) E A x εx. regardless of whether it was generated by a classifier. We will focus primarily on confudence calibration, which only considers the proba- bility assigned to the predicted class; however, we provide analogous results for full calibration in App. E.3 The following definition is standard:
Definition manfidence-calibration A distribution over prediction-output pairs (c,u)= EAXX
EK is perfectly confidence-calibrate if
[ cs+ cx+] = 0 where k* < argmax Cj. (+.y)~
E
kE[K]
The definition depends crucially on the distribution D. In this work we take 'D to be the evaluation distribution of interest (e.g. TriviaQA, GSM8K, etc), unless otherwise specified.
From Language Model to Categorical Predictor For a given prompt x, we obtain a distribution over K categories by pushing- forrnn--fowarr"" the LLM's output distribution po(: [x) via the function B,- Specifically, the distribution over categories Te == tp Bu (- 2) assigns to each category k E [K] the sum of probabilities of all strings that B₂ maps to that category:
(Batp=)(k) Pr [Bx()) = k]
Σ (z)==
po(z x)).
(I)
s~po(-la)
This process transforms the original prompt-answer pair (z. y) from the dataset 'D into pair suitable for calibration analysis: (BrtPo, (y)), where Brip= is the model's predicted distribution over categories and (y) is the ground-truth category. Now, we say that the model pe is B-confidence- calibrated the induced distribution over (Brtpz. (v)) is confidence-calibrate confidence-calibratton means if the generated and ground- answers are both post-processed by B, then the resulting K K-way- wlaysslassifier is confidence-cen callbrraee
That is, B-
The model pe is B-comfidence-calibraeed with respect to distribution D if the induced distribution over pairs (B(t),,B2yy)) E AK [K] is perfectly confidence-calibeee calibrated (per Definition (7)
Definition 2 (B-comfidence--caliraatiee
Our entire framework is well-defined for any arbitrary computable function B, though we usually choose to be a semantic-collapsing function. In general, an LLM might be B-calibrated for some choices of B, but not others one goal of our theory is to understand why.
3 THEORETICAL MECHANISM
Our conjectured mechanism for emergent calibration builds on the work of Blasiok et al. (2023 2024] which connects the statistical property of calibration to the optimization property of local loss optimality. The core intuition is that a miscalibrated model implies the existence of a "simple" per- turbation to the model that would reduce its test loss. For example, suppose LLM is semantically miscalibrated in the following way: on questions where it is 70% semantically-coniide average only 60% accurate. Then, an obvious way to improve the LLM's test loss is: whenever the original LLM was 70% semantically confident, it should downweight the probability mass it places on all strings its majority semantic class, thereby decreasing its confidence. We argue that base ILMss, trained to minimize cross-entropy loss, should not leave such "easy wins" on the table, and thus should well-calibrated.
it is on
This example reveals some of the subtlety in the LLM setting: unlike standard classifiers, the LLM does not explicitly output its [semantic] confidences. Thus to implement such a loss-improving perturbation during pretraining, the LLM must implicitly "know" its semantic confidence for a
We use "I" as the standard notation for the mathematical pushforward of a measure by function. E.g. for function B and distribution p. the notation Btp denotes the distribution of )x)))=xp
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B
D
LLM "knows" own distribution - B
LLM "knows" its intermediate B-confidences
Thm
LLM can easily Iearn all Wa Va-perturbations
for base LLM
LLM is Wg-locally- luss-optimal
Thm
LLM is B-confidence- calibrated
Claim 10 experimentally testable)
Conjectured Mechanism for Semantic Calibration. Implications have varying levels of support: the solid blue arrow ( (+*) has formal proof: the dashed blue arrows (==*) have proof's of "morally similar" (but weaker) implications. Claim I0 encompassse the full chain of implications, and has experimental support.
Figure 3:
given question even before generating its answer-in order know what type of upweighting/- downweighting of answer strings is required. In settings where the LLM does not "know" its semantic confidences (informally). we may expect poor calibration will see this aspect both our theory and experiments. A technical overview of our results is in Sec. 3.1 followed by formal
theorem statements Sec. 3.2 and See. 3.3
All proof' are deferred App.
3.1 CONJECTURED MECHANISM: OVERVIEW
Fig. illustrates our conjectured mechanism. There are three main steps in the conjecture, with different degrees of evidence for each step. For the first step, we have a fully rigorous proof. For the other two steps, we have partial theoretical evidence: proofs weaker claims which are "morally similar" to our conjectured claim. Finally, we have experimental evidence for our overall conjecture (presented later Sec. 5). We outline each step below, following Fig- Fig-3Jire from right-to-left.
(C) (D): The first step of our argument, described in more detail in Sec. 3.2 is a general equivalence between calibration and local loss optimality. We say LLM is locally-loss-opiine its test loss cannot be improved by post-processing its output distribution via any function some given set of perturbation functions (formally, Definition 5 For particular choice of perturbation functions, this turns out to exactly characterize B-calibration. We prove Thm. 6 that for any choice of collapsing function B, E-coofiddenee-caliiraton is equivalent local-lsssoptiimality with respect to a corresponding family of perturbations, denoted WB. Roughly speaking, this family WB consists of perturbations like our earlier example: "If the B-class collss-confideece was 70%. then downweight the probability of generating all strings in the majority B-class." Overall, Thm. tells we can equivalently
if
us that if we want understand when LLMs are R-comfidence--caiibuutt understand which types of perturbations L.MSs are loss-optimal with respect to.
(B) - (C): At this point, invoke informal assumption proposed in Blasiok et al. 2023) and likely folklore much earlier: we assume that base LLMs are nearly locally-lossootimal on their pretraining distribution, w.r.t any perturbation that is "casy" for the LLM to learn. We state this assumption more precisely as Claim 16 in the Appendix. Blasiok et al. (2024) Theorem 1.2) offers partial theoretical justification for this claim, by proving that, intuitively. if small models represent a set of perturbations, then ERM over a family of slightly larger models yields local loss optimality w.r.t. these perturbations; this serves a approximate representational analog of the desired assumption. The intuition is that pretraining should not leave any easy wins on the table: simple (i.e. casily-learnable perturbation could have improved the test loss, the LLM would have learned it during training. We agree with Blasiok et al. (023) that this assumption is plausible, because is fairly weak; it does not require that models are globally optimal in any sense.
(A) - (B): From the above two points, we can conclude that base LLM will be B-confidence- calibrated if the corresponding perturbation family Wp is simple for the LLM to learn. But when is WB simple to learn? This is subtle because the perturbations WB are defined over the sequence-leve probability distribution but LLMs must implement perturbations by modifying next-token probabil- ities. For example, in order to implement a perturbation such as "increase the probability of ulti- mately generating Paris-type answer". the model must begin by deciding how to adjust its first
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token probabilities in order to achieve this. We bridge this gap in Thm. 9. by proving represen- tational analogue of the implication (A) -- * (B) of Fig. 3 we show that if the LLM "knows" its own induced distribution over B-classes at each intermediate point during generation (even the very beginning), then it can implement the associated family of perturbations Wa in a "simple" way. (Notably, this does require the model to know the correct answer's B-class, only that of its own generation.) Formally, we prove a circuit-complexity version of this: the next-token probabilities of the perturbed model can be computed with shallow circuit given oracle access the interme- diate B-confidence functions, and the original next-token probabilities. In practice, we will focus primarily on the model's ss ability to predict its B-distribution the beginning of generation (before outputting the first token). Intuitively, this is more likely to hold for straightforward questions such as "What is the capital of France?" than questions requiring many steps of reasoning- we will say more about this in the following section.
Putting everything together. the overall mechanism predicts that base LLM will be B-confidence- calibrated if the L.MM "knows" the distribution of B-classes of its own answers (i.e. if it can be LoRA-adapted immediately output this B-class distribution, given only the question). When B is a semantic collapsing function, this theory naturally suggests a number of practical predictions about which models and tasks should be semantically calibrated, which we explore and test experimentally in Sec. []] The next several sections give the formal theory supporting the mechanism we have just outlined.
3.2 B-CALIBRATION AND LOCAL LOSS OPTIMALITY
We now setup and establish the equivalence between calibration and local loss optimality (Thm. 6 We use the sequence-level cross-entropy loss, which decomposes into the standard autoregressive
next-token log-loss: B(ra))=(((() p=)] E( Eie(N] log Pe(y: V<r,,) We will use the following notion of perturbing a distribution, known as exponential tilting (Cover & Thomas 1999 Chapter 11), which turns out to be the appropriate notion for the cross-entropy loss.
Definition (Perturbation operator). Given distribution f E A(V*) over sequences, and signed measure w E RIV"1, define the perturbed distribution (f * w) E A(V*) as:
Vz 43 VNV
(f+ == softmax (w(z) - log [(])).
Next we define a specific class of perturbations which characterize B-comfidence-calibrttion In- tuitively, these perturbations modify the probability of the most-likely B-class, by modifying the probability of each string x according to (only) its B-class (2). The formal definition some- what technical, based the language of weighted calibration developed in Gopalan et al. (2024).
Definition 4 (Semantic Perturbation Function Classes). Given an arbitrary collapsing function (x)z) E [K], we define the class WB of perturbation functions w(x, E RII* follows. These functions w(2.pz) generate a perturbttion vector based on the prompt x and the model's s predictive distribution Pa-
- WB = {w| 3r [0, 1-1, e V w(x p=)(2] (T=lk²)) 1(B
where Te = Butpas and k" + argmax [k).
kE[K]
Finally, we define local loss optimality with respect to arbitrary perturbation class W.
Definition 5 (W-local loss optimality). We say that po is W-locally loss-optimal if Yu E W: (2.0)DD
E
[C(y.P))]
E IE [C(y. Pa. * w.)) where wz = uo(x Px.) Pa = po( x))
(2.0)>
We now state the main result of this section (see App. E for all proofs).
Theorem 6 (Equivalence of Calibration and Local Loss Optimality). Given a model pe. collapsing function B, and a distribution D, the model po is perfectly B-comfidence-caliben if po is Wn-locally loss-optimal on D.
on 'D if and only
Remark 7. Thm. 6 states a simplified version of our full theoretical results, for the sake of clarity. Thm. 0 only characterizes perfect confidence-calibrate but it is possible to show a much more

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robust equivalence: it turns out that model is "close to" B-calibrated if and only if it is "close to" " locally- poss--opimaal in the appropriate sense. We state and prove this generalized version as Thm. 36 in App. E where also generalize allow any arbitrary proper-loss l, and any notion of weighted-calibattee (including canonical calibration and confidence calibration).
3.3 SPECIALIZING AUTOREGRESSIVE MODELS
remains to understand when the perturbation class WB easy for an LLM to learn (box (B) in Fig. B). Although we cannot currently fully answer this question, we can gain insight by studying a simpler question representation: when is a perturbation class WB "easy" for the LLM to rep- resent (for example,
small circuit top of the original LLM)? The main remaining challenge is that perturbations are defined on probability distributions over sequences (Definition 3). whereas autoregressive models must implement perturbations token-by-token. Fortunately, for perturbations in WB. it turns out the perturbed next-token distribution can be expressed as a simple re-weighting of the LLM'' original next-token distribution. This re-weighting is governed by set of scalar- valued functions {9:}. defined below. We call these functions "intermediate B-confidences"" be- cause gi (zgz x) is the probability mass the model places its most-likely B-class, given both the question æ and the response prefix zgi generated so far. Thus, the difficulty of representing the sequence-leve perturbation reduces to the difficulty of representing these intermediate confidences values during generation.
Definition& 8 (Intermediate B-Confidences). For a given function B : V x VN - [K] and model pe. we define the intermediate B-confidences as the sealar-valued functions {g. (0.1.....
g((26()) = Pr
[2 (z) = k*] where k* + argmax(BBatp.)lk s-po)
RE(K]
We will informally say that the LLM "knows" its intermediate B-confidences if the functions gi have a simple representation (e.g. each is computable by small circuit on top of the LLM). In that case, we show Thm. that for any perturbation w WB, the perturbed model pe w has an htw-sliightyymmr-commpp
representation than the original model Specifically, the perturbed model can be computed by composing circuit C with the functions Jt- Explicit formulas are provided in App. E.6.3.
Theorem 9. For all functions B : V x VN + [K] and all perturbations w E WB, there exists small circui B C.. such that for all models pe : V - A(V~), ), all x E V", , V all i [N), and with P. == P- 2). Ws == w(x, Pa), the perturbed model x + Pa w2 satisfies
(p2 * wz)(22 2<5) & Cw (((=)s;))) (x))
(3)
where the constant of proportionallt is independent of zi, a := Pa (2: zez) is the original next- token probabilities, and go, It are the intermediate B-confilences of Desinition
Putting all the theory together, the message is: if the LLM "knows" its intermediate B-confidences, then perturbations Wg are easy to implement, and we should expect emergent B- B-calibration.
EXPERIMENTAL PREDICTIONS: WHEN ARE LLMS CALIBRATED?
Our main empirical question is: Under what conditions and for which functions B should we expect
a pretrained LLM to be
The theory of the previous section suggests an answer: we should expect emergent B-calibration when the autoregressive B B-confidences (Definition are easy for the LLM learn. We can simplify this into an experimentally (i.e. does it "know") the distribution Ba of its answers post-processed by B? Practically, we can operationalize "casy for the LLM to predict" by training small LoRA top of the base LLM to predict the B-class of the answer.
heuristic: for a given question x, is it easy for the LLM to predict
Claim 10 (Main, heuristie). Let (a,y) ~ be a distribution on question-answer pairs, let B V x VN + [ be a collapsing function, and let po(z x) be an autoregressive language model
"Specifically, an arithmetic circuit of constant depth and O(K) width.

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trained on D with cross-entropy loss. Then, po will be B-confidence-catiraace on 'D if the function
G:V- + AK defined as
G:I Bxty= is "aasy learn" for the LLM (e.g. with a LoRA adapter)
In words: the LLM should be able to accurately estimate the distribution over semantic labels (z) under its own generative process, given the question æ.
Finally, we specialize Claim 10 to the practical case of semantic calibration- that is, we let B be function that collapses long-form answers into semantic equivalence classes, yielding the following: Corollary 11 (Main, heuristic). LLMs trained autoregressively with cross-entropy loss will be mantically calibrated on in-distribution data if: the model can easily predict its own output distri- bution over semantic answers, given only the question.
Corollary leads to the following predictions, which we verify experimentally in Sec. 5
Prediction 1: Semantic calibration emerges from standard pretraining. When B is semantic-
collapsing function, we expect it to be easy-to-predict in many settings: Claim 10 only requires that the LLM intuitively "knows" what types of semantic-answers is likely to output for given question. Thus, we should expect emergent semantic calibration for large class of pretrained LLMs, a remarkable fact not previously understood.
Prediction 2: Instruction-tuningg can break calibration. We only theoretically predict calibration in models trained autoregressively with cross-entropy loss, that is, standard pretraining or SFT. (Cross-entropy loss is required to connect calibration with local-lsssoptimality in Thm. 6 We have no reason to expect calibration in models trained in other ways, including Instruct models post-trained with RLHE, DPO, or RLVR - although our theory does not preclude it.
Prediction 3: Chain-of-though reasoning (CoT) can break calibration. To satisfy the conditions of our theory, the distribution over semantic classes must be easy for the model estimate, even be- fore generating the first token. In hard CoT setting such as math problem-solving, the model usually does not know what its answer will be until it has finished "thinking". Therefore, CoT is expected to break our mechanism for calibration. Notably, what makes CoT powerful (allowing the model leverage more compute produce a better answer than it could have produced immediately) exactly what makes our mechanism of calibration fail.
EXPERIMENTS
In this section, we experimentally test the predictions of our theory on real models and datasets. All of our experiments include 5-shot examples in the prompt, and use temperature T = sam- pling. We compare three different prompts, designed to elicit different styles of responses from the model: "concise" (answer in single word/phrase), "sentence" (answer a complete sentence), and "chain-of-thought (cot)". The few-shot examples are formatted in the desired style (e.g. for the "sentence" type, the few-shot examples have complete sentence answers). To measure calibration error, we use the SmoothECE metric introduced by Blasiok & Nakkiran 2024). For lack of space, full experimental details are in App.
5.1 EXPERIMENTAL RESULTS
We evaluate semantic calibration of Qwen, Gemini, Mistral, and Llama-family models, of varying sizes from 0.5B to 72B, for base and instruct variants, using each of the 3 response styles, on open-ended question-answer datasets: GSM8K (Cobbe et al. (021), Mpennathinstrruct22 (Toshni- wal et al. 2025). TriviaQA (Joshi et al. 2017). SimpleQA (Wei et al. 2024), MATH500 (Lightman ct al. 2023). and TruthfulQA (Lin al. 2022b). This yields over 650 evaluation experiments, which we compile into Fig. 4 by overlaying their reliability diagrams. The box-plots in the bottom row of Fig. 4 show the distribution of calibration errors in aggregate for each dataset and configuration. We will use this condensed figure to discuss our experimental predictions. We expect our theory to apply on all datasets except, notably, TruthfulQA: This dataset contains common human miscon- ceptions, and thus violates our in-distribution assumptions (see Remark 12] The full list of models is in App. D.3 and disaggregated results are reported in App.
Prediction 1: Semantic calibration emerges from standard pretraining, Our theory predicts that base models, in non-CoT settings, should be semantically calibrated. The top row of Fig.
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GSM8K
OpenMathinstruct
Math500
TriviaQA
SimpleQA
TruthfulQA
beae concir
Semauni
1.
Accunees
baac iniieci stentie Inatructict
Confidence
Confidence
Confidence
Confideace
Confideace
bare actienec
++
basrnce
Calibration Erooe
Calibration Erroe
Calloration Eroor
Calloration Error
Calloration Eroor
Calloration Eroor
Semantic Calibration of LLMs. We evaluate Qwen, Gemini, Mistral, and Llama- family models, with 6 configurations for each model: (model-variant) response-style) E (Base, Instruct] x [Concise, Sentence, CoT}. First row (predicted calibrated): Reliabil- ity diagrams of all configurations predicted be confidence-calibrated according base models with concise or sentence response types. TruthfulQA, a dataset of common miscon- ceptions, is the exception: it violates the in-distribution assumptions of our theory, and is poorly calibrated. Second row (not predicted calibrated): Configurations which need not be calibrated according our theory: post-trained instruct models with any response type: concise, sentence, tfain-of-thoughee and base models with chain-of- of-in-of-thouuht. Third row: Box plots summarizing the distribution of calibration errors for each of the 6 configurations. Only the first two configurations (base-concise and base-sentence) are reliably well-calibrated, as predicted by our theory. Note, we only consider chain-of-then of-hhuhf-t for the math datasets. TruthfulQA reference answers are available only in the sentence form, only report results for sentence response-style
Figure 4:
theory:
shows reliability diagrams for such models we evaluated (configurations base-coneise and base- sentence), and we observe nearly all of these experiments are well-calibrated. Notably, semantic calibration does not depend significantly on model size for base models: even small models (S 1B) remarkably calibrated; see App. C for more in-depth look at this aspect. Models are also well-calibrated regardless of the response style ("sentence" vs. "concise"), supporting our theory that semantic calibration depends not the specific phrasing of the answer, but rather whether the model "knows" its semantic class distribution before starting to generate.
Prediction 2: Post-training can break calibration. The middle row of Fig. includes reliability diagrams for instruct post-trained models, for all three response types. Many of these settings are miscalibrated, typically overcontident (i.c. curve below the diagonal), as expected from a reward- maximizing RL objective. Fig. 5 takes a closer look at the effect of different types of instruction- tuning on calibration. We compare three models from the same lineage: a base model (Mistral-7B- v0.1), version of post- post-trainee via instruction supervised finetuning (SFT, zephyr- 7b-s----------fll) and version post-trained via both SFT and Direct Preference Optimization (DPO, zephyr-7b-dpo-full) Rafailov et al.) 2024). The DPO model (not trained with a proper loss) is significantly miscalibrated, while the SFT-only model and the base model (both trained with proper losses) are better calibrated.
break calibration. The middle both base and instruct models, which are poorly calibrated in the math settings (GSM8K, OpenMath- Instruct, MATH500). Base-cot responses underconfident (above the diagonal), while instruct-cot are underconfident for GSMSK, but overconfident for OpenMathlnstrrur,, see Fig. 9 Notably, this miscalibration is inherent to math: base models are calibrated when asked to provide the answer immediately (base-conciss and sese-sentence), but become miscalibrated when allowed to reason.
Prediction 3: CoT reasoning
of Fig. 4 shows CoT with

