Inv-Entropy: A Fully Probabilistic Framework for Uncertainty Quantification in Language Models
Authors: Haoyi Song
Ruihan Ji
Naichen Shi
Fan Lai
Raed Al Kontar
Venue: NeurIPS 2025 poster ()
Keywords: Uncertainty Quantification
Probabilistic Modeling
Random Walks
Perturbations
Large Language Models
Primary area: deep_learning
TL;DR: Uncertainty Quantification for LLMs

Abstract: Large language models (LLMs) have transformed natural language processing, but their reliable deployment requires effective uncertainty quantification (UQ). Existing UQ methods are often heuristic and lack a fully probabilistic foundation. This paper begins by providing a theoretical justification for the role of perturbations in UQ for LLMs. We then introduce a dual random walk perspective, modeling input–output pairs as two Markov chains with transition probabilities defined by semantic similarity. Building on this, we propose a fully probabilistic framework based on an inverse model, which quantifies uncertainty by evaluating the diversity of the input space conditioned on a given output through systematic perturbations. Within this framework, we define a new uncertainty measure, Inv-Entropy. A key strength of our framework is its flexibility: it supports various definitions of uncertainty measures, embeddings, perturbation strategies, and similarity metrics. We also propose GAAP, a perturbation algorithm based on genetic algorithms, which enhances the diversity of sampled inputs. In addition, we introduce a new evaluation metric, Temperature Sensitivity of Uncertainty (TSU), which directly assesses uncertainty without relying on correctness as a proxy. Extensive experiments demonstrate that Inv-Entropy outperforms existing semantic UQ methods.

Inv-Entropy: A Fully Probabilistic Framework for Uncertainty Quantification in Language Models
Haoyi Song University of Michigan haoyiso@umich.. edu
Ruíhan
Naichen Shi Northwestern University naichen. Shiinorrthestterneeee
University of Minnesota j10002340umn edu
Fan Lai
Raed Kontar" University of Michigan al alkontarrmiicoo edu
University of Illinois Urbana-Chann Thbana-Champaign
Abstract
Large language models (LLMs) have transformed natural language processing, but their reliable deployment requires effective uncertainty quantification (UQ). Existing UQ methods are often heuristic and lack probabilistic interpretation. This paper begins by providing theoretical justification for the role of pertur- bations in UQ for LLMs. We then introduce dual random walk perspective, modeling input-output pairs as two Markov chains with transition probabilities defined by semantic similarity. Building on this. we propose a fully probabilistic framework based on inverse model, which quantifies uncertainty by evaluating the diversity of the input space conditioned on given output through system- atic perturbations. Within this framework, we define a new uncertainty measure, Inv-Entropy. A key strength of our framework its flexibility: supports vari- ous definitions of uncertainty measures, embeddings, perturbation strategies, and similarity metrics. We also propose GAAP, a perturbation algorithm based on genetic algorithms, which enhances the diversity of sampled inputs. In addition, we introduce a new evaluation metric, Temperature Sensitivity of Uncertainty (TSU), which directly assesses uncertainty without relying on correctness as proxy. Ex- tensive experiments demonstrate that Inv- unv-Entrppy outperforms existing semantic UQ methods. The code reproduce the results can be found at Latn/UUUnDSSSSSimooaaaa//aaaaarn hwwpaaittttttunniiiiaaaaea
1 Introduction
Large language models (LLMs) have demonstrated remarkable success in various natural language processing tasks, such as text generation, question answering, and summarization [Brown et al. 2020 Chowdhery al. 2023 Touvron et al. 2023). These models have pushed the boundaries of what is achievable in language understanding and generation [Chung et al. 2024 OpenAl 2023 However, despite their impressive capabilities, significant challenge remains: LLMs tend to hallucinate, generate confidently wrong predictions [Maynez et al., 2020 Zhang et al. 2024b). This is a serious concern in applications where reliability paramount, such healthcare, autonomous systems, and legal domains, where incorrect outputs can have dire consequences []i et al.] 2023). Addressing these challenges essential for ensuring the reliable and responsible deployment of LLMSs scale, ultimately unlocking their full potential in real-world applications.
"Corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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A central step in addressing these limitations is developing effective measures for UQ, enabling LLMs acknowledge their confidence in a generated output. Existing UQ approaches rely predominantly on heuristic consistency checks. Most commonly, they use the model's own generation likelihood or perplexity as a proxy for confidence, or they measure dispersion across multiple sampled continuations
(e.g. via n-gram overlap or embedding-space variance) [Mudumbai and Bell 2024). Such likelihood- measures, however, lack a grounded probabilistic justification. Also,
based and sampling-base token-level probabilities are known to dramatically under-estimate uncertainty as models can be "confidently wrong" (Jiang et al., 2021] and they are often inaccessible in black-box LLMs.
To probe deeper LLM brittleness, recent work has turned to pnput-petturbation UQ methods. Variant prompts are created through paraphrasing. adversarial token insertion, or temperature shifts. Output sensitivity is then quantified as an uncertainty signal [Gao et al. 2024 Tuna et al.) [222] SSeböökk
at. 2019]. For instance, [ao et al. [2024) randomly perturb both prompt wording and sampling
temperature, then aggregate variation across outputs to flag unstable predictions. (Tuna et al. [2022]
applies adversarial paraphrases to uncover "blind spots" where small comantic-preseryy edits trigger
large output changes, while SSebööck et al. [2019) uses systematic character-level and word-level corruptions to map regions of high model vulnerability.
Despite these advances, an ab initio probabilistic framework has not been established, as existing methods mainly rely on heuristic score functions for UQ. To address this, we introduce Inv-Entropy. a fully probabilistic framework built on candom-walk theory that offers a new perspective: it learns the statistical connections between LLM inputs and outputs. This is accomplished through structured perturbations, which simultaneously capture input variability and the influence of different inputs on model predictions. Our contributions are summarized as follows:
1. We present the first work in UQ for LLMs that adopts a fully probabilistic framework. grounded in random walk theory. This framework is highly flexible, and its probabilistic nature allows for the use of various UQ measures. It is also intrinsically capable of handling black-box models, as it does not rely on token probabilities.
2. We introduce an inverse perspective that quantifies input diversity given an output, inspired by "Asymmetry in Semantic Variability" (defined later). Extensive simulations highlight its advantages, especially for short-form answers where traditional methods often struggle.
3. Theoretically, our work provides a principled foundation for using perturbation- based methods for UQ, justifying the recent momentum behind such approaches.
4. We propose GAAP, novel perturbation algorithm that enhances input sampling diversity. Empirical results show that GAAP significantly improves perturbation- based UQ
5. Finally, we introduce new evaluation metric. Temperature Sensitivity of Uncertainty (TSU) capable of evaluating uncertainty without relying on correctness as proxy. This enables evaluation of UQ on any dataset, even when labels are unavailable.
For the remainder of the paper, we use [E|-] to denote expectation and V denote variance. We note that a more detailed related work section can be found in Appendix A
Original Question
Perturbated Question
Perturbated Question 2
Parturbated Question 3
1950 Arns Knuumlone signedd lucrative contract to model
What Anros Kournilenn sign profitabiis agreemment showcase 19999
1960, Anna Koum kave surred west paying contract promosc which product?
Anne Koumilooss intedd financillyy uily reverding deal
Perturbation
endorse
LLM
LLM
LLM
LLM
Scors
Soons Swiimsut
Sporto Hauctono
Sporn Swimeuh
Busrond heue
Ad
K-Swiis
Warches
Iaaue
houe
Misleuding Low Unsertaintty
High Uncertainty Captured by Perturbation
Figure 1: Toy example highlighting the importance of perturbations. The original question is from TriviaQA Joshi et al. 2017]. and the correct answer is "bras." The responses are generated by ChatGPT-3.5 Turbo. Input perturbations reveal hidden variability that multiple sampling (i.e., replications) alone fails to capture, as replication alone can be confidently wrong.
2

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2 Perturb-then-Quantee
2.1 Why perturb the input?
We begin with a simple illustrative example in Fig. to highlight the importance of imput perturbations. In this example, when the input question is simply replicated (as is common in much of the existing literature [Lin et al. 2024)) ) all generated responses are identical. This misleadingly suggests low uncertainty even though all answers are incorrect, thus providing false confidence. Replication alone therefore insufficient to capture the underlying variability. In contrast, applying input perturbations yields diverse of responses across semantically equivalent prompts, enabling more faithful characterization of uncertainty. The importance of such perturbations is not only evident from this example but also supported by simplified theoretical argument in what follows.