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GSM8K
OpenMathlnstruet
Math500
TriviaQA
SimpleQA
TruthfulQA
Calibration Error
0.3
Model
Mistral-7B-vO.1
0.2
xephye- zephyr-Zhodpo-ful
0.1
0.0
Figure 5: Calibration error for three models based on Mistral-7B-vO.1 pretrained only, instruction-
snpervised-finetunee and IPO--fietunne.. Here, "sentence" response style, see Fig. for others.
Quantitative Learnability Probe. Claim suggests an explicit experiment to predict when a base model will be B-confidence- calibrated for a given choice of B: can the model "easily learn" the function G + B.APa mapping a question æ to the distri- bution over the model's s own semantic answers for that question? We top of
Calibratio Ervor
0.2
test this by training small LoRA (Hu et al. (202)) the model, to directly generate the semantic class distribution BatPa For example, in CoT settings,
0.0
when prompted with the question this would require the LoRA to "short-circuit" the reasoning steps, and immediately generate the final answer that the model would have produced with COT. Notably, this does not require the model to produce the correct semantic answer, but just match its own generative distribution. In Fig. 6 we train rank-8 LoRAs on Qwen2.5 models of varying sizes, on GSM8K.
Figure 6: Testing Claim T0
across Qwen2.5 models.
We then compare each LoRA's KL gap optimality (x-axis) to the underlying model's calibration error (y-axis). The correlation agrees with our theory: models which can easily predict their own semantic class distribution (low KL gap) also well-calibrated. Full details in App. D.2
Discussion: Calibration in LLMs other deep networks. One may wonder why the state of calibration in LLMs seems significantly different from calibration non-LLM deep networks (e.g. image classifiers), Specifically, deep network classifiers sometimes severely overconfident, and sometimes well-calibrated, depending on the specific network (e.g. Guo et al. (2017) vs. Minderer 202T On the other hand, all base LLMs tested were well-calibrated (in non-CoT settings) there was sigmificant calibration difference between models. This difference between LLMs and classifiers due to differences in training practices: when training LLMs, practitioners monitor the test/validation loss closely, and stop training before the test loss overfits (increases). On the other hand, when training classifiers, practitioners care about the test classification error, and often continue training
test loss increases. Most trained LLMs therefore locally-los-optimme w.r.t. test loss (and thus calibrated), but trained classifiers might have high test loss (and thus be miscalibrated). This perspective the calibration of deep networks was articulated in Section 1.1 of Blasiok et al. (2023).
CONCLUSION
We find that base LLMss, despite being trained with token-level syntactic objective, are remarkably calibrated with respect to the sequence-level semantics their generations. Our central contribu- tion is principled mechanism behind this emergence, building on recent theoretical connections be- tween calibration and loss-optimality (Blasiok et al. [203 2024). This theory provides unified lens through which to understand the nuanced calibration behavior of models in practice, distinguishing settings which are calibrated from those which are not. Among limitations, we only propose one possible mechanism for calibration; is possible that other types of calibration (e.g. verbalized cal- ibration) emerge for yet-undiscovered reasons; other limitations discussed in Appendix [.. More generally, work be a step towards understanding the formal structure of L.MMs' output distribution.
ACKNOWLEDGMENTS
We are grateful for discussion and feedback from Parikshit Gopalan, Eran Malach, Aravind Gol- lakota, Omid Saremi, Madhu Advani, Etai Littwin, Josh Susskind, and Russ Webb.
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APPENDIX CONTENTS
Additional Related Works
17
B Extended Discussion and Remarks
17
B.
1 Limitations
17
2
Potential Extensions
18
3 Technical Remarks
18
Additional Experimental Results
20
D Additional Experimental Details
22
D.I Visualizing calibration: reliability diagrams
23
D.2 LoRÅ Fine-Tuning
23
D33 LLMs evaluated
25
D.4 Prompts
27
Theory
30
Quick Reference
E,2
Weighted Calibration
30
Equivalence between B B-calibration and weighted calibration
30
E.3.1
Full Calibration
31
E 3.3
Confidence Calibration
32
E,4
Equivalence between Ceighted-Calibrann and Local Loss Optimality
33
Proof of Thm.16
34
E.6
Autoregressive Settings
34
E.6.1
Weighted Calibration
34
E.6.2
B-Calibration
35
E.6.3
Proof of Thm. 9 Simple Circuit B-Contfdence--Pertubbaiion
36
E.7
Quantitative Bounds on Multi-Class Calibration and Post Dost-Processing Gap
37
E.7.1
Specialization to cross-entropy loss
38
E.8
Conformal Prediction via Weighted Calibration
40
E.8.
Conformal Prediction from Full Calibratonn
40
E.8.2
Conformal Prediction from Weighted Calibration
40
Disaggregated Reliability Diagram Results
GSM8K
42
F2
Onsstuutl
48
F3
TriviaQA
54
SimpleQA
60
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A ADDITIONAL RELATED WORKS
Recalibration Methods. A number of prior works study methods to improve the calibration of
LLMs, ranging from temperature-scaling at inference-time (e.g.. Xie et al. 2024 Shen et al. 2024) to training calibration-specifie probes that predict correctness (Mielke et 2022) training with
calibration-imppe improving regularization terms (Wang et al. 2025). Other approaches attempt to cluster
questions and predict per-cluster accuracy (Lin al., 2022a Ulmer et al. (202)) or make use of
the fact that ensembling models tends to improve calibration (Jiang et al. [023b] Hou et al. 2024).
Probabilistic approaches (such as Bayesian deep learning, or evidential deep learning) have been
found to often yield better calibration (e.g.. Li et al. 2025 Yang et al., 2024a).
Sampling-based Confidences. A number of prior works have proposed sampling-based ap- proaches to defining LLM uncertainty. Both |Wang et al.] (2023) and [Wei et al. (2024) sample multiple answers per-question, and define confidence swer. Wei et al. (2024) additionally groups answers together by string-matching, which allows for some degree of semantic equivalence. This approach was extended and popularized by the notion of semantic entropy (Farquhar et al., 2024). Semantic entropy clusters sampled answers together by semantic content, and then measures the empirical entropy of clustered answers. Recently. [Lamb] et al. 2025) define Empirical Semantic Confidence, which is essentially an empirical version of our notion of semantic confidence. Note that one distinguishing aspect of our formalism is, we param- eterize the notion of calibration by the choice of collapsing function B. This allows to develop somewhat more general theoretical insights, which are not tied to a fixed notion of semantics.
the frequency of the most-common an-
Factors which Harm LLM Calibration. Various factors have been observed in prior work to harm LLM calibration. is well-known that RLHF often harms calibration multiple-choice QA settings (Kadavath et al., 2022 OpenAl 2023). Other RL post-training methods such as DPO have also been observed to harm cali bration (Leng et al. 2025 Xiao et al., (025). Some studies have also found chain-of -ifii-offttt of thought responses to harm calibration, agreeing with our results (Zhang et al. (024). However, we warn that not all of these works use the same notion of confidence and calibration as we do, and so are not directly comparable.
Related Technical Tools from Prior Work. The connection between local-loss-optimaliy and calibration was theoretically studied in (Blasiok et al.) 2023), which proved a version of Thm. 6 for binary classifiers, and was inspiration for our theoretical results. Moving from binary classi- fiers to LLMs posed three main technical challenges. First, in binary classifiers there is essentially only one canonical notion of calibration, and so Blasiok al. (2023) only required one notion of local-loss-optimaliy. However in our LLM setting, there are many notions of calibration (parame- that
terized by functions B), and so we needed identify the "right" notion of local-loss caal--ooss--plallii is also parameterized by B. To do this we observed that B-calibration can be written as type of "weighted calibration," a notion introduced in Gopalan et al. (2024). Second, we needed to gener- alize the I-dimensional results of BBtaioo et al. (2023) to higher dimensions, to handle multi-class settings. This turned out to be a straightforward though somewhat technical generalization, using the Savage representation of proper losses (Savage 197I). Third, and most significantly: unlike classifiers, LLMs do not output their predicted probabilities explicitly. Rather, they implicitly define probability distribution via their next-token predictions. This difference between implicit and ex- plicit probability distributions required a number of conceptual adaptations to the theory of Blasiok al. 2023), which guided our definitions of the perturbation operator and W--lccalllossoptimae (Definition
B EXTENDED DISCUSSION AND REMARKS
B.I LIMITATIONS
Types of Calibration. One limitation of our paper is that we focus on a very specific type of calibration, which is essentially sampling- based notion (B-coofidence-ccllirtttoo that other types of calibration (c.g. verbalized calibration) also emerge for certain types of LLM training; we consider this possibility interesting but out-of-scope for the current work.
It is possible
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Practical Implications. Our work is primarily scientifically motivated, and so we do not fully ex- plore practical considerations or implications. For example, we do not consider the computational efficiency of our confidence measurements. This is a limitation to using such measures in practice, since computing semantic confidence requires sampling an LLM multiple times for the same ques- tion. We consider translating our scientific results into real-world improvements to be important direction for future work.
Datasets. Although we evaluate on variety of different models, we only evaluate on 6 selected datasets. We chose these datasets to cover a diversity of domains and problem difficulties, from questions about with open-ended answers, since calibration of multiple-choice datasets is already extensively studied (Kadavath et al. 2022] Zhu al. 2023). Although we do not expect our results to depend signif- icantly on the choice of dataset, it is possible that certain other datasets have different calibration behavior; this is limitation of our experiments.
to mathematical reasoning problems. Further, we chose datasets
Remark 12. Notably, there are some datasets which we would expect to behave differently, such as TruthfulQA (Lin et al.) [022b)) which is dataset containing common human misconceptions. This dataset fails to satisfy the "in-distribution" requirement of our results (e.g. Claim 70,, and so is consistent with our theory for models be miscalibrated.
B.2 POTENTIAL EXTENSIONS
The theoretical framework described here is fairly general, and extends beyond the setting of confidence-caliirratin in LLMs. Briefly, since most of our theory is stated in the language of weighted calibration (Gopalan al., 2024), it applies to any property that can be written as weighted calibration. This includes slightly stronger notions of calibration, such as top-label calibration, and also includes conformal conformal-prediction type of guarantees (more details in App. [E..T See et al. (2244) for a number of properties which can be expressed as weighted calibration, and App. E.8 for the connection to conformal prediction. Our general theoretical results appear in App. E
Intuitively, the high-level message of our results is that if a model is trained with a max-likeliinee likelihood / log-loss objective, then we should expect it satisfy weighted calibration for "simple" family of weight functions. The appropriate notion of simplicity depends on the model architecture; simple weight functions should roughly correspond to easy-to-learn perturbations to the model's output distribution. At this level of generality, we expect some version of our results to apply even for real- valued density models, such as continuous normalizing flows (e.g. Zhai al. (2025)) ), which are also trained with the log- ligliiill objective. That is, we should expect such normalizing flows to also exhibit certain (weak) types of calibration. We believe this a promising avenue for future work.
B.3 TECHNICAL REMARKS
We collect several technical remarks regarding the theory of Sec.
Remark 13 (Heuristic Simplifications). In translating the theoretical results of Sec. too the practi- cal heuristic of Claim T0 we took several steps which we describe more explicitly here. First, Thm. 9 is about ease of representation, but in Claim [[we we chose to use ease of learning. This is both more practical (since learning can be directly tested) and, we believe, more natural (since then both the premise and conclusion of Claim T0 involve the learning procedure of the LLM).
it is sufficient for the functions { of Definition g to be "easy to learn" " for the LLM, for all prefix lengths [N]. Claim 10 deviates from this in two ways. First, instead of considering all prefix lengths i, we only consider the empty prefix (i = 0) ie. the model's 's distribution given only the question. Intuitively, the prediction from the empty prefix is likely the most challenging, and practically, this simplification means that only one comple-to-implene semantic confidence function (gu), Claim considers learnability of the entire semantic distribution (B.tpz). Practically, this improves robustness of the empirical estimator, since the KL divergence can be estimated from samples. Empirically, did not find these simplifications significantly affeet the conclusions.
Now, Thm. 9 suggests that for B-confidence-caliirait
probe is required. Second, instead of considering learnability of only the
Remark 14 (Multicalibration) One detail of the theory worth discussing further is the role of the distribution D. For clarity of exposition, we described the theory as if there is only one distribution DD
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of interest, but in reality, we evaluate calibration across mulriple distributions (TriviaQA, GSM8K, etc). and we pretrain on yet another distribution. Moreover, we find that a single model can be simultaneously calibrated across many evaluation distributions.
Formally, requiring B-calibration across maltiple distributions simaltaneously can be thought of as a multi-calibratinn property [Hbbert-Johnson et al. [2018) Suppose for example that the pretraining distribution D some mixture of disjoint sub-distributions: D a D1 + ax2 + +.... Suppose we are interested in B-calibration simultaneously for distributions D₃ and D2. Then, it is possible to show a generalization of Thm. O
A model is ceaaffidence-calibrate across both D1 and D2 if and only if it is Sccally-loss-optimal on DD w.n.t an expanded class of perturbations WB.
Informally, the class of perturbations Wi is essentially the usual class WB (of Definition aug- mented by indicator functions 1 E Dit, 1x E D2) for membership in each sub- dib-iistributton.
We will not get into the technical details, but using this version of Thm. C it is possible to carry out the remaining steps of the argument from Sec. Band Fig. 3 Applying the same heuristics, for exam- crcofidenne-caalibaane
ple, we would conclude: an LLM will be simultaneously
on distributions D1, D2 if it is easy for the LLM to (1) estimate its own distribution on B-classes and (2) identify DD
samples as either
2D2-
The second condition is likely be satisfied in all our experiments, since all our evaluation datasets are distinct and easy to identify. Thus, the predictions of our theory remain unchanged, justifying our choice to avoid discussing multicalibration in the main body.
Remark 15 (Full calibration). At first glance, it may seem that a minor generalization of our mech- anism (Fig. 3y would also imply full B-calibration (ie.,, canonical calibration of the B-induced classifier), rather than just confidence-calibrutin. After all, Thm. Aormully generalizes to arbi- trary weight families W (see Thm. 27. including the family corresponding full B-calibration (defined as Wffull) in Definition 20 However, full B-calibration too strong a property to hold in genera So, which part of our argument Fig. breaks for full calibration? The culprit the heuristic step in Fig. 3 The weight family wiffull) relevant for full calibration is, roughly speaking, B
"too large" for the same heuristic hold.
To better understand why the heuristic fails, here is more general version of the heuristic step in Fig. 3 which we believe is plausible for arbitrary weight families W.
Claim 16 (heuristic, informal). If perturbation family W is easy-to-learn for a pretrained LLM, meaning: for all perturbations E W, the LLM ps V + A(VA) can be easily LoRA-fine- CeRA-ffinetunee -une to match the distribution of a perturbed- model G: V + A(VN),
G: P. * W2 = pol- | x)) * w(x,px
(4)
then pe will be W-loocall-looss-pptiee w.r.t. its pretraining loss.
In other words, if all perturbations in the family W can be "easily learnt," then we should expect the LLM to be loss-optimal w.r.t. W. Claim 70 is essentially a special case of this more generul claim, for the specific class WB relevant to B- B-confidence calbbraaaii.iii.innn............
If we believe Claim T6 we can see why our mechanism would apply to confidence-calibration but not to full-calibration: For confidence-calibraaion
the perturbation class WB (Definition) 4 is simple enough to be learnable, while for full calibration, the corresponding perturbation class (fulll) (Definition 20, is too large to be efficiently learnable from samples. To gain intuition for this, it helps directly compare Definition to Definition 4 From this discussion, possible to extend our results to certain types of calibration which are weaker than fallnffll--oaa full-calbrration, but stronger than confidence-calinn caaiiidenneealbaatioon.n. We leave this direction for future work.
can see it is likely
FFor example, when K (the number of B-classes) is large, full B-calibration would be computationally intractable even estimate (Gopalan al. 2024
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GSM8K
OpenMattinstruct
TriviaQA
SimpleQA
1.0
base-concise
iese-srmtene
Calibratio Eror
instruct-contise
Calhrratd
pnstruct-eeneence
0.5
hwse-cott
calllo, crcot upper bound
Calibntiion Ennr
Overonffiden
1.0
Underconhtent
Caléoratoon Enoor
0.5
0.0
0.5
1.0
0.0
0.5
1.0
9.0
0.5
1.0
0.0
0.5
1.0
Semantic Accuracy
Semontic Accurary
Semantic Accurncy
Semantic Accuracy
base-concise
base-senteacee
instruct coscise
instruct-cat
Average Overoonfsenne
Averuge Overconfidence
Average Uvereoudieacee
Average Overoondidenee
We plot Calibration Error vs. Semantic Accuracy for all models in Fig each dot represents separate model. First row (predicted calibrated): In the settings our theory applies, we see no correlation between the model capability (semantic accuracy) and the calibration error. Second row (overconfident): Configurations which we empirically observed to be mostly overconfident. The dashed line illustrates the upper bound on the calibration eттоr w.r.t. the accuracy, for maximally overconfident predictor. We see little correlation between semantic accuracy and the calibration егror beyond what is dictated by the upper bound. Third row (underconfident): We see little correlation between calibration and accuracy, except near the extreme when models approach perfect accuracy. TriviaQA and SimpleQA plots empty because there configurations. Fourth row: The distribution of average overconfidence across models, for each configuration; positive/negative values indicate over-lunderrooffidenee..
Figure 7: Effect of Scale:
underconfident
C ADDITIONAL EXPERIMENTAL RESULTS
Due to their volume, disaggregated reliability diagram results are reported separately App.
Model Scaling Effects. Here, we aim to explore the effect of model sealing (parameter count, compute, data) on calibration. Since information about training details are most often not publicly available, we use the model capability (measured with accuracy) as a proxy variable for model scale. In Fig. 7 plot calibration error (smECE) semantic accuracy. For base models without ch ooouggttt (first row), we see no correlation between model capability (semantic accuracy) and calibration emror. This is consistent with theoretical predictions, which have explicit dependency on model scale or capability. It is worth noting that prior works have observed that calibration of base models can improve with model scale for other notion of calibration: next-token
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prediction in multiple-choice question setups (Kadavath al. 2022 Zhu et al. 2023 Plaut al. 2023). We do not find such improvements for semantic calibration of base models.
For instruct models and base models with chain-of-thouggh we empirically observe that some con- figurations overconfident, while other underconfident,, and divide those configurations into separate rows in Fig. 7 The dashed line illustrates the upper bound on the calibration error w.r.. the accuracy for overconfident configurations, which is dictated by the behavior of a maximally- overconfident predictor that puts its entire probability mass on single choice. For the overconfi- dent configurations, we see little correlation between calibration error and accuracy beyond what dictated by the upper bound. For the underconfident configurations, we see also little correlation overall, except for in the high-accuracy regime: calibration error tends to decrease when models approach perfect semantic accuracy] However, it is not clear whether this is robust phenomenon.
GSMBK OpenMathhnstruutt
TrivisQA
SimpleQA
Calleution Eroor
Resgoni style
Calib Calibration aitin Etror
Resporzie style:
Calibation Esoor
Respons style
Model
Mistral-B--ol1
hepbyrr/b-shhffill
sephyys dh- Full
0.0
Figure 8: Calibration error for three models based on Mistral-7B- v0.1: pretrained-only, instruction- SFT model (zephyr-7b-sft-ful)) DPO model (zephyr-7b-dpo-ff and SimpleQA for the "cot" response style. The "cot" result for Mistral-7B-vdl for OpenMathIn- struct is missing due to the model not terminating generation within its maximum context length.
We did not evaluate TriviaQA
GSMsK
OpenVaathnstruutt
TriviaQA
SimpleQA
base concise
base-sentence
base-cot
cutract-coaccie
sentence
instruct-cat
-10 -0.3
8.0
-1.0
-0.5
0.5
1.0
-1.0
-0.5
-100
-0.5
0.01
0.5
Average Overovefidenos
Averige Coemeonfideee
Average Dverenffilnce
Average Overannfidene
Distribution of average overconfidence (or "mean non-absolute calibration error") for data in Fig. Positive values indicate overconfidence, negative values indicate underconfidence.
Figure 9:
'Notably, this is not mathematically necessary. Since semantic accuracy involves only the argmax- probability class, is possible for predictor to be perfectly semantically-accuaate while having high cali- bration
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D ADDITIONAL EXPERIMENTAL DETAILS
Datasets. We focus on open-ended question-anseee answer (QA) settings, since calibration for multiple-
choice QA is already well-studied (Kadavath al. 2022 Zhu al. 2023), and special case of
our results. We evaluate on: GSM8K (Cobbe et al. 2021 OpenMathInstruutt2 (Toshniwal et al.)
2023), TriviaQA (Joshi et al.] (017), and SimpleQA (Wei et al. 2024). from Huggingface datasets Wolf et al. 2019 Lhoest et al. 2021
Models. We evaluate on models including the Qwen, Gemini, Mistral, and Llama family, of sizes from 0.5B to 72B. The full list of models we evaluate App. D.3 We use vL..M (Kwon et al.) 2023 for inference.
Prompt format. See App. D.4 for the exact phrasing used in prompts. All of our experiments include 5-shot examples in the prompt. We use three different prompt types, designed to elicit three different styles of responses from the model: "sentence", "concise", and toaai--of-thuugh (cot)". The few-shot examples are formatted in the desired style (e.g. for the "sentence" type, the few-shot examples have complete-sentence answers). For instruct models, in addition to formatted few-shot examples, the prompt also includes explicit formatting instructions. The "concise" prompt type encourages the model respond with just the final answer (a single word, phrase, or number). The "sentence" prompt type asks the model to answer each question in complete sentence (making it likely to phrase the same semantic answer in different ways, so the B-collapsing function essential for a meaningful notion of semantic calibration). The "cot" prompt type elicits chain-of- -f-huughtth reasoning from the model; this prompt type is only used for math datasets.
These prompts are typically successful in eliciting the desired type of responses from the model. However, in some cases observed models (especially Qwen models) produce "chain-of- th-uuhhhht responses even when prompted to reply in a single word. To exclude such cases, we exclude any responses for the "concise" prompt on math datasets which are too long (heuristically, more than 15 characters before the first newline).
The semantic collapsing function. Recall, the function B is intended to collapse semantically- equivalent generations into a single class, an idea proposed by Kuhn et al. (2023). We implement the function B with two-stage procedure as follows.
The first stage is canonicalization: we extract short "canonical form" answer from the LLM's response. For "concise" and "cot" prompt types, this is done via simple string parsing (for "cot", extracting only the final answer). For the "sentence" type, we Instruct) prompted to extract a short-answer from the generation, given the question as context. The prompts used for canonicalization for math settings. We also normalize strings at this stage, converting to lower-case and stripping spaces, including math-specific normalization for domains with LaTeX outputs. Specifically, we use the MATH string-normalize al. (022).
strong LLM (Qwen3-14B-
in App. DD.4 Prompt 4 for non-math settings, and Prompt
from Minerva, given in Listing 1, Appendix D.I of Lewkowycz
The second stage, used only for non-math settings, is semantic clustering: we prompt an LLM judge (Qwen3-14B-o Instruct assess whether two responses to a question are semantically equivalent, 1 This is necessary for mon-math settings to handle irrelevant differences in canonical forms (c.g. "Seattle, WA" vs "Seattle"). The prompt used for semantic equivalence is Prompt 6 in App. D.4 For math settings, the second stage is unnecessary, since the first stage already outputs a number symbol that can be directly compared.
and use the output to cluster responses
Measuring calibration. We first produce an LLM-induced semantic classifier, following the ex- perimental procedure described in Sec. and illustrated in Fig. T For each dataset, we take 10K ran- dom evaluation samples (or the entire dataset for those with fewer than 10K total samples). For each question, we construct the appropriate 5-shot prompt, sample M 50 responses from the LLM at temperature 1, and then apply the semantic collapsing function (described above) to each response. The semantic confidence is defined as the empirical frequency of the plurality semantic class, and the semantic accuracy is the 0/1 indicator of whether this plurality class matches the ground-truth' semantic class. This yields, for each question, pair of (semantic-confidencce, semantic-accuracy)
TThis is slight variation of the wwy entailment method used by Farquhar al. 2024
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[0, 1] {0, 1). We then evaluate the calibration of the resulting classifier over the entire dataset of questions using SmoothECE (smECE, Blasiok & Nakkiran (2024)), paarrtticllly-pinnippie ver- sion of the Expected Calibration Error (ECE). We use the SmoothECE implementation provided by:
e/ Fiaabbbrioon
D.1 VISUALIZING CALIBRATION: RELIABILITY DIAGRAMS
We follow the guidance of Blasiok Nakkiran 2024). and visualize calibration using kernel- smoothed reliability diagrams.
Reading the Diagram. Fig. 2 gives several examples of reliability diagrams. The solid red line is the regression line, an estimate of (u()) == E[semantic accuracy | semantic confidence = c). The black cross is the point (E (E|semantic confidencel, E E/semantic aceuracy]) [0, x [o, 1], that is, the average semantic confidence and accuracy. The gray histograms at the bottom of the plot visual- ize the density of semantic confidences. We plot two overlaid histograms, one for the confidence distribution of correct predictions (i.e. the confidence of samples where cemantic-accuracy= )), and another for the confidence distribution of incorrect predictions. The width of the red regression line varies with the overall density of semantic-confidences..
Implementation Details. For reliability diagrams, we use the implementation of relp pl with minor modifications: we use a fixed kernel bandwidth = 0.05 for the regression line, and we visualize the density of confidences using histogram binning with 15 constant-width bins.
(https: :tttpss:/gitee ///gthub.. com/ apple people/mmlccae mml-maallbatiii
To compute the scalar SmoothECE (smECE) metric, we use the original implementation of relplot without modification (including its automatic choice of bandwidth).
D.2 LORA FINE-TUNING
To test Claim 10 more quantitatively, we train LoRA version of the LLM to explicitly learn the function defined Claim 10 We do this as follows. Let pe be the base model. Instantiate a rank=8 LoRA adapter (Hu al. 2022) on top of the original model ps. which denote Ps.
We train Pa behave as the "semanticallyycollaay"
version of pe- That is, when prompted with a question x; the model pa should generate a distribution on answers b which imitates the base
model's ' semantic answers B2 (2):
(b |x)
Pr sr-po(-12)
(z)2) b] (Br#p=)
(5)
Since our implementation of the collapsing function B produces string outputs (canonical answers), we can train Pa as standard autoregressive model. Explicitly:
1. For each question in the dataset x, sample the original model 50 times, and apply the collapsing function B each generation. This produces 50 samples b(r, of question x and canoer b, for each original question x, effectively expanding the original dataset size by 50 times.
2. Train Ps with the standard autoregressive objective, the prompt-completion pairs {(x, b:)} from above. That is, train po to complete prompt x with generation b,
Our training procedure is similar to the procedure used to train "P(IK)" in Kadavath et al. 2022 in that we also train on an "expanded" training set defined by base model samples. Similar to Kadavath et al. (2022), we do this mainly for convenience.
For GSMSK, we hold-out 2000 questions for evaluation, and use the remainder for training as above. We train all models on an 8xA100 node for epoch on the expanded dataset, using the SFTTrainer implementation from Huggingface TRL (von Werra al. (2000) with the following parameters in Table [ Note, we shuffle the expanded training set manually beforehand, so we do not ask the dataloader to shuffle.
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Table 1: Hyperparameters for Supervised Fine-Tuning (SFT).
Parameter
Value
Training & Hardware
num train epochs
1
per devi crain batch si ze gradient_accume atliaaiinn sts.osstepss (Effective Batch Size)
4
2
64 8 GPUs 2)
bf16
True
Optimizer & Scheduler optim
adamw.torch
learning_rate
5e-5
weight_decay
0.0
warmup ratio
0,05
(LoRA) Configuration
use-peft
True
lora.r
8
lora.alpha
16
lora_dropout
0.0
lora porattarget.modulee
all-linear
task.type
CAUSAI_IM
bias
none
Data Handling
packing
False
catalcader.shuffle
False
After training, evaluate how closely Eq. (5) holds, by estimating the KL divergence between RHS and LHS of Eq. (5). This KL measures how well our LoRA pa matches its training distribution. Conveniently, the KI. can be written the difference between the negative-log- of ps and the semantic entropy the original model po:
Gap to optimality == KL( (BrtPe) || pol. |x))
(6)
E z~D x=pp(=(a)
log Po (B(2) |x]]
H (B.HPt.)
Somantic catropy po
Eval NLL loss
This is particularly convenient because the eval log-loss is standard metric tracked during training. Note that for our purposes, it is important to compute the unnormalized log-loss (i.e., not normalized by sequence-length).
In Fig. 6 plot the KL gap of Eq. on the x-axis, and the SmoothBCE of the original model on the y-axis. We evaluate base models: Qwen2.5-(0.5BB, 1.5B, 3B, 7B, 14B). with all three response styles: concise, sentence, cot. This results 15 points plotted in Fig. 66 colored according to response style using the color scheme of Fig. 4 We observe that, consistent with Claim [0] configurations where semantic class distribution is easy-to-learn (low KI gap) also have small calibration error. The points with high KI. (and high calibration error) are the chain-of-thought experiments, as well as the small 0.5B model with the "sentence" response type.
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D.3 LLMS EVALUATED
Below, we list all models evaluated in this paper. All were obtained from HuggingFace.
Table Pretrained-only base models evaluated in this paper. Models sharing a prefix and reference are grouped.
Family Prefix
Model Suffix
Reference
gemma-2-28 gemma-2-
(Gemaa Team al., 2024
germaa 11mma--2-2777 22-27b
google/
germa 1omma----1--p
Demma-3-4b-pt.
GGmma Team al.
2025
gemma-3-12b-pt
gemma-3-27b-pa
Owen2.5-0...
Qwen2 . 5-1.5B
Qwen2. 5-3B
Qwen2 5-
Yang et al.
2024c
Qwen2.5-144
owen2.5-322
Qwen2.5-72B
Qwen/
Qwen2.5-Mathh
Yang et al. 2024b
Qwen2.5-Mathh-725
Qwen3-0. 6B Base
Bieen-1.7B-Baal
Qwen3-4B-Base
Yang et 2025
Swen3-8B-Base
Qwen3-14B-Baz Base
Mistzal-7B-vo.11
(Jiag et 2023a
Mist:ral7B--vO.3
mistralai
Mistral--Maal1-2BBBBass
(Mistral Al Team
2024b
Mixtral-8x7B-vo.1
Misstall Al Team
2023)
Llama-3.1-B8
neta-llama
Grattafiori al. 2024
Llama-3.1-
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Table Instruction models evaluated in this paper. Models shar- ing prefix and reference are grouped.
Family Prefix
Model Suffix
Reference
gemma-2-2b-
genma-2-9hh
Gemma Team et
2024
genma 12mmaa-2-- 22-22------- it
google/
genma conma-3-1b-ir genma-3-4b-it
Gemma Team et
2025
gemma-3-12b-it
gemma-3-27b-
Owen2.5-0. 10t.5-00..-IInstue..
1wen2.5-1.3B-Instzze
@ven2.5-33-Instruuct
Ow
Yang et 2024c
(ven2.5-14B-Instruu
Swennttt-72-IInstruuct
Owen/
Yang et al.
2024b)
Owen3-C.66
3wen3-1.77 7B
Qwen3-4B
Yang et al.
2025
3wen3-8B
Qwen3 -4BB
Qwen3 32B
Missraal-7B-Instrrc---000
Jaang et al.
2023a
71sstrall-7--Instratt--000-0
mistralai/
Ministral-AB--nstttot-----
MMistral AI Team
2024a
(Mistral Al Team
2024b
Nouus Research
2024b
NousResearch/
Nous Research
2024a
alignment-
zephyr-7b-Gpo-full
Tunstall al.
2023
handbook/
xephyr-7b-sft-full
Llama-3.1-9B-Instruees
meta-llama/
Grattafiori et al.
2024
1lame-3.3-778B-Instrucc
microsoft/
phi-4
Abdin al.
2024
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D.4 PROMPTS
We use different prompt styles: concise, sentence, and chain-of-thought (cot). All prompts use few-shot examples from the dataset. We describe the prompt formatting here by way of example, using our prompts for the GSM8K dataset. For base models, we use the full prompt text as context, while for instruct models we format the few-shot examples using the model-specific chat template (per Huggingface).
Prompt: shows the "concise" prompt for GSM8K. This prompt style uses only the final answers provided by the dataset (excluding any chain-of- -fhhouuhh.
Prompt shows the "sentence" prompt type. This prompt formats the few-shot answers in com- plete sentences, and also includes instructions to format answers accordingly. Note that we inten- tionally varied the sentence structure of the few-shot examples, to encourage the model to use diversity of phrasings. This makes the "sentence" responses more syntactically complex than the "concise" responses, though not more semantically complex - thus testing the limits our theory.
Prompt 3 shows the "cor" prompt type. This includes reasoning and formatting instructions, as well as sew-shot examples that include reasoning traces (provided by the dataset).
The prompt formatting for other datasets follow the same conventions these GSM8K examples. We exclude the "cot" prompt type for non-math datasets.
Prompt 1: GSM8K-coneise
Questioons Natalia so clips May. How nany clips did Natalia sell aitogether in April and Hay? Ansver: 72
friends Aprill
half meny olips in
Ocestion: Nong carns How much did the earn?
babysitting. Yesterday, she just minutes babyeisting.
Anaver: 10
coste Betty only sho reeds. Hez pareats decided to cive her 915 for tast purposa, and het grandpareats twice as mach aet parents. How much nore money does Betty need to auy the wallet? Ansver:
Betty saving noney
noney
Question: Julie is reading 120-page book, Yesterday, she was able to read 12 pages and today, she read tuice as many pages ea yesterday, If she wenta to read hale of the remsining ng pages tenorrox, how many chould she read?
Quastion:
Janas writes J-page letter zo 2 dicfcrent triends twize week, How nany pages does he
write
Answer:
Question:
[QUESTION]
Prompt 2: GSM8K-
Anaver the followw's quest fon sirgle brief but corglete sentence. Quastion:
Natalia sold, clips
hez resends April, and then
many clips in
May. How nany clipe did Natalia sell aizogether in Agril and Hay? Anaver: ia sold 12 clips in April and May combined.
single guastion: Nena carns 512 on hour for babysitting. Yesterday, she just did 50 niautes of babysisting. How much did she eaun? Winn
Ansver tollexing question
complete
Ansver:
$10 yesterday.
Ansver the tolloxing question in single briet but corplete sentence. Question: Betty is saving money for new wallet which costa $100, Berty has only halt of the noney decided cive 915 Betty to acy
needs. Bes
her grandparcats twice
parents. How much more money Answer: Betty needs 65 nere to buy the vallet.
beiet complete sentence. quastion: Julie reading 12C-page book. Yesterdav, she able read tvice as nany pages as yastorday. if she vants to read half of the remaining peges tonozrox, how many pages chould she read? Answer: vould
Ansver tollexing question
pages today,
read
Anaver the folloxing question in single brief but complete senteace.
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Queation, ion: Janea writes S-page lettez co 2. diCCucent Irienda tvice neak, How doea be Anoaees danes wriles pages per year.
Answer the Folliww quest single brief but corglete sentence.
lons (QUESTION]
Anavez:
Prompt 3: GSM8K-cot
Answer the folloxing question. To do that, first reason sbout it by seying "Ressoningi" and then derive answer. After concise answez to the question. Qoestion: Katalia e clips to 48 of her friende April, and then she sold half as meyy clippe in Nay. HoN nany clips did Natalia sell aitegether in lpril and Hay?
you weite <D0xE>,
is: and write short and
Natalia sold 48/2 <c98/2-245022 clips in
c<4B-20=729972 clips altogether in April and Kay.
By answee 1s: 724DCNE>
Ansver follexing cuestion.
saying 'Reasoningg' and "Ry answer and shart quastion: Neng carns 912 hour for babysltting. Yesterday, she just did 50 ninutes of babysisting. much
derive ansver. concise answez to the question. last, write <D0XE>,
that, when you done,
carn? Anaver: beaaoning: Weng earns 12/50 9ec12/50-0.290.2 par minute. Working 50 minates, she corned 0.2 x 50 2::00.230-100-100 answer is: 10:DCNE>
AnSVEE tollexing cuestion. do that, derive the answer. After that, wher you are done, write "By answez is: and write stort and concine answez to the quastion. last, krita <DONE>, quastion:
reason sbout by saying "Reasoning:"
5100. tor tast purpose, as her parants. ноw nuch noze noney does Betty need to buy the wallet? Answer: Reasening: In the begiariat, Betty has only 100 / 2 9ec100/2-50--50. Betty's grandpazent
which
only grandpareats twice mach
reeds. Hez parencs decided give
2cc15-2-309930.
This moans, Batty needs By answoE is: 5<20KE>
15 1-1200-50-30-1--550 noze.
Ansver
saying 'Reasoningg' and that, whon you done, "yy answer and shart concise answez to the iasttion.iaat, welte <Doxe>.
derive answer.
quastion: Julle is reading 12C-p2qe book. Yesterday, she able read 12 pages and today, sha nany peges yeaterday, many pages should she zead?
the rensining pages Lomuzeus,
Ansver: Reasoning: Maila read x 2 <<12-2-2(0924 pages today. (112124556236 pagea aince yeaterday. read. since she wants <<120-26-34228 remaining pages tomozzoN, cc84/2-42>42 pages. acaver 19: (2<DONE>
Su alw able Lu sead Lotal 12
120
Anaver folloxing queation. do that, Cirst ceason about by saying "Bessuninge" of and derive the answer. After that, wher you are done, wzite Myy ansvez is: and write sbort and conci answes to the ion. krite showts. Janea write year?
3-page
disnet Ieienda Lvioe souk. many pages
Anavers : Bessonings the writes each friend 3:2acc3s2e6boe pages weok Bu he waites 2226635-24129912 pages every veek
urizes 21s52ecce225222222533 pages year My anaver lss 524xDONE>>
Anauer the folllww' question. do that, First about by saying ' 'Ressonings' and then that, you concise answes to question. <DONB>. Question: (CURSTION) Anaver:
iss
Prompt 4: Canonicalization
Coastion:
"/QUEST:ON."
Response: "[RESFONEE
Your
this
Zollow rulas:
Keep only the zore answer Io.c.. runbor, a name, or a short phrase)
Remove all extre words and filler.
abbreviations
"UaM rUnited States America"). thair words (e.g.> "eighe" "g").
numbers with digits, Foz locationa, output enly the highes: pprccision part le.g. (Seaitle, Hashington" >> (Beactle")
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For dates, unleas othervise apecified, fozmat WYY--WE-D (e.q- "Auguat 1999* -> +1930-08-011 apecirios, nake up unapeci fled informat lan.
If
"Aaqust" "2003" 2300*).
No explaining or zeasoning. Outpet the coce ansuez only.
TF the response dows not address the unclanged.
if
what return the
alter neaning
reaponae, aven
incorsect.
bo not infer inssing informatinn only rephrase what 18 given in the reagonse,
Prompt 5: Canonicalization (math)
"[RESPONSE
Your Lask is to culuen vonlys the uoem ansves fron this reaponze.
Follow these sulea:
only
number LateX tave. 10.5r (Mrac(t) value of variable, only gutgut value atwil (e.g. (10r).
It the anawer
Write all numbers with digita, not as words (0.9.) "eiche' => (Bt).
Renove all
words and filler, raapanse
explaining reasoning
not contain
value, you are unsure to do, retarn
reaponse unchanged.
Never aller the value of the reaponse, even if is incozewal.
infez nissing infornation)
Prompt 6: Semantic Equivalence
You will be given a queation, and Lau pussible respomes. Your Lask is to deteznine whetlees Lhe Lwo anavera are sanancioally consiatent, i.e., shether the two zesponses aqres what the answez to the quastion
Quastion: [QUESTION]
Jeagonze 1: (NRSPONSEE)
Sesponae (R2SPOK3B2)
Are Lhese two re responses swnand ieally al aligued resporaes to the question? Respand only with withes Uhe atring "Yos" the string "No".
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E THEORY
E QUICK REFERENCE
For convenience, we give references to proofs of theorems from the main body.
Thm. is proved in App. E.S
Thm. re-stated and proved as Thm. App. E.6.3
Proving these theorems involves some additional theoretical machinery. which we develop in the remaining sections. The following sections restate some of the notation and definitions from the main body for convenience.
E.2 WEIGHTED CALIBRATION
A key object in theory is the notion of weighted calibration, from Gopalan et al. (2024), which is capable of expressing many different types of calibration. We use a version of this definition suitable for our LLM setting, stated below.
Definition 17 (Weighted Calibration, Gopalan et al. (2024), ). For a set of weight functions w V x A()*) + R and distribution D over pairs (x, y) E V x V model pa is perfectly W-wwighted-calibral on 'D if:
KØ Pai wx, P.))
(x
where Ps = pa(: x) E A(V*) c RIvl1 is the model's output distribution on input æ, and ÿ E {0, 10,131MM is the one-hot-encoding of
E.3 EQUIVALENCE BETWEEN B-CALIBRATION AND WEIGHTED CALIBRATION
Here we prove that several kinds B-calibration (including B-confidence-calibraan and full B- calibration) can be characterized in terms of weighted calibration (Definition 17
Notation and Setup There two relevant output spaces: the space V of long-form answer strings, and the space [K] of semantic answer classes. Let M := | will be convenient to identify strings VN with index in [M), and will abuse notation by writing E [M].
To simplify some of the proofs, will rely E V we denote its one-hot representation as ÿ E {0, 1JMM For a given prompt æ E V", the model's distribution over completions is pa(- (- x) E A(V*) R& which treat vector embedded in R We write Pa := po(- x) for convenience.
explicit one-hot representation. For string
A collapsing function B V' VN + |K] assigns to each prompt E V and long-answer y E V a B-class (y) E [K]. Moreover, the function along with the model po induces a distribution on classes [K] as follows. For given input E V", we take the model's distribution po x)) and push it forward through B, to obtain categorical distribution n. defined as
Rg == Butpo | x²) E Ak.
(8)
Explicitly, the probability assigned to category E [K] is:
(c(c) (B.dpa) (c)
Pr (-pp(-]a)
(B = =]=
Σ
Pe(z x))
==B=(s)=o
This fosh-forward operation can be written in matrix form. Define the collapsing matrix B, as:
B {0, K0
Balk. [[Bb())=k)+
(10)
Then the pushforward distribution and ground-truth semantic class can be expressed as π = BaP E AK.
B (Ba(v) Eg.
Thus, matrix-
vettox vector multiplication exactly implements the pushforward operation:
(x=)== = Σ
(o(z a:) Bayzka.
(11)
(=)=
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E.3.1 FULL CALIBRATION
Definition 18 (Full Calibration). A distribution DD over prediction-outpu pairs (c, v) E AK x &K perfectly calibrated if the expected error, conditioned on the prediction, the zero vector:
(c. W - e ccl
(12)
Note that since y and are both vectors in IR this subtraction is well-defined.
Now, we apply this template to our setting. We say a model is B-calibrated if the distribution it induces over the collapsed, semantic categories is itself perfectly calibrated.
Definition 19 (B-Calibration)) A model pe is B-calibrated
distribution 'D if the induced distri-
bution over pairs (Tz, Bz(y)) is perfectly calibrated according to Definition 18 Here, Ta BB
takes the role of the prediction c, and the ground-truth category B2 E [K] takes the role of the outcome Formally:
E
[x Tz n.l = O.
(13)
(I.D)~~
Following our convention, the scalar (z) E [K] identified with its one-hot vector in EK to perform the vector subtraction.
Now, we provide results for B-calibration that are analogous to Definition and Thm. for B- confidence- confidence-calibration calibration..
Definition 20 (Semantic Perturbation Function Classes). Given an arbitrary function B,(z) E [K], which we think of a semantic collapsing function, we define the B-induced weighted function class (a class of perturbation functions w x. Pa) that generate perturbation vector based on the context x and the model's predictive distribution pz):
w ={u { w(x Pv: [=]] (T= (Ba))) some K - [],
(14)
Intuitively, every sequence is assigned weight based on its semantic category (=) E [K], and the weighting scheme itself can adapt based on the model's 's overall categorical prediction Tg-
Lemma 21. Let Wifill) be weight function defined by w(z, px)x] T( [B,,)) Its corresponding vector representation is given by (
Proof. We will prove the equivalence showing that for any sequence V the z-th component of the vector B ( is equal to T(Tx) (z)]. Let T(Ta), which is a vector in R
Now, we want to analyze the components of the vector BB
For any V the z-th component of v is given by the definition of matrix-vector multiplication:
K [u] B u] B
K
le.k k. up
Uk Σ (B) uk
where the last equality is by definition of B.: see Eq. (10). The indicator function 11 [[(B,(=)=*) is non-zero for only one value of & in the sum, namely when K is equal to the category of the sequence 2, i.e., k = (z). Therefore, the sum collapses to a single term:
[w]= 1 (B
Σ
uh (B.())
Be(+)st
Substituting back the definition u = T(Tx). we get: [p], = ((Tx)[B((x)] This expression matches the definition of w(2, [2] exactly.
Since this holds for all sequences z, the vector (Tx)) is the vector representation of the function w(x, P-)
With the definition of the weighted class and vector representation, we can state the main equiv- alence theorem.
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Theorem 22
IB-Calibration as Weighted Calibration), A model pe is perfectly B-calibrated if and only if is perfectly -weiightd--allibrrted
Proof. We start from the definition of B-calibration, which (as established in Definition 18 is for-
mally expressed as vector condition:
(ee Tx TT 0.
By the properties of conditional expectation, this holds if and only if for all functions AK 7 1-1.1] holds
[(eB.()) T2, ((ma)) O.
(15)
Substituting the matrix representation into Eq. (5):
E [(e n2: (TT))) =
E (B - BuPA, T (BBP2))) = 0
E (B, ū Px); r (BePx))) = 0
E (ū Pz. B ( (B=P=))) =0
From Lemma 21 the term ((app) is precisely the vector representation of the function
w(x, Pa) from Definition 20 Thus, the condition is equivalent to:
k Pz, w(x, Px) = for w W (full)
which is exactly the definition of wifuill) weigghte--callibattonn see Definition
E.3.2 CONFIDENCE CALIBRATION
We first define the standard notion of confidence calibration, a weaker form of calibration that fo- cuses only the model's top prediction.
Definition 23 (Confidence Calibration). A distribution D over prediction-output pairs (c, y) E AK x &K perfectly confidence-calibrated calibrated if. conditioned on the model's top predicted proba- bility, that probability matches the expected outcome. Formally,
lyk-
Ck-
ck- = where k- = argmax Ck- kE(K]
(16)
(c.g)>
Now, we apply this concept to our LLM setting. A model is B-confidence-callbraeedd the categori- cal distribution induces is confidence-calibrated.
Definition 24 (B-C PB--Coniidene-CCee Confidence-Calibrattonn) A model is B-conffdeenc--calibrated on distribution D if the induced distribution over pairs (Ta, Bu(y)) is perfectly confidence-calibratee according to Definition 23 This requires that, for k* argmax [)] TT (k),
[1) k*} (k())) Tz (k*)]
(17)
(r.y)~
We re-state Definition here for convenience:
Definition 25 (Semantic Perturbation Function Classes), Given arbitrary collapsing function (z(z) [K], define the class Wa of perturbation functions w(x, P&) E RIV* follows. These functions generate a perturbation vector based on the prompt x and the model's 's predictive distribution pz:
WB { w 3r [0,1]
Vz E V w(x, pa)[2] (Tx((+)) - (x)
where == Butpas k² := argmax T (k:).
kE(K]
Using this definition, we have the following equivalence.
Theorem 26 (B IB-Confidence- Calbbbrtttiiiiiiion as Weighted Calibration). A model po perfectly confidence-calibrated if and only if is perfectly Wis-weighted-calibbueed
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Proof. The model is WE- Wy--eiighee-caliirar if, for all w E Wg, the following holds:
(i Pz: wa Px)M = 0. (r.
E
For given w defined by function [0, 1] [-1, since ÿ is one-hot vector with in the coordinate z y, the first term evaluates to
G. w(w((pp)))
Σ Ø| w[)) pL) w(x p.)
(18)
Substituting the definition of w: w(x, Px)lul = (v+) (Bx())=++) where uy == Tz (1*).
The second term is (px, ((T E3 Ps (2) P=] Substituting the definition of Σ Pa( E Pz(z) (T (v ((m_())=))))
=T Σ (2
-
Putting these together, the weighted calibration condition becomes:
E [T (w:)) [(B_())=A~) - (x=) [ = >>> E (x.g)~>
[T (e+) (1(B,(0)=+))- -wū]] = 0.
(r.y)~D
This condition must hold for all functions expectation, this is true if and only if the term being multiplied by the arbitrary function of o has conditional expectation of zero. This gives us:
00 - [1,, By the properties of conditional
E [1 - ] 0,
which is precisely the definition of B coaffddence-cllbrattin
E.4 EOUIVALENCE BETWEEN WEIGHTED-CALIBRATNE AND LOCAL Loss OPTIMALITY
For the log-loss f) == - En ys log(fe), we can analyze perturbations more easily through its dual representation. The dual loss, which operates logit vector is defined
(" 2) log Σ t
yT and V softmax(2) y=
The primal and dual views are connected by the variable mapping x log f, which provides the key equality f(y.f) é" (", 2). This relationship allows us translate complex perturbations in the probability space into simple ones in the logit space. A multiplicative re-weighting of the probabilities, defined as f w = softmax(log f + w) softmax(2 + wr), equivalent simple additive perturbation the logits. Therefore, the loss of the perturbed model can be expressed in either world:
ely, f * w)
(v, + w)
(19)
Loss on perturbed probabilities
Loss on perturbed logits
Theorem 27 (Equivalence of Calibration and Local Loss Optimality). Given a model po- distribution D, and a family of weight functions W (Definition [77. the model po is perfectly -aweiigted-calibbraeedd on 'D if and only if it is W-locally loss-optimal on D.
Proof. We apply the first-order optimality condition to the dual loss e (y, 2) with a simple additive perturbation w on the logits z. With the perturbed loss function, for > 0.
((E) = E* (yy 2 =w) and dL
de (E) = (V (y.
aw), w)
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By local loss optimality
0 ((e) (()) dC
E de (0) d()) - (V 77_6"(px (y, 2), w)
(V
The same reasoning replacing w by -w, we also have (V=l* (y, =), w) 0. Thus
e (v. =) S lx (y. 2 +w) = (V=l* (v. 2), w)
The opposite implication follow from convexity, have:
e* (y, + w) 2 E- (v, + (V. z), w).
Thus, if (v, 2), w) = 0 holds, the inequality simplifies to: l (y, + w) 2 2).
Taking the expectation on both side
E (x.0)~D
[1y
f
E (a.
[e f w]] (a.y)~
(y. 2)] v
E (r.y)~
[e (y, 2 + w)]
(f
y, w)
l (y, w)
(a.0)~
A model is calibrated under the log-loss if and only if its expected prediction error f y orthogonal
to any systematic perturbation w of its logits.
E.5 PROOF OF THM. 6
We can now combine the above ingredients to directly prove Thm. 6 from the main body.
Proof. Recall we have a model pa. a collapsing function B, and a distribution D. We have the following equivalences:
is B-coonfidenc--caliiraedd on D is We--weighted-caiibraee on D
(by Thm. 26
py is "yg-locally-loss-ppimee on
(by Thm. 27
E.6 AUTOREGRESSIVE SETTINGS
Recall the definition of the perturbation operator, Definition
3
f[z] exp(w(=))
V2 V VVV; (f ( [=] = softmax( [[2] + log f(zl) Σ Even f[z*] exp(w[a1]
(20)
highlighting that this transformation multiplicative reweighting of the reference distribution f by e followed by renormalization to get a valid distribution. Applying it to the next-token setting, obtain significant simplifications for both full and confidence calibration.
E. WEIGHTED CALIBRATION
Lemma 28 (Autoregressive Decomposition of the Perturbation). For any position i, the perturbed conditional probability of the next token is the original conditional probability multiplied by ratio of "lookahead expectations":
Bey agi)
exp((wx(2s)) {>1)]]
(px * wx) z2e) = P2 22))
(21)
Ezgrep ses)
exp w. (Zetr b2)))]
Proof. Let 7:
By definition of conditional probability,
(Px * W (254)
(px * fo* (x) xez)
(22)
(px wx (zes))
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Expanding the perturbation operator and applying (a(2ga = (= 22)).
(px * w (zs() = Z
Σ
Pz P=(2gi, ((((3))))1))
221
[e (e
Z
AsampelHagd
Similarly,
Po(2cs)
(p * w., (=<e))
l (
(23)
Taking the ratio and canceling Z,
Pa (2gi)
(lsss We ([
(pz *zx
=<s)
Pa (2es))
(xgr) (e
E Ba (1 (e
= Pa( zei)
E Beg e (Zc
E.6.2 B-CALIBRATION
The general decomposition in Lemma 28 is insightful but computationally intractable, as it requires summing over all possible future sequences. We now show that for our specific class of semantic perturbations WB. this complex ratio simplifies dramatically into small, efficient arithmetic circuit. The key to define two "autoregressive B-confidence" vectors that can be tracked during generation.
Autoregressive B-confidence Given a model Pa and a semantic mapping B,s we define:
1. The initial B-confidence (o(x) E Ax. which is the model's overall predicted distribution on the K categories before generation begins. This corresponds to the B-induced pushfor- ward distribution Te B.
(0(x)) b = Pr [x) =
(24)
znpp
2. The conditional B-confidence (r, zgi) E Ax. which is the model's predicted distribution on categories, conditioned on having generated the prefix agi
Pr 9s (z, [b] x'ropal agg)
[Ba( b).
(25)
Theorem 29 (Simple Circuit for B- B-Perturbations). For any perturbation Wn (defined by scaling function T), the perturbed next-token probability is proportional to the original conditional probability multppiidd by simple circuit Cw:
(px * (ea zed) x Palzs zes) (golg)z) g;(x, zgi)),
(26)
where the constant of proportionality does not depend on 24 and
K
Cu 9i) = Σ exp((T(0)()) [b.
(27)
1=
This circuit has constant depth and width linear in K.
Proof. From Lemma 28 we know that
Exppa w (a(s
(px w.) Ree) = p2((2 (22))
(28)
Ez (eec) m0 (Fcr12)
For E Wa. by definition, (x) ([
where go(x)) Bydpz.
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Expanding the expectation,
(=gu(=))))]
(e
3~p2(1155)
K
Pr(B)(Zghz2) = b] =
K (1 2g e [1o(+)))
The denominator is an expectation over 2 Px(- which depends only on the prefix zei and
not on the choice of 25. Hence it is a constant with respect to 2; and can be absorbed into the proportionality. Therefore, (Px (2)((1 zz3) x P=(4] Rex) (xx)()(0()))) =gx))- L
E.6.3 PROOF TIM.. 9 A SIMPLE CIRCUIT FOR B-CONHDENNI-PEETTRRRNNN
The circuit for general B B-perturbations involves K-dimensional inner product. For the more stricted class of B--oofidence-pont peaennncepprtuubatioon.s WB. the structure simplifies even further to a trivial scalar arithmetic circuit. First, we define the key sealar quantities needed.
Definition 30 (Autoregressive Top-1 Confidence). Given model ps and mapping Bas let Brtps be the initial categorical distribution, and let k. = argmaxxe(K]] (Tz) be the single most likely category. We define:
1. The top confidence value vi E [0, 1}. which is the model's confidence in this top category:
0 = (T2)
2. The conditional probability of hitting the top category, 9: probability of eventually generating sequence in category k*, given the prefix agii
(conf) (a, 2g1) E [0, 1). which the
Pr (conf) (z. 2<s) (lpgs)(
(Bl(zs = k*).
(30)
With these scalars, the autoregressive update becomes simple linear transformation.
Theorem 31. For any perturbation E Wa (defined by function T). the perturbed next-token probability proportional to the original probability modified by a simple scalar circuit C..:
(px * (2) zes) x Pz xex) Ow (canf) (t,zz))).
(31)
where the circuit Cw is a linear function of 9 (conl),
Cv.(v,) == 1 (cxp(r(v)) -
(32)
Proof. By Lemma 28
(pu * fa*waale wx) Ree) pzlas Zes)
E (-pp((12s))
(xp((ux())
(33)
For w E WB we have
W/ (2) = c, (1)x())=k1) with eo == T())).
exp(ws (z)) 1 + (exp(cx) 1) ((B, =k)
Taking expectation under p- =gg)) yields
1 + (exp(cx) - (r( = b* 22 = 1 (cxp(T(v")) - 1) g, (x, 2g:).
(conf)
(34)
By Lemma 28 the perturbed conditional probability is the original pa pz(2; (2, &22) scaled by the ratio of this term to an analogous denominator depending only on the prefix &eg. Since the denominator is independent of 2,, can be absorbed into the overall proportionality constant.
n
36

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Published as a conference paper at ICLR 2026
E.7 QUANTITATIVE BOUNDS ON MULTI-CLAST CALIBRATION AND POST- PROCESSING GAP
Beyond cross-entropy loss, we provide in this section generalization for the class of proper loss functions and quantitative bounds relating post-processing and calibration gap. The main result in this section, Thm. 36 should be interpreted as generalization of Theorem E.3 in Blasiok et al. (2023) to the multi-class setting, and robust version of Thm. [7] it essentially states that model is "close to" Wa-iiighted-calibbraeedd if "close to" W-loss-optimal.
First, we recall a standard result on convex representation of proper losses (Savage 1971 Schervish 1989 Gneiting& & Raftery, (007).
Definition 32 (Savage representation). A loss function {ei,...,exx) x Ax - R is proper isf there exists convex function Φ A - RR such that
fyy v) = -0(v) + ( vo
(35)
Next, define the convex conjugate db 6", a dual variable, and the dual form of the loss.
Definition 33 (Dual loss). For proper loss f with potential & in Definition 32 define:
Convex conjugate:
vb(u) == o² (u) = sup ((u, v) <(v)), oEAR
Dual variable: dual( == Vo(v),
Dual loss: (\ (g. == (b2) (u. 2).
Remark 34. The dual parameterization of Definition 33 satisfies:
1. Agreement between primal and dual losses: ((w) (y, dual(v)) E(y, v).
2. Probability - dual map: dual(v) Vo(u) for all Ak.
3. Dual + probability map: v = (d(dual(v)) for all E AK-
Definition 35 (Generalized dual calibration and post-processing gap). Let W be class of finctions W: : X x R > KR and let 'D be a distribution over & x (eI,...,cky
For predictor f X + Ag. let g X + R be its dual representation such that f(x) Vx(((()))
Vx
(36)
Define for shorthand
A (w)= (( (wV- [(x),, w(r, g {x))))]
C(h) = B(er)) (2(1) (y, hx)))
(37)
The dual calibration error of g with respect to W
CF(g: W) = sup |[w)]]
(38)
wEW
The dual post-processing gap of with respect to a functiion class H. is
Gap(g: H) C(g) inf L(h).
(39)
kcH
Theorem 36 (General relationship between calibration and post-processing). Let 1b : RRK - TR. be differentiable and X-smooth i.e. Vú is X-Lipschitz. Let W be class of bounded functions w : X R + g with |mmm| S For w E W and 3 E 1-1A,, define the perturbed dual predictor
(v) = g(x) + 3 g()))
(40)
Let Gw (gw w E,W, B 1-1-111,1/|]] Then, for every g X RRK and distribution DD
(E)G: 2 < x Gap(g: S CE(g; W).
(41)
Proof. By the definition of g(#),
L(g) L(gw) B[V(o(z)) - (v.g(a)) - (x)( + (x))]
-1(((o(()) (x) + Bly, w(1, ((x)){
37