To build theoretical insight into the importance of perturbations, we leverage a key property specific to language: semantic equivalence. An ideal LLM should respond consistently semantically equivalent inputs. We use this property, in a simplified setting, to construct imput perturbations within equivalence classes, which allows reason about uncertainty. We start from a proof-of-concee example where the target function f* is single output function f IR -> IR. We lack access f*, but have access to an approximation f: IRd R, such as a pre-trained model. At first sight, it seems hopeless to quantify the alignment between f and f* without knowing f However, the semantic equivalence class relative to an input To provides set [xro) S IR such that f=xx" provides valuable information for UQ as shown in the subsequent argument and Figure
expose inconsistencies in model behavior and formally
(zo), Va E I(xo). The equivalence class
The shape of ((ro could be complex for general f". However, in a small neighborhood of zo where (xo) * 0, ((ro) should be locally close the tangent space of f More specifically, {zo (1 mT.r) 2; R²,, where f"
define tangent invariance set Zangen((Z)
(("( Intuitively, T. acts the orthogonal projector onto the subspace orthogonal to (fo)))). Taylor expansion shows (zn) 2 I(zo) when restricted to the small neighborhood
[zo)]
of zg- To generate algorithmic insight, we further define probability density
Irangent (zro): we
use P(x;; (io ar) to denote the probability density function of x' = Fo + (T "/vr+))= where
sampled from d-dimensional isotropic Gaussian distribution (O, o o²). It is easy to verify that P(: o) is a Gaussian distribution supported on Iraggen (Xo) whose mean is Zo-
The following Lemma shows that estimate the angle between V fzx) and V (x) by examin- ing the variance of fxx',, where x' is sampled from P(x/;ær,)) o).
Outputs change
Perturb
Figure 2: Left: Conceptual illustration of level sets of the ground truth f We perturb the input mo to æ1, 22. along a schematic isocontour such that ((((r)) f(( .... Right: Conceptual illustration of level sets of the model f. The deviations of f(x)) for from f(ou) reflect the model's uncertainty around To.
Lemma 2.1 Assume (1) f is twice differentiable, and both [=)(()) and (x)(l))) are bounded for all x E Rdd and (2) V ( ((x) # 0 and V (xx) + O. Then, for sufficiently small a, we have V P [f(2(((())))
| [(0)) sin' (Vf(zo), V + O(a),
(1)
where (11 U2) = arccos
0g || 12
denotes the angle between two vectors U1 and 02-
Equation shows that the variance of function values of f on the invariance set is indicative of the alignment between V and V When the variance is larger, if respects the invariance of f less,
3

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which in turn implies larger misalignment between ff and Vf* and larger uncertainty. The full proof of Lemma i. relegated to the Appendix B
Despite its simplicity, Lemma 2.1 demonstrates two important elements of perturbation- UQ: input perturbation and output variance evaluation. In what follows, we introduce concrete designs that implement these elements in the LLM context to effectively characterize uncertainty.
2.2 A probabilistic framework via dual random walks
The variance estimate Lemma presents useful insights yet is oversimplified to realistically model LLMs, whose outputs are token sequences rather than single scalar. Also, it is difficult to sample exactly from a semantic equivalence class. To tackle these challenges, we introduce a probabilistic approach inspired by Markov chains.
We Sag denote semantic input LLM f. and Syo + ((S.)) denote one of its possible corresponding outputs. Because f is typically stochastic, the output Sno is random variable. Next, consider semantic embedding function that maps both inputs and outputs into continuous space X, such that (Io. 36) = (v (s.ss,, (8Sr))) We also assume perturbation algorithm Per(S2,) that produces finite of perturbed inputs:
Per(Saa) {5 Soy..... S.n, }
whose detailed implementation will be described in Sec. 2.6
Applying the embedding function (S..)) we obtain the embeddings:
X, = { .i...., where x; (S,,).
One corresponding output embedding set hence given as
Yn = (1o- ,1,..., BnJ, where y; ()S) = (((s.))))
Samples in X,, and Yn one-to-one correspondent. We thus estimate the structural similarity of the samples in the two sets. the following, we propose probabilistic approach to characterize the structural similarity through the lens of a random walk.