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Published as a conference paper at ICLR 2026
By convexity and X-smoothness of for g(x), z' gw((z) and w2 w(z, g(x))
AB2
(Vb(z), Brea) - (x(z) - (b22) & (Vo(z), Bwx) + 2 ||
(42)
Since f(x) Veb(g(>)) and ||w=11 I 1, this yields
x - L(g) - L(gw) S B (ww).
8 (ww)
(43)
Lower bound. For E W, set
(ww) /> (which lies in [-1/X, 1/A).. Then
1
Aww)2 s f(g) Llgw).
(44)
2X
Taking sup yields wEW
(CE(g:W)) < A Gap(s: 2
(45)
Upper bound.
For 9w Gw, since |3| 1/A
C(g) - L(gw) S 3 ( < 1 x
1
[b(w))
(46)
Taking sup gives weW
A Gap(g: Gw) Combining the upper and lower bounds proves Eq. (1
CEig: W).
(47)
Remark 37 (Tighter exponent under strong convexity). If. in addition, sb is u-sttrongyy convex for some μ > 0 i.e.
(1²) (Z2) + (Vu(z), - 2) + #1| 211?
then one obtains matching upper and lower bounds. In this case, both inequalities in Thm. 36 become quadratic in the calibration error:
P.
(CE(g; WI)² Gap(g; S (CEG: W))
(48)
2Xa
2p
That is, the dual pest-processing gap and the squared dual calibration error are equivalent up to constants determined by (J,
E.7.1 SPECIALIZATION TO CROSS-ENTROPY LOSS
For completeness, we summarize the standard facts about the dual parametrization of the negative log-loss Table
Table 4: Duality relationships for the Negative Log-Loss (Cross-Entropy) proper scoring rule.
Primal Proper Loss (fnn)
E(y, -EK V/ log i/
Convex Function (Φ)
o(v) = EkI log(u:) (Negative Entropy)
Convex Conjugate (Ø*)
(²(2) = log
E> exp(zi) (Log-Suu-Exp))
Dual Loss (fe)
Ee ( 2) = (=(x) yTz
Dual Mapping (V
VΦ = softmax(2)
The log-sum-exp function o* (z) log
K E exp(z+) is 1/4-smooth, as shown in Beck &
Teboulle (2003) and Nesterov (005)) Thm. 36 applies with X 1/4. Moreover, to translate
38

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Published as a conference paper at ICLR 2026
the result into the notation of our main theorems, recall the relationship between the primal predic- tion fx) and its dual representation g(z):
f(x) V (g(x)) softmax(g(11)
g(r) = lo)(
The perturbed loss can then be expressed in terms of the dual variables. The dual loss on perturbed logits g + w is equivalent the primal loss on the perturbed probability distribution * w.:
(y. 9 + w) (v softmax + w)) = (y * w) where f w softmax softmax(log(f) + w).
39

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Published as a conference paper at ICLR 2026
E.8 CONFORMAL PREDICTION VIA WEIGHTED CALIBRATION
Here we observe that conformal prediction guarantees can be expressed type of weighted cali- bration (Gopalan et al. 2024), for particular weight family.
Recall conformal prediction asks for model F(x) which outputs set of labels, with the guarantee that this set contains the true label with high probability. Specifically, a conformal predictor has coverage if:
Pr ly F(x)) 2 2,00
(49)
For an introduction to conformal prediction, see Angelopoulos et al. (2033) or the lecture notes of Tibshirani 2023).
E.8.1 CONFORMAL PREDICTION FROM FULL CALIBRATION
Given a standard predictor f. which outputs a distribution on labels, one natural way to construct conformal predictor Fa is: given input T, and prediction f(r), output the of highest-predicted- probability labels which sum to total probability 1 - a. This means, outputting the K most-likely classes according to (zx) where K is chosen per-sample based on the predicted probabilities.
The first observation (which folklore) is: if the predictor of full-calibration, then the induced conformal predictor Fa is correct (i.e. has coverage a). This statement not very relevant in practice, since full calibration is often too strong to hold. However, can achieve the same result with weaker notion of calibration. This is straightforward result; we sketch the argument below.
perfectly calibrated, in sense
E.8.2 CONFORMAL PREDICTION FROM WEIGHTED CALIBRATION
Suppose f ; + AN is perfectly weighted-calibrate (in the sense of Gopalan et al. Lemma 38.
2024 with respect to the following family of weight functions w(f) E RN
W: W== (w(l) = E [0, (+1))
(50)
Where 1T E {0.1) 1 is the indicutor-vector for set of indices T, and the set contains the highest- probability labels, defined
{f) max(t:
Σ f1lf/2 (E[N]
21 (the threshold probability, given f)
Ta(f) = (i: f. (f)}
(The set of top-class indices, for given level a))
That is, suppose:
E
Kl - f(x), w(f(x))) =0
[a,y)~D
Then, the induced conformal predictor Fa, of f is valid at all coverage levels a.
Proof. (Sketch) Notice that by construction, (f, (r.(f) 2 1 1-a. Therefore by calibration we must have: (y, 1T,())) 2 a.
Moreover, the set Ta(f) is exactly the output of the induced conformal predictor Fa, given base prediction f. Therefore
Prly E Ta(((x)) = Elv, 1Ta0
(51)
2 1-a
(52)
By the general connection of Theorem 27 if a model f is W-locally-loss-oppinal w.r.t. the weight class of Equation (50), then the induced conformal predictor F,, has coverage for all [0,1],
40

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Published as a conference paper at ICLR 2026
F DISAGGREGATED RELIABILITY DIAGRAM RESULTS
In this section, we report disaggregated reliability diagram results for individual configurations we evaluated. The plots are displayed follow:
the right three columns present results for instruct models,
the left three columns present results for the corresponding base models.
In some cases, there are multiple instruct models trained from a single base models, hence for some base models, their results are being presented multiple times.
Some instruct models do not have public corresponding base model-in those cases, the left three columns of the row are empty.
5 TriviaQA and SimpleQA were not evaluated for the CoT response style. The figures start the next page. For a quick references:
As discussed in the Sec.
GSMBK App.
OpenMathInstrucct in App. F22
Trivia0A App.
F3
SimpleQA in App.
F.4
41