We define two Markov chains, M, and My, over the same state set S = {0, 1, 1.....,... n). where each state i corresponds to perturbed instance Sr,- The chains differ in their transition dynamics: MM built from similaritics among input embeddings Xn. and My from similarities among output embeddings Yn. Let USimilarity (2, a'): : X' x X + I denote non-negative similarity function that measures the closeness between embeddings in X. Using this function, we define the transition matrices Pa., Py 1((++))((+1) for Mr. and My elementwise as:
ASimilarity (Zi, xz))
OSimilarity (2i, Mj)
Py[i)]]
(2)
O== (Siillarity (344
k= USiiilryity (3/s ykk)
The two transition probability matrices characterize two random walks in the space of + pairs (zi, Vi ==o- where the transition probability from i to is higher if their input or output semantic features are closer. Notice that variety of well-established methods have been proposed for (Similarity some of which are deployed in our numerical studies (see Sec.
At its core, 22 defines stochastic dynamics on the S, capturing the similarity structures both the input and output spaces. These dual dynamics uncover meaningful semantic patterns that can be leveraged for UQ There various ways characterize uncertainty by examining the alignment between the two induced graphs (Vishwanathan et al. 2010]. In what follows, we construct framework tailored for discrete input and output spaces rigorously define uncertainty.
2.3 Constructing the distributions
We use X and Y to denote discrete random variables whose supports are X,, and Y.. respectively. There are many possible ways to define possible distributions of and Y. In the following, we will introduce one design of P(Y) and P(X YY) based on (2). For notational simplicity, we denote the uni- form distribution over all states S by π Unifeem which is given by π Uniform
x+1

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The marginal distribution The random variable Y corresponds to the LLM's response to question that perturbed from the original question z0-- We define the marginal distribution of Y as:
P(Y = Us) Unikorn
Pulbi
GSimilaity (3i, Mi)
(3)
Qyamitrity (9h, Hi)
where notation [j] denotes the j-th element of vector.
The distribution (3) has an intuitive interpretation: we randomly sample a point uniformly from the state space S, then randomly transit the sample with P, for one step. After the transit step, nodes whose corresponding output samples are surrounded by many similar outputs are assigned higher probability, while isolated nodes with fewer similar neighbors are assigned lower probability. Therefore, the mass is concentrated on regions of high semantic density in the output space.
The conditional distribution
Now, we introduce the conditional probability P(X = 2, Y yj)
OSiillarity (I, 2Kk)
Similarity (y Mkb)
P(X = =8==x/Y= = =y,) (PyPa)li,
(4)
(imillriyy (Tm T&k
OSimitarity (1/z D1
where notation [i denotes the (j, i)-th entry of the matrix PyPs (x+++))((+1)))
Essentially, 4 defines the conditional probability
Py LLM Correspondence
Ps
through composite transition dynamics the state space S. We design a two-stage random walk: first, states transition under p,, capturing output similar- ities; then, they transition under Pz. capturing in- put similarities. Since both P. and Py operate on the same state space S linked by the LLM, (4) es- tablishes probabilistic bridge connecting similarity structures in the output and input spaces (see Fig.
We explicitly model the conditional distribution of inputs X rather than outputs y for two main rea- sons. First, the perturbed samples generated by (er(S may differ semantically from the original input. Reweighting these samples using both input- and output-space similarities ensures that semanti- cally consistent perturbations are emphasized while spurious ones are nown-weighted Second, LLMs exhibit an inherent semantic asymmetry between in- puts and outputs: many distinct prompts can lead to similar responses, whereas small input changes typically cause only small to modest output variations. Modeling P(X Y) therefore provides more stable and informative view of uncertainty by capturing the diversity of possible inputs that could relate given output.
Figure 3: Kandom-walk transitions under-
lying P(X YY PyPa. Highlighted blue paths show two representative transitions (one through i: and through n), each following Pv
LLM
IK Pe Fi:-
Vj
Uk
Our goal is to use the information encoded in Y to guide the conditional distribution of X. This coupling between input and output similarities allows the model to assign higher probability to inputs supported by multiple semantically consistent input-output pairs, yielding a more faithful representation of uncertainty. Based on the defined P(Y) and P(X Y), we can then derive P(X) & Uniformpp y. and P(Y X) via Bayes' theorem. Such formulation provides a flexible foundation for defining diverse uncertainty measures, including divergence-bast metrics (e.g., KI. or Wasserstein distances), entropy- based quantities, and other probabilistic constructs. In the next section, we introduce an entropy-based metric as concrete example.