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F.1 GSM8K
GSMBK
Qwen/Qwen225 family
buse-concise
base-senterice
base-cot
instruct-concise
instruct-senlence
instruct-col
Qwen2 5-0.5B
Qwee2.5-0.5B-tmmse.
Semanti Accuncy
1.0
sm&CIT 0.002
sMOCE 10.12
smbCK 02275
smbCT 0.311
smbCI 00.12
smbCE 2222
0.5
Qwen2.5-11 5-1.BB
Qwen2.
Seuanto Accuncy
1.0
SmBCC 0.027
smBCZ 10002
emFCE 200
MMBCC 00357
xmBCE 200125
sm X
Qwen2.5-22 5-3B
Qwen2.5 5-3B- 1B-nsttrcct
Secanti Accuray
0.5
0.0
Qwen2.5-77 B-BBB
Qwen2 5-7--nsstuttt
Semazte Accuraey
1.0
SIIECE 00207
MIBCCE 2,0052
SIECE SATEE
SIBLLE 0.357 0.357
smBCCE 700322
SmECE 20.06
0.5
0.0
Qwen2.5-14B
QwenZ.5-1 14B 4B-Instruct
Semaatic Jccuncy
1.0
SOLBCE: 0.050
smBCE $.020
smECE 10,095
SIECEE 0.381
smBCE (0.312
smBCE (0.012
0.5
@wen2.5-28 5-322B
Qwen2.5 Owen2.5-322B- 1--.----------seee Insttrctt
Semante Accuniy
0.051
0.0
Qwen2.5-P77 5-72B
Qwen2.5 Siven2.-72BB. 7-2.5----2-B-rt
Semanttc Accuracy
sm.CC. (0040
smbCE 100056
XmDCE FRUAHT
RMBCC 00,88
smDCC G00252
SMÉCE
0.5
0.0
0.0
0.5
1.0
0.0
0.5
1.0
08
0.5
1.0
0.0
0.5
10
0.0
0.5
1.0
0.
0.5
1.0
Confidesce
Confidence
Semantic Confedence
Semantic Confidence
Coefidence
Confidene:
42

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Published as a conference paper at ICLR 2026
GSMBK
wenut Qwen3 family
base-concise
base-sentenee
base-cot
instruct-
Instruct-sentence
instruct-cot
Qwen -.o----a-----eeeee
Qwen33-0.6B
Semante Accurayy
smECB
0.5
0.0
Qwen3-1.2 1--------Baas
Qwen3-1.77 1-77B
Sccantic Accurayy
SEDECC FAAEEBB:00000 00.021
SMECE
smECB 0.379
0.007
0.0
Qwen3-4B Baen3--B-Baase 4B-Base
Qwen3-4B smFCF 0. $49
Semantic Accuncy
1.0
smFFC:: 01021
smFCF. : 022
SnRCE WATTEX
smBCE, 999
smRCT: (.096
0.5
Qwen3-8B- BBess--B--Base Base
Qwen3-8B
Secannte Accuracy
SIECE
SmEEEE 0.0L1
0.5
0.0
Qwen3 14B-Base
Qwen3- 14B
Semaatie Accursty
smFCE. 0.005
smECE 0,022
snBCE: 14150
SmBCE: D 3999
XmFCB. :0..99
smFCT: 01012
Qwen3-32B
Semantic Avcuracy
SMBCE: 0.332
amECE 0.238
SIIBCE: D.OLI
0.0
Semantic Confieence
Semantic Confidence
Semantic Coufidence
43

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Published as a conference paper at ICLR 2026
GSMBK
google/genmaa- family
base-coneise
base-sentence
base-cot
instruct-concise
instruct-senlence
instruct-cot
gemmaa-2-2 2-2
gemma--2220 2-2b-i smbCC 002263
Semastic Accunay
smbCI 0.005
smOCE 100057
smbCE 10.108
smbC 10336
smBCE Ho.RR)
0.5
0.0
germaa
gemma
Scmanti Accuracy
1.0
SmBCCT 10OT6
SmbCCe (0011
smiLt
SMACTT & 10.199
SmbCCE, 10225
smFCSs C000T9
0.5
gemmna-2 2-27b
gemma 2 2-27b-it
Scoandi Accuracy
SmBCCE
SmFCE
MIBCCE
smFCE
Semantic Confidence
Semantic Confidence
Semantic Confidence
Semantic Confidence
Semantic Confidence
Semancic Confidence
google/genma-3 family
base-concise
base-sentence
base-cot
instruct-concise
instrucc-senttent gemma- 3-1bitt
instruct-cnt
gemma-3-1b-pp
Semantic Accuncy
2.0
SuLBCE: 0.046
SmBCE 00040
amECC :0. L6162
SIUECE 0.332
smaBCE (0.226
smbCC 30.256
2.5
gemmaa-3-oe 3-aeea3--------
3-4b-it (0.118
Semaatic Accurity
ambC
gemman-3 3- 12b-pt
gemma- 3- 1-1--itt
Semastie Accuncy
smBCC. 00025
smOCE (0005)
SmbC
SMECE 0.387
smbCC 10:373
smDCC 70.012
0.5
0.0
gemma 27b
gemma- 3-27b-it
Senante Accuncy
1.0
SmBCC. 10:051
smbCk. 100017
SMFCE Seg
MIACC 10.44
SmbCC (0.37
smfC' GRUDT?
0.5
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0,5
1.
0.0
0.5
1.0
0.0
0.5
1.0
Semantic Coufdence
Semantic Confidence
Semantic Confidence
Semantic Coufdeace
Semantic Confidence
Semantic Confidencs
44

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Published as a conference paper at ICLR 2026
GSMBK
mistralai/ Mistral family
buse-concise
buse-sentennee Mistral-7B-vO.
base-cot
instruct-concise
instruct-senlence
instruct-col
Mistral-7B-Jnstruce 7B-Jnstruct-vO.1
Semante Accurcy
7.0
smiEC' 0.035
SmECE 00037
smEC'
22227 22227
sunEC' 0.060
smbCE b.051
smbCS 1925
0.5
Mistral-7B-vo.
7B-Instruct-vO..
Semuntie Accuncy
SmBCI 0.017
smECE 10026
amEC 222
smbCT 0.232
smbCZ 70.94
smBCB 19260
Ministral - &B Instruct-2410
Semantic Aceurcy
Hiistral-Small-22 Smalll224B-Base-25500 250--aa5-221
Mistral Small-24B 240---2tttt--nt----5555 Instruct-2500
Semsnnte Accuncy
0.5
0.0
Mistral Mistral-Large Large-Insttute--22222 Instruct--2111
Secaantk Accuracy
SMFCEE 10.33
SmFCCE GALLO
umnFr P30331
Mixtral-8x77 Bx-----
Mixtral Iast--ututtt--..
Secanik Accuracy
1.0
SIFCCE 10OESS
MIFCCE 200014
SMFCCF 00355
SmFCCE 702%
unFFC TMTT
0.5
0.0
0.0
0.5
1.0
0.0
0.5
L.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
Semantic Confddence
Semantic Confidence
Semantic Confidence
Semantic Confidence
Semantic Coufidence
Semantic Confidence
45

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Published as a conference paper at ICLR 2026
GSMBK
Qwen/ Seeem//wen2.5-MMM
family
base-concise
buse-sentenice Swwen.55-Math-aa
instrucc-senleece
Qwrnn2.5-Math 1et.e----aa--------eeeeee 1.58-Instucct
Sccantic Accurry
00.41
MmRCE. 0.,00
kmRCC, (0.06)
S 5- Math-7B
Qwen2. 5 M-Math- MatthhB--Instiie 7B-Instruct
Semeztio Accunny
000057
SaaBCE 0..87
00,202
0.5
0.0
Qwen2.5 Math-72B
Qwen2.5 Swren2.--Math-225 Mwwh-Ma2--aaattttttt
Scmantic Accunay
1.0
Sm3CCT c 10:59
smbCE. 10003
SMFCE SOMES
SMACT & TUASE
smbCC (035T
smFCCs 20.39
0.5
0.0
0.5
1.0
0.0
0.5
1.0
0.9
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
Semantic Confidence
Semantic Confidence
Semantic Confidence
Semantic Confddence
Semantic Confidence
Semantic Confidence
GSMBK
NousRescarcl/Nuu-- Seormnes-2-Mixtree -xtttal----BBBBB- family
base-concise
base-sentence Miixtral-&x7B-v..11
base-cot
instruct-concise
instrucc-sentencce
instruct-cot
Senante Accuray
1.0
smBCC. 0.935
smOCE 00.014
smBC 2.25
SMECT 0.001
smDCE: (00009
smBCE 70.072
0.5
Mixtral- Mixtral-8x3B-vol 8x7B-v0.1
Nous Nous- Haaez-Heeee-222 2etrmee---Mixtal--eee -ixttral-&w/B-DDO
Samantic Accuncy
SMLCC:: 0.003
smBCE 0.014
smaC CCEAA
smbCE: 0.101
smBCE::D.. 10.13
smbCs 1o
A
0.0
0.5
1.0
0.0
0.5
0.9
0.3
0.0
9.5
0.0
0.5
0.0
0.3
Semantic Confidence
Semantic Confidence
Semantic Confidence
Semantic Confidinee
Semantic Confidence
Semantic Confidence
GSM8K
alignment wwwdboakkinaaaa..e.mmomooooooaee Bew.ool///eepyy----e fumily
base-concise
base-sentence Mistral-7B-vil
base-cot
instruct-concise
instruct-sentence 2ephyr-7b-dpo-full cmBC8.9920
Instruct-cot
Semuatt Accuncy
smBCC 01,055
smBCE (0007 (8007
amEC 2.0.220 2017
Km.CE 00322
smBCE 7o 19:
Mistral-7B 7B-v0.1
zephyr-7b 7b-sft-uull
Scoantle Accuracy
0.087
smBCE 0.087
Confidence
Semsntic Corfidence
Semantic Confidence
Semantic Confidmne
Semientcc Confidence
Semantic Confidence
46

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Published as a conference paper at ICLR 2026
meta-llama/Llamm- family
base-concise
base-sentennce
base-cot
instruct-concise
instrucct-enttenee
instruct-col
Llama-3.1-8B
Llama-3 1.1-aa....---B111 3.1-sB-Instruct
Semantic Jccurcy
SuLBCE: 0.040
smBCE: (.023
smbEC 0:203
suaCEE 0.278
smBCCE: (0.09
smbCa 1
0.5
Llama-3. 1.mma-33117000 1-70BB
Llama-3, 3.1 7-78--nstruce
Semantic Accunit
0.0
Lluna-3 3-1B
Llama-33 3.3-70B-Instruct
Senanti Accuncy
Sm.CC. 00:25
smOCE 10.014
SEACE 0033
smbCC (0:00
smOCE FOUUDS
0.5
X
0.0
0.0
0.5
1.0
0.0
0.5
1.0
0.
0.5
1.0
0.0
0.5
1.
0.0
0.5
1.0
0.
0.5
1.0
Confidence
Corfidence
Semantic Confedence
Semantic Confidence
Confidence
Confidene
GSMcK
microsoft/phi-- family
base-concise
base-sentence
base-cot
instruct-cot
phi-4
Semantti Accuracy
38.05F
"0000
47

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Published as a conference paper at ICLR 2026
F.2
OPENMATHINSTRUCT
OpenMathlnstrutt
Qwen/ family
buse-concise
base-sentenice
base-cot
instruct-concise
instruct-senlence Qwen2.5-0.5B-Imsstte
instruct-col
Qwen2.5-58 2.re22.5-00.B
Semanti Accuncy
1.0
sm&CIT 0.035
smOCE (0.65.
smbC& 11..19
smbCT 0.005
smDCI (0.225
smbCS n0.13
0.5
Qwen2.5-1 5-1.BB
Qwen2.5-1
Seuanto Accuncy
1.0
SmBCC 00.91
smBC2 10.16
AMECE POUM
MMBCC 0.349
smBCE 10004
smBCC 70--30
Qwen2.5-23 5-3B
Qwen2.5 5-3B- I--nstrucct
Secanti Accuray
0.5
0.0
Qwen2.5-7B
Qwen2 1-0-255-----iiee II-Instr
Semazte Accuraey
1.0
SIIECE 00014
MIBCE 200025
SIEECE 1OL16A
SIBLEE 0.397
siBCCE 70.351
SmEECE 70.165
0.5
0.0
Qwen2.5-14h
QwenZ.5-1 4B-Instruct
Semaatic Jccuacy
1.0
SILBCE: 0.016
smBCE (0017
smECE 10.25T
SIEECE: 0.399
smBCE (0.359
smBCE 10.038
0.5
Qwen2.5-522 5-32B
Qwen2.5-3222 10102-----2---nstrree Inn-trett
Semantc Accunit
0.0
Qwen2.5-728 5-7BB
Qwen2.5- 20en2255722BInstruee
Semantic Accuracy
sm3CCE (1017
smbCZ 100053
EMBCE 1OME
RMACE 00385
smDCC 10:350
smBCC 700233
0.5
0.0
0.0
0.5
1.0
0.0
0.5
1.0
0.
0.5
0.0
0.5
1.
0.0
0.5
1.0
0.
0.5
1.0
Confidesce
Confidence
Semantic Confidence
Semantic Confidence
Coefidence
Semantic Confidence
48

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Published as a conference paper at ICLR 2026
OpenMathinstrutt
Qwent Qwen/Qwen3 family
base-concise
base-sentenee
base-cot
instruct-comee
instruct-sentenn senrece-sentence Qwen3--0.B
instruct-cot
Qwen3--O.d O.o---------aeeee
Semante Accurayy
0.5
0.0
Qwen3-1.785 1-7---1---Bata
Qwen3-1.77
Sccantic Accurayy
SMECE
smECB 0.374
SmaCCE 0..055
0.0
Qwen3-4B 4B-Base
Qwen3-4B smFCF 0. 858
Semantic Accuncy
smFEC:: 0.019
smFCF. : mm8
SIECCE: 1..26
smBCE, 0.428
smFCE: (1085
0.5
Qwen3-8B- Beess--f--Bsse Base
Qwen3-8B
Secannte Accuracy
0.321
SELECE:
SIDECE:
0.5
0.0
Qwen3 14B-Base
Qwen3- 14B
Semaatie Accursty
smFC.: 0.051
smECE :
snBCE: II
MmBCE: 0-420
XmFCB. 0.555
smFCCF 1-135
X
Qwen3-32B
Semantic Avcuracy
smBCE: 0.432
amECE 0.258
SIIBCE: 0.073
0.0
Semantic Confieence
Semantic Confidence
Semantic Coufidence
49

---

Published as a conference paper at ICLR 2026
OpenMaathnntruct
google/gemma- family
base-coneise
base-sentence
base-cot
instruct-concise
instruct-senlence
instruct-cot
gemmaa-222 2-2b
gemma-22222 2-2b-it smbCC 10.177
Semastic Accunayy
smECI 0OOT
smOCE 000000
smbCE 10.146
smbC 00392
smBCE HQ.0K]
0.5
0.0
germa-
gemma- 2-9h-it
Scmanti Accuracy
1.0
SmBCCT 1001S
SmbCC DOO2
SMILIS DO
SmbCCE 10255
SmFCSS CRUDST
0.5
2-27b
gemma 2- 2-27b-it
Scoande Accuricy
SmBCCE
MmFCE.
MIBCCE
RmFCE: 2,22
Semantic Confidence
Semantic Confidence
Semantic Confidence
Semantic Confideace
Semantic Confidence
Semancic Confidence
Opthnnatitttrutt
google/gemma-3 family
base-concise
base-sentence
base-coo cos
instruct-concise
instruct-sentenn gemma-- smBCCE (0.117
instruct-cot
gemma-3-1Bes
Semantic Accuncy
2.0
SuLBCE: 0.040
smBCE (0053
amECC 10.253
SuaBCE 0.399
amECE :0,
2.5
gemma--3-Be Battnna-----bee
3-4b-it
Semaatic Accunsty
(0,00
x
gemman-3 3- 12b-pt
gemma- 3- 1-12-ittt
Semastie Accuncy
sm.CC. 0.029
smOCE 10.052
SMDCE (RURO
SMECE 0.387
smbCC 10:380
smDCE 10.169
0.5
0.0
gemma pt
gemma- 3-27b-it
Senante Accuncyy
1.0
SmBCC. 10,020
smBC3. plose
MINCC 10374
SmbCC. 20385
smfCCS DRU2EN
0.5
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
Semantic Coufdence
Semantic Confidence
Semantic Confidence
Semantic Coufddace
Semantic Confidence
Semantic Confidence
50

---

Published as a conference paper at ICLR 2026
OpenMatthnstruct
mistralai/ mistralaiMMittral Mistral family
buse-concios
buse-sentencee Mistral-7B-vO.
base-cot
instruct-concise
instruct-senlence
instruct-col
Mistral-7B-Insttuen 7B-InstructtvOO.
Semante Accurcy
7.0
smiECE 0.824
SMECE 0.014
sunKCE 0.080
smbCCE 00055
0.5
Distral-7B-v0.3
Mistral-7B 7B-Instruct-v0..
Semuntie Accuncy
SmBCI 0.025
smECE 00000
smEC& 1011 sm.16-10.11
SmbCC 0.39
5mCCE 00.03
smBCB 70.
Ministral -n- &B Instruct-2411
Semanbic Aceuncy
Hiistra--Small--a 24B--aas--2501
Mistral Small-24h Bonall-2BB-Instruce Instrucc-2200
Semannte Accuncy
0.0
Mistral Mistral-Large Large-nsstrutt-eeee Instruct-2211
Secaann Accuracy
SmFCCE GOLI2
umFCC 20TR
Mixtral-8x/P -Bx--o0---7--vo111
Mixtral- Sarral-xxx---nsteee---e Instruct--O..1
Secanik Accuracy
1.0
SmFCCE 10020
MIFCCE (0044
AIFCCE (ALD0)
SMFCCF 10:44
smFCCE (020
unFCC 2OUDW
0.5
0.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
Semantic Confddence
Semantic Confidence
Semantic Confidence
Semantic Confidence
Semantic Confidence
Semantic Confidence
51