2.4 Inv-Entropy via bootstrapping and Monte Carlo
We next introduce how to leverage the probabilistic framework described above define our UQ measure, denoted as Lnverse-Entrops (Inv-Entropy))
5

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Algorithm 1 Iny-Entropy overall framework
Input: (Sngs Syo), ), f. 1. Per, and @Siillrrity
2: Perturb: Use (S) and t to obtain X,, Compute Pz using 2
3: LLM Generation: Input each question in Per(Sza): r times into f and obtain (R, {
4: for b = to B do
5: Sample Y,
(30
(b)
6: Compute p(b using
y 7: Compute p() (x, (b:)
(P()) P.) (from (4) for V
8: Compute (X,, Y:!
using 5
9:
end for
10: Compute (x|r) using 6
Inv-Entropy A natural measure of uncertainty is the conditional sample entropy (X,,, Yn)). which connects the similarity of the input set X,, to the similarity of the corresponding output set Yn:
H(x, Yn) A - trace log(P,P:)) = Σ P(r ws) log P(#: vs).
(5)
i=0
where ④ denotes the Hadamard product. This quantity captures the degree of alignment between P. and P.. High entropy indicates that semantically similar inputs yield divergent outputs, that semantically distinct outputs correspond to similar inputs, both revealing uncertainty in the model's behavior. Note that Inv-Entropy represents unnormalized entropy measure, designed to capture not only the dispersion of P(x; ys) but also its magnitude, thereby jointly modeling these two sources of uncertainty.
That said, it important to realize that X,, does not map to a unique Y, due to the inherent stochas- ticity of f as LLMs can produce different outputs for the same input. Thus, beyond perturbations that capture epistemic uncertainty, replications can help account for aleatoric uncertainty arising from sampling randomness. To this end, each perturbed input Sz. E (er(S2.)) can be queried r times, yielding replicated output set R, = (yA.1..... Dirr) for each question I;, where yt = e (f(..))
Bootstrapping & Monte Carlo With this setup, we are able to generate B bootstrap samples (b)
(b) } for E {l,..., B}, each corresponding to Xn, where y/ R2 is drawn y (vo y'
with replacement for E (O,....n), n.), yielding a transition matrix p(a), Each bootstrapped output embedding set Y!!²). together with the input embeddings Xn, defines a probabilistic instance of our uncertainty measure, Inv-Entropy. The final UQ estimate is obtained by averaging Inv-Entropy over the B bootstrap replicates:
& 1 Σ B
A Y)
HX,, YY
(6)
Our overall framework in given in Algorithm I is important to emphasize that key strength of our framework lies in its flexibility: it accommodates arbitrary choices of embedding function tb perturbation strategy Per, and similarity metric asimilarity. allowing it to adapt different tasks and model architectures. Moreover, [[ Y) represents just one possible uncertainty measure; many others can be defined within our probabilistic framework. For instance, our numerical studies evaluate alternatives such as the Wasserstein distance between the marginal distributions P(X) and P(Y).
2.5 Insights into Inv-Entropy
To provide some insights into our UQ measure, we introduce two parameters: eg E (0, 1]. which controls the input perturbation level, and ey [0, 1], which controls the output dispersion level (defined via ASimilarity below). The corresponding transition matrices are then readily derived as:
[1,
i=
(n + 1) nE;
j.
@Similarity (21,
1 E2y i
fz (n + 1) ne;
i j

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where 2 E {a, 8). With this, the joint conditional probability can be written as: P(z, 133 Ez Cy) Pa(C)))a(Cs))
++n=ne; tRe We can then show that for the same input perturbation level 6x, larger output dispersion by corresponds to higher uncertainty, consistent with the definition of Inv-Entropy. To see this, notice that the derivative of P(r; y/3 <z; fy) with respect to 6y is aP(silpees.e))
> Now, if we assume all P(x; 16s fas Exrty