---

Published as a conference paper at ICLR 2026
OpenMathhnstruet Qwen/ Setee//wen2.5-MMe 5-tth family
base-concise
buse-sentennce
instrucc-senlence
instruct-cot
Sween.5--Math-Ba
Qwen2255-Math Watth-1.88-nnstrer
Sccantic Accurcy
MmRCE. (0.015
SmBCC:: 0.065
kmRCC, 0.,00
Swen2.5 5- Math-7B
Qwen2 5 5-Math-Maa---t- Maath----Inattee 2B-Instruct
Semertlo Accunny
00.85
200018
00.85
00.154
0.5
0.0
Qwen2.5 Swen2.5-Math--ae Mathh-2B
QwenZ.5 Srren2.5-Math-2225 M-t-t--at--------- Math--22B-Instruct
Scmaniic Accunay
1.0
Sm3CCE (ooSs
SmOCE. 10.045
SMACT PESCCE:BO FUAST
smbCC 10322
0.5
0.0
0.5
1.0
0.0
0.5
1.0
0.9
0.5
1.0
0.0
0.5
3.0
0.0
0.5
1.0
0.0
0.5
1.0
Semantic Confidence
Semantie Confidence
Semantic Confidence
Semantic Confidence
Semantic Confidence
Semantic Confidence
OpenMathinstruct
NousRescarch/Nour Honeeacrrccnoooo---ee--ee-e--- M-Miiitaa-----xta-----aa- -xr7Bal----BBBBB family
base-concise
base-sentence
base-cot
instruct-concise
instruct-sentencce
instruct-cot
Mixtral-8x7B--a B--0-00
Nous-Hermes-22-5 Mileemmmer---Mitttt-------------
Senante Accuray
1.0
smBCC. 0.980
smOCE. 1006
smOCE (0.000)
SMECT 0.046
smDCE. (00687
smBCC 70.18)
0.5
Mixtral- Mixtral-xx/B.vo. 8x2B-v0.1
Nous- Nous- Hermes-2 Seemnns----iiniee --M--tral-&/BB-DPO
Samantic Accuncy
smLCCE: 0.020
smBCE 0.043
smÉCE 10.080
smbCC: 0.089
smBCE (0009]
smbCE 00.099
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.3
0.0
9.5
0.0
0.5
0.0
0.3
Se Semantic Confidence
Semantic Confidence
Semantic Confidence
Semantic Confidinee
Semantic Confidence
Semantic Confidence
OpenMathinstruct alignment www.bhunndnaaannee Sewundboookzepphr-PPP fumily
base-concise
base-sentence Mistral-7B-vvu.
base-cot
instruct-concise
instruct-sentence zephyr-7b-dpo-ttl
Instruct-cot
Semuntt Accuncy
smBCC 00,024
smBCE. 38001
Sm.CE 00339
smBCB 18.180
smBCB 70.322
Mistral-7B-vOL
zephyr-7b 7b-sft-full
Scoantle Accuracy
0.014
smBCE
Confidence
Semantic Corfidence
Semantic Confidence
Semantic Confidmne
Semientc Confidence
Semantic Confidence
52

---

Published as a conference paper at ICLR 2026
Open
Satzzlllm///llam--- fumily
base-concise
base-sentencce
base-cot
instruct-concise
instrucct-enteenee
instruct-col
Llama-3.1-8B
Llama-3 3.1-aa3..--BBB--1 3.1-sB-Instruet
Semantic Jccurcy
SILBCE: 0.015
SmBCE 0.021
amEEE 10, 108
SIBCEE 0.164
smBCCE: (0.147
smbEC. (0.294
0.5
Llama-3. 1.mma--3117000 1-70B
Llama-3, 3.1 3.1-708-ITaae 708-Instruct
Semantic Accunit
0.0
Lluna-3 3.1-70B
Llama-3 3.3-70B-Instrucc
Senanti Accuncy
SmbCC: 00018
smbCE 10005
eMBCE -0.053
REACE 00333
smbCC 00335
smBCE 10.L77
0.5
x
0.0
0.0
0.5
1.0
0.0
0.5
1.0
0.9
0.5
1.0
0.0
0.5
10
0.0
0.5
1.0
0.
0.5 Semantic Confidence
Confidence
Corfidence
Semantic Confedencs
Semantic Confidence
Confidence
OpthMathlnstruct
microsoft/phi-4 family
base-concise
base-sentence
base-cot
instruct-cot
phi-4
Semanttie Accuracy
"0000
0.0
53

---

Published as a conference paper at ICLR 2026
F.3 TRIVIAQA
TrivisQA
Owen/Qwen225 family
base-concise
base-sentencce
base-cot
instruct-concise
instruct-sentence
instruct-col
Qwen2.5-0.5B
Qwen2.55-0 5.B.-----.-ttttrccttt
1.0
Seoantic Accuracy
ManBCE. 00.991
SmICEE 0.089
smaCEE 00.48
smaRCE 0.028
Qwen2.5-11 5.55BB
Qwen2.5-1 1-5.--111------- JB-Instruct
Somaniic Aocursyy
0.5
0.0
Qwen2.5-3B
Qwen2.5-3B
Semastio Avcuncy
1.0
S 0.01
SMECE: 0.825
SmnAC 0.081
smaCEE 0.087
Qwen2.5-22 5-7B
Qwen2.5-7B4 I-sntsuucc
Semaatic Actuncy
SmFFC
smBCE: 0,000
smFCF:
0.112
0.0
Qwen2.5-146
Qwen2.5 Inn-sssrruuc
Sccantic Avcuacy
1.0
smTCCT 11,000
smBCC: 0,030
emTCC: 11:81
smBCE: 0.704
0.5
Qwen2.5-32B
QwenZ.5 2- 5-32--532---2 2-w.5--------stte- I-nttruuct
Senantic Accuncy
sanBCE: 00,30
0.089
smBCEE 00.95
0.110
0.0
Qwen2.5-72B
Qwen2.5 5-2.25--7--- 72B--Intruct
Semanti Accurayy
1.0
SEFECE 0.005
SmICCE: 0.036
SMICE: 0.081
SIICCE 0.080
0.5
54

---

Published as a conference paper at ICLR 2026
TrivisQA
@wen/ Qwen3 family
base-concise
buse-sentence
base-cot
instruct-coneise
instruct-senlence
instruct-col
Qwen3-0.688 (ee033--.B-BBB-. Base
Qwen3-006B
Somanii Accuncy
amLCB: 00000
saECD: 9,030
0.5
Qwen3-1.7Bk B.B--eee
Qwen3-1.. 3.77B
Semantcc Accurayy
cm
9,621
0.5
Qwen3-4BB -B--eee
Qwen3-UB
Semante Accurzy
snECE:
0.5
Qwen3-sB. -B-Bese Base
Qwen3-SR 8B
Secantie Accuracy
0.028
SOECE:
0.5
Qwen3-14BB 14B-
14B-Base Base
Qwen3-14B 14B
Semantic Accurac
SmFCF.F
SnECE:
0.5
Qwen3-32B
Semanbi Accurazy
0.0
0.5
0.0
0.5
1.0
Semantic Confidence
Semantic Confidence
55

---

Published as a conference paper at ICLR 2026
TriviaQA
google/genme-- family
base-concise
base seseesentence
hase-cott
instruct-condise
cestrrcc--entence
instruct-cot
gemmaa-22 2-2b
gemmu--2220 1ormmu-2-2b-it
Semantic Jccuracy
1.0
smBC. 11,02
smICE: 1.087
smECC 1..41
smBCE: 0.945
0.5
gemma-22 2-9h
gemmuu-2 1aamma---------
Samaztic Accuncy
0.5
0.0
gemmaa-2
gemmaa- 2-27b-it
Semantic Accuncy
1.0
SERECE (1262
SMICE 10030
SIICCE 10008
SIICEE 00075
0.5
Semantic Confidence
Semantic Confidence
Semantic Confidence
Semantic Confidence
TriviaQA
google/gemma-3 family
base-concise
base senteeee
base-pot
instruct-concise
instrucc-sentence gemma---1b-r
instruct-cof
geinmu-3-11 1-1-pp smBCE: 9.003
Semantic Securkty
smFCC: 00.91
emFCEE 10
smBCC 0.306
gemmaa-3 3 3-4b-pt
gemmaa 3-4b-it
Semante Accuncy
Rm 1..09
smBCC: 0.0%6
Sm 0.306
Sm 0.337
0.5
0.0
genuma-3 1-10-8----1200 pt
gemmaa-3112 1-mmma----2bitt
Semant: Accunsy
SIBECE: (0.41
SIICCE 0.035
SIICCE 0.150
SmiECE 0.136
0.5
gemma-3- 3-27httt
Semantk Accurayy
0,085
0.100
0.5
0.0
0.5 Semantic Confidence
0.5
Confedence
Confidnnce
Semantic Confidence
56

---

Published as a conference paper at ICLR 2026
TriviaQA
mistralai/ mistralai/Mistrall family
base-concise
base- sese-senteecee
base-cot
instruct-concise
Instruct-sentence
instruct-cot
Mistral-7B- -B-.001
Mistral-7B -B--nttruutt-v..1
Secannte Accuracy
SIBECE 0.035
SIIECE 0.036
SIEECE 0.047
SuaECE: 0.046
0.5
0.0
Mistral-7B-vO..
Mistral 7B-Instruct--0.3
Semantic Accurccy
SEHBLEE 00,22
smBCE 0.085
smBLEE 00.995
suaBCE 0.089
Ministrul-8B- 4B-Inssrucct-211
Semantic Accurcy
smLCD 0.045
smbC0: 9.029
Mistral-Small Base-2501 1
Mistral Small-24B Baaall-2BB-Instruue Instruct-2501
Semaatic Accuncy
SunECE: 0.042
0.032
0.5
0.0
Mistral-Large- Langg---sstuuttttaaa Instruct-2411
Semantic Accurccy
emECT: 11:08
smBCE : 0.064
Hiixtral--x7B-vvlll
Mixtral Mixtral-8x/BB 1x
Semuntl Accuncy
Cm 0.301
smiEC 0.033
Sm 0.090
SmiECL 0,081
0.5
0.0
Confidence
57

---

Published as a conference paper at ICLR 2026
TrivisQA
Qwen/Qwenz Swen2.5-Math family
base-concise
base-sentence Qwen2.5 5-aatt---hhh1-- 1.5
base-cot
instruct-concise
instruct-sentence
instruct-cot
Qwen2.5 1.te:l5--Maa-------mmtte
Semantic Accuacy
SEMECC 0.072
SIIECE: : 0.039
SuaECE 0.097
0.5
Qwen2.5-Maai Matth--B
Qwen2 -aaaaa-B-Innstaann
Semaztlc Becuracy
Imm.CCC 00336
smICC : 9.022
smdCC 0.379
smBCE 0.327
Qwen2.5 Owen2.5-Malle 5-Math-728
Qwen2.5 5aatn----Mtt--2BB5 72B-Instuct
Semaatec Accuncy
S 0.324
smicC 9.030
SmICD 0.058
SmiCCL 0.313
0.5
0.0
Confidence
Confidence
TriviaQA
NousResearch/Nousn Haairrd//nNo--eeeee-------- M-Minns----iittl----- family
base-conciis
base-sentence
base-cot
custruct-conciie
instruct-sentence
instruct-col
Mixtral-8x7B--v..
Hous-Hermes-22 1asess---Miis --ixxal--xx7BBB 8x77-SFF
Senantc Accuray
0.281
0.081
SILECE: 0.066
0.052
0.5
0.0
Mixtral-8x7B--a 1..1
Mous-Hermms-22-- Mixtral-8x7B-DDD
Scnartle Accuncy
S C. 0.301
smiCC 9.053
Cm 0.059
smiCC 0.051
0.0
0.5
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
Semantic Confadence
Semantic Confidence
Semautic Confidence
Semantic Confidence
TriviaQA
family instruct-concise
base-concise
base-sentence Mistral-7B-vo.,
base-col
instruct-sentence zephyr-7b-dpo
instruct-cot
Semante Accunit
SEEECE IISR5 1.0
suICCE: : 0703
suBCCE: : 10,915
Mistral-7Ba 7B-v0.1
zephyr-7b 7b-stt full
Semazt Accurayy
1.0
CC 0.005
SmLCU 0.036
SC 0,002
ECL 0,036
0.5
0.0
58

---

Published as a conference paper at ICLR 2026
TrivisQA
meta- Baraalllmma//aaa Llama-3 family
base-concise
base sentene
base-cot
instruct-concise
instruct-senteenc
instruct-col
1lama-3.1
Semaztic Jecuracy
1.0
SmalC 0.338
smbCI 0.851
smlCC 0.356
smiCC: 0.839
0.5
Llama-3 3.1 1-70B
Llama- 3. 208-Instruct
Somaniic Accuney
0.5
0.0
Llama-3 3.1-70B
Llama-3 3.3-m---33-7---- 3.3--ot-Iatttuctt
Semantic Accuncy
1.0
SMACCI' 11:44
Sm..I: 10.066
amFCCF IUSKS
0.5
Semantic Confidence
Semantic Confidence
Semantic Confidence
Semantic Confidence
TrivinQA microsoft/phi-4 family
base-concise
base-sentence
base-cot
instruct-concise
instruct-sentence phi-4
instruct-col
Semanbbr Arcstrnc
amBCE: 0,002
0.346
0.0
0.5
1.0
0.0
0.0
Semantic Confidence
Semantic Confidence
Confidence
Semantic Confidence
59

---

Published as a conference paper at ICLR 2026
F.4 SIMPLEQA
SimpleQA
Qwen/Qwen225 family
base-conciie
base-sentencce
base-cot
instruct-concise
instruct-sentence
instruct-col
Qwen2.5-0.5B
Qwen2.55-0 50...---r.tt
1.0
Seoantic Accurccy
sanBCE. 0.822
SIICE: 0.062
smBCE: 0..41
smaRCE 0.069
Qwen2.5- 1.5B
Qwen2.5-1 1....-111---11. AB-Instruct
Scmantic Accuracy
0.001
0.301
0.066
0.5
0.0
Qwen2.5-31
Qwen2.5-3B
Semantic Accuncy
1.0
S 0.033
SMACE: 0.072
2.229 0.299
SmaCE: 0.184
Qwen2. 5-78
Qwen2.5-22 7-s-nutuuc
Semantic Accunay
smFFC
smBCT: 0.072
smFCF:
0.0
Qwen2.5-148
Qwen2.55 14B-Instruct
Sccantic Avcuray
1.0
smTCC 1.005
smBCE: 0.080
emTCC: 1..89
smBCE 0.305
0.5
Qwen2.5-25 5-3BBB
QwenZ.5 2- 5-222-532B---- 2-t.:55-3---nstreee In-strutct
Semanto Accuniy
Sa:BCE:
0.074
smBBUE 00.200
0.288
8.0
Qwen2.5-72B
QwenZ.5 72B--nsttuct
Semantic Accuray
1.0
SEIECE 0.041
SIICCE: : 0.059
SEICCE: 0.506
SIICCE 0.280
0.5
Confidence
60

---

Published as a conference paper at ICLR 2026
SimpleOA
@wen/ Qwen3 family
base-concise
buse-sentence
base-cot
instruct-coneise
instruct-senlence Qwen3-0.6B
instruct-col
Qwen3-0.688 (eeenn--.6B--aee Base
Somanti Accuncy
cmICB: 0.028
#aLCD:
0.5
Qwen3-1.7B4 Bwen3-i.7B-Buse
Qwen3-11m 1.7B
Semantc Accuray
cmECD
smBCB
0.5
Qwen3- -B-Bess
Qwen3-UB
Semante Accuray
snECE:
0.5
Qwen3-SBB B-Base
Qwen3-SB BBBB
Secantie Accuracy
00.033
SDECE:
0.5
Qwen3-14BB 14B
14B-Base Rase
Qwen 14B
Semantic Accuracy
SmFCF.
SnECE: 0.657
0.5
Qwen3-32B
Semantic Accurazy
0.0
0.5
0.0
0.5
1.0
Semantic Confidence
Semantic Confidence
61

---

Published as a conference paper at ICLR 2026
SimpleQA
google/genums-2 family
base-concise
base sentenen
base-coo cot
instruct-condise
instruct-sentence
instruct-cot
gemmaa-22 2-2b
gemmu--2220
Semantic Jecurncy
1.0
smBC. 1.046
smICC: 0.6b9
SmTCC 0.369
smBCE: 0.314
0.5
gemman-22 2-9h
gemmur-2 2-ae-----90 2--9biit
Samantic Accuncy
0.5
0.0
gemma--2227
gemmaa- 2----tttt
Semantic Accuncy
1.0
SERECE D,265
SIICE 10057
SEICEE D.250
SIICE 00233
0.5
Semantic Confidence
Semantic Confidence
Semantic Confidence
Semantic Confidence
SimpleQA
google/gemma- family
base-concise
base senteneeeeeeeee geinmu-3-1e 3-1-ppt smBCE: 0.094
base-pott
instruct-concise
instrucctsentence gemmaa---1-i
instruct-cof
Semantic Securaty
smFCC: 0.040
emFCE 0.277
smBCC : 0.193
gemmaa33 3 3-4b-pt
gemmaa 3-4b-it
Semante Accuneyy
smBCC : 00635
SmCCD 0.339
MmBC 0.347
0.5
0.0
genuma- 12b-pt
gemma-3 1--mma---12b-tt
Semant: Accuncy
1.0
SEIECE: 0.044
SIIECE: 0.040
sIICCE 0.363
SmIECE 0.354
0.5
gemma-3- 3--mma---227-p
gemma-3-22 2-27htt
Semantk Accuracy
0.392
0.5
0.0
0.0
0.5 Semantic Confidence
0.5
Confedence
Confidence
Semantic Confidence
62

---

Published as a conference paper at ICLR 2026
SimpleQA
mistralai/Mistee family
base-concise
sese-sentence Mistral-7B-
base-cot
instruct-concise
instruct-sentence
instruct-cot
Mistral-7F -i--rttruttt-
Secannte Accurac)
SIIECE 0.045
SIIECE 0.046
sMECEE 0.095
SIIECE 0.093
0.5
0.0
Mistral-7B-vd.
Mistral 7B-Instruct--0.3
Semantic Accurccy
SEnBLE 00.47
sMBBCE 0.054
smBLEE 10..63
suaBCE 0.154
Biiisstull-B--nsttreeee
Semantic Accurcy
smDCD 0.057
smbCC: 0.000
Disstrl-Smaall-ar Base-2501 1
Mistral- Small-24B- Fmall-24BB-nnstreef Instruct-2501
Semaztic Accuncy
SunECE: 0.043
0.001
0.373
0.437
8,5
0.0
Mistral-Large Laattta--Larg--nstrutt---- Instruci-2411
Semantic Accurccy
emFCC 112766
MmBCE 0.21
Mixtral-8x/hi -x1
Mixtral Mixtral-8x/BB-in -.
Semuntl Accuncy
Cm 0.353
smiEC 0.009
Sm 0.324
SCE 0.251
0.5
0.0
Confidence
Confidence
Confidence
Confidence
63

---

Published as a conference paper at ICLR 2026
Simple]A
Qwen/QwenZ 5-Math family
base-concise
mase-sentencee Qwen2.55 5-aath---hhh1-- 15
base-cot
instruct-concise
Soeen.5-MMath-11 Saauhh---B-Insstron instruct-cot
instruct-senfence
Semantic Accuacy
SEIECE 0.082
SIIECE: : 0.042
sMBCE 00.49
SuaECE 0.070
0.5
Qwen2.5- -
Qwen2 5-Math-TT--nnnn -B-nstruudt
Semaztlc Becuracy
smnDCC 0.302
smBCE : 9.06]
smdCC 00,30
smaCC 0.307
Qwen2.5-M8 5-Math-728
Qwen2.5 5-atoa----tth-28B5 I2s-ssruutt
Semaztic Accuncy
D 0.010
smiCC 9.039
Sm 0.250
smiCC 0.320
0.5
0.0
Confidence
Confidence
SimpleQA
NousResearch/ Hernes-e-emetittttteet -xx7B family
base-conccis
base-sentence
base-cot
instruct-concicn instruct-sentence
instruct-col
Mixttral-x7B-vo0
Nous-Hermes-2 2- Mixtral-8x7B
Senante Accuray
SILECE:
0.138
0.5
0.0
Mixxtral-8x7B-xll
Nous-Hermes- M-Mixtaal---x7---DDDD
Scnaztle Accuncy
C E 0.003
smiCC : 0.009
Cm MalCCEDb.. 0.111
smiCC 0.324
0.0
0.5
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
Semantic Confadence
Semantic Confidence
Semautic Confidence
Semantic Confidence
SimpleQA
family instruct-concier
base-concise
base-sentence Mistral-7B-vo., smICCE: : 07046
base-col
instruct-sentence Sephhy--b-dpoo
instruct-col
Semante Accunit
1.0
SEEECE I1:45
B 6..66
SuECCE : 0.164
0.5
Mistral-7B-
zephyr-7b 7b full
Semaztk Accurayy
1.0
CC 0.915
SmLCU 0.906
SC 0,000
ECU 0.067
0.5
0.0
Confidence
Confidence
64

---

Published as a conference paper at ICLR 2026
SimpleQA
meta- Saae--lla//Laaana family
base-concise
base sentence Llama-3
base-cot
instruct-concise
instruct-senteenc sentence
instruct-col
Ilama-3.1 -BB-Instruct
Semaztic Jecuracy
1.0
smalC 0.808
SmbCC 0,838
smlCC 0.180
smiCE: 0.339
0.5
Llama-3 3.1 1-70B
Llama- 3. 1 -7B--uustuuc
Somaniic Accuney
0.331
0.5
0.0
Llama-3 3.1-0B
Llama-3.3 1.3mm--333-700 3.3-tuutt
Semantic Accuncy
1.0
SmRCCF (0:51
smBCT (XX))
MmECC: 0,300
0.5
Semantic Confidence
Semantic Confidence
Semantic Confidence
Semantic Confidence
SimpleOA microsoft/phi-4 family
base-concise
base-sentence
base-cot
instruct-concise
instruct-sentence phi-4
instruct-col
Semmnntr Arcanncy
amBCE: 00.264
SMIRCE 0.156
0.0
0.5
1.0
0.0
0.0
U.5
Semantic Confidence
Semantic Confidence
Confidence
Semantic Confidence
65


Reviews:
[{"review_id": "NTp6CT6MRH", "reviewer": "Reviewer_ohGr", "summary": "The paper studies whether base LLMs (trained only by next-token prediction) are calibrated not just at the token level but at the semantic level for open-ended QA. It proposes a formal framework—$B$-calibration—that treats calibration with respect to semantic equivalence classes induced by a collapsing function B. Empirically, the authors estimate “semantic confidence” by sampling multiple generations for a question collapsing each answer to a semantic class (using an extractor) to obtain a distribution over classes, and then checking if confidence matches accuracy (standard reliability notion). The main claims are: (1) base LLMs show strong semantic calibration under their sampling-based procedure; (2) instruction-tuning breaks this calibration; and (3) chain-of-thought (CoT) reasoning also breaks it. A diagnostic LoRA experiment supports the mechanism: when a small LoRA can easily predict the model’s own semantic class distribution (low KL gap), the underlying model is better calibrated.", "questions": "Q1: OpenMath-Instruct2 is a dataset generally used for training (as it has more than >1M samples). Any reason to choose a train set for evaluation? How many samples did your test set of OpenMath consist of? I would rather suggest using MATH [2] for evaluation. I don't expect models to be able to solve questions from MATH in just a single word or sentence so I assume this might not follow the properties required by your theory but I still would be interested to see whether base models end up *not* being calibrated on MATH.\n\nQ2: The authors note that some datasets (e.g., TruthfulQA) likely violate in-distribution assumptions and may behave differently. This is a great point and I actually would be interested to see whether the predicted breakdown on TruthfulQA (or something else) happens empirically.\n\nQ3: For a given model and task (prompt), was the semantic classification performed on just 3 samples (1 for each prompting methodology)?\n\n[2] Hendrycks, D., Burns, C., Kadavath, S., Arora, A., Basart, S., Tang, E., ... & Steinhardt, J. (2021). Measuring mathematical problem solving with the math dataset. arXiv preprint arXiv:2103.03874.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "The paper is a pleasure to read and provides a nice interplay of theory guided empirical experiments to show the existence of semantic calibration in base models.  As I am not a theory person and don't have a background in multi-class calibration and its relation with loss functions, I am unable to assess the correctness of the main theorems. However, I was able to follow the intuition which broadly makes sense to me and connects well with existing literature. The main conceptual takeaway from the paper is useful and easy to follow.\n\n### Originality\n\n1. Provides a clear, general formalism—$B$-calibration—that parameterizes semantic calibration by an arbitrary collapsing function, decoupling the theory from any single definition of “semantics.” \n2. The theoretical link between $B$-calibration and local loss optimality over $W_B$ perturbations offers a principled mechanism for when semantic calibration should emerge in maximum-likelihood–trained LLMs.\n3. The paper focuses on open-ended Q/A which is important and useful going forward as prior work on MCQ is already well-studied. \n\n### Quality\n\n1. The mechanism yields intuitive predictions (base models should be calibrated; instruction-tuning and CoT harm calibration), and the paper reports experiments that align with all three.\n2. The LoRA diagnostic is a thoughtful probe: correlating the ease of predicting the model’s own semantic class distribution with empirical calibration quality is a useful sanity check that directly targets the proposed mechanism.\n3. The empirical measurement procedure for semantic calibration is clearly articulated: sample multiple completions at fixed temperature, collapse to classes, compute confidence and accuracy, then assess calibration. This is methodologically straightforward and reproducible in principle.\n\n### Clarity.\n\n1. The paper is very-well written and easy to follow. The writing is to the point and backed with relevant prior works. \n\n2. The sections are neatly organized and I thank the authors for providing an opening summary for each section on the content about to follow (giving clarity to the reader) and often providing intuition in simple words. \n\n3. The paper carefully motivates why token-level calibration is insufficient for long-form, open-ended tasks and explains the need for semantic calibration. The role of $B$ and the sampling-based estimator is described concretely. Fig.1 clearly shows the contribution of the paper in a simple and useful manner. \n\n4. The “from calibration to local optimality” storyline and the three-step mechanism (two proved, one heuristic) are presented in an accessible way.\n\n5. The plots for Section 5 are beautiful and experiments are well-executed. \n\n### Significance.\n\nIf base LLMs are often semantically calibrated under this operational definition, then knowing when to trust base-model answers via sampling becomes a practical tool, and the observation that instruction-tuning and CoT can degrade calibration is a nontrivial, actionable takeaway for practitioners who care about reliability.\nI look forward to seeing follow up work based on the concepts established in this work, especially towards verabalized confidence.", "weaknesses": "I don't have any issues with material covered in the paper which is thoroughly presented. The authors have presented a solid work worthy of acceptance. The only drawback I see is the limited applicability of this theory and the exclusive focus on their B-confidence calibration which the authors acknowledge in Section B.1.\n\nWhile this work greatly extends our understanding of calibration on a semantic level for base LLMs, the semantic level is still limited to at most a sentence-level (as longer reasoning would fall into the CoT category). The authors state in footnote 3 that, \"We will however assume that the latter holds (which is plausible if each evaluation distribution is a reasonably-sized sub-distribution of the pretraining distribution on which local-loss-optimality holds).\" \n\nThis assumption assumes evaluation distribution to be already covered in the pertaining distribution. While this often done targetedly by model practitioners, it is unreasonable to expect test query to be already covered by training data. LLMs may be used for OOD tasks and also benefit significantly from inference time-compute (as it is unreasonable to expect an LLM to produce an answer, especially for challenging tasks, in a single forward pass). Thus, to me, it is not surprising that token-level calibration can lead to sentence-level (in this work, *limited to a single sentence*) calibration as the model is saying the same thing but instead of spiting it out directly, it just says the same thing in a sentence. What is more interesting is whether models are/can be made calibrated on long-form generations [1]. I say this more towards future work as the paper is already dense and contains a sizable contribution!\n\nMinor: some plots are missing in Appendix figures like on page 44 for phi-4 and page 46 for Qwen3.\n\n[1] Band, N., Li, X., Ma, T., & Hashimoto, T. (2024). Linguistic calibration of long-form generations. arXiv preprint arXiv:2404.00474.", "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": "tHWujtzdRU", "reviewer": "Reviewer_SbxL", "summary": "The paper answers the question whether large autoregressive LLMs provide meaningful confidence about the semantic content of their long-form outputs (not only next-token probabilities). This is important both scientifically (do models know what they don't know?) and practically (calibrated confidences enable safer decision-making).", "questions": "1. Experiments use four datasets (GSM8K, OpenMathInstruct-2, TriviaQA, SimpleQA). The theory relies on an \"in-distribution\" assumption (Claim 10 / Corollary 11) and the paper notes datasets like TruthfulQA may violate that and be miscalibrated. But many real-world culturally/evolving phenomena (memes, internet-slang, adversarial misinformation) are OOD and precisely the settings where calibration matters most.\nThe practical value of \"semantic calibration\" is highest in long-tailed, dynamic, and OOD settings. Showing calibration only in relatively conventional in-distribution QA may somehow limit applicability.\n2. The empirical semantic confidence measurement requires many samples per prompt (M=50 at T=1) and then applying the B pipeline, which may be costly and may be impractical for real-time use. \nIt is suggested to provide experiments or analysis on how calibration estimates change with M (e.g., M=5,10,20,50) and temperature — to show whether cheaper approximations are viable. \n3. It is not yet clear which aspects (objective mismatch, dataset selection, reward shaping) cause the breakdown. The DPO result is interesting (DPO model miscalibrated vs SFT-only less so) but needs deeper analysis. You may add some controlled ablations within a single model lineage varying (a) SFT size and loss (proper vs improper), (b) small RLHF/DPO steps with controlled reward shapes, and (c) temperature / decoding strategies. Report which factor correlates most tightly with calibration degradation.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. Parameterizing calibration by an arbitrary collapsing function B is a neat, flexible formalism that connects sampling-based semantic confidence to established calibration literature. This lets the authors transparently say which semantic granularity they evaluate.\n2. There is a solid theoretical contribution linking diverse prior work.The equivalence (Thm.6) between B-calibration and local-loss-optimality is a meaningful bridge from optimization theory to semantic calibration; Thm.9 provides a plausible route for autoregressive implementation. \n3. Multiple model families, base vs instruct variants, three prompting styles, and the LoRA probe present a coherent story that ties theory and practice together.", "weaknesses": "1. The chosen datasets primarily focus on an in-distribution assumption, which may limit the generality of the proposed method.\n2. The claim  \"instruction-tuning breaks calibration\" may require deeper analysis.", "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": "s2zxU7lOSf", "reviewer": "Reviewer_2hGk", "summary": "Authors proposes a theory explaining how large language models (LLMs), despite being trained only on next-token prediction, can still exhibit semantic calibration—the ability to assign meaningful confidence to the content of their answers. The authors introduce B-calibration, a framework that measures calibration with respect to any “collapsing function” B that maps textual outputs to semantic classes. They prove that B-calibration is equivalent to local loss optimality, meaning a model is calibrated if its loss cannot be reduced by simple perturbations of its output distribution. Empirically, they show that base LLMs (e.g., Mistral, Qwen, Llama) are semantically calibrated across open-ended QA tasks, while instruction-tuning and chain-of-thought reasoning often break this calibration—since these processes alter the model’s ability to anticipate its semantic output before generation. The findings suggest that semantic calibration naturally arises from likelihood training, providing a principled mechanism linking uncertainty estimation to optimization dynamics", "questions": "The paper assumes that base LLMs are “locally loss-optimal” with respect to simple perturbations. Could the authors provide empirical evidence or a small-scale ablation demonstrating this property, even approximately?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "It introduces the notion of B-calibration—a general framework for defining calibration over arbitrary equivalence classes of outputs—which unifies token-level and semantic-level calibration under a single formalism. This perspective reframes an underexplored question (“can base LLMs meaningfully assess confidence in their answers’ meanings?”) into a rigorous, testable problem, providing a clear theoretical bridge between semantic uncertainty and local loss optimality. \n\nThe work’s quality is high: the theoretical arguments are well-grounded in recent calibration theory, the proofs (especially the equivalence theorem and its autoregressive extension) are technically sound, and the experiments are extensive, covering multiple datasets, model families, and prompt styles.", "weaknesses": "Although the authors present calibration as emerging from the ability to “predict one’s own semantic output distribution,” this remains correlational. The LoRA probe experiment (Claim 10) demonstrates correlation between learnability and calibration but not causation.\n\nThe paper claims novelty in unifying calibration and loss-optimality, but related work in multi-calibration and conformal prediction is discussed mainly in the appendix. Bringing these connections into the main text—perhaps as a discussion of how B-calibration extends or simplifies existing frameworks—would situate the work more clearly within the calibration landscape.", "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": "W0oNUDQDZC", "reviewer": "Reviewer_mX9b", "summary": "This paper introduces semantic calibration for LLM, where a model’s probability over semantic answer classes matches its empirical accuracy. It proposes a theoretical analysis to link calibration and local loss optimality. Experiments show that instruction tuning and Chain-of-thought (CoT) hurt LLM’s ability of semantic calibration.", "questions": "Please refer to my comments in weakness.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. This paper proposed semantic calibration instead of calibration over token-level confidence.\n\n2. The theory can explain the relation between loss optimality and calibration performance.\n\n3. The observations are very interesting. It is quite surprising to me that instruction tuning and CoT hurt the performance of semantic calibration.", "weaknesses": "1. The conclusion that CoT breaks calibration is drawn mainly from math datasets.\n\n2. There is a misalignment between theory and experimental results. The theory aims to reveal that a LLM is calibrated if it’s locally optimal under its loss function, but it cannot explain the interesting observations as (1) base LLM is the best for calibration, (2) instruction tuning and CoT hurt the performance of semantic calibration.\n\n3. Any method to alleviate the drop of calibration performance for instruction tuning or CoT?\n\n\n4. Some Typos:\n\n     4.1 In abstract, we introduce “B-calibration,” a notion... ->  we introduce “B-calibration”, a notation of…\n\n    4.2 In Equation 1, is it formal to use math notation like $#$?", "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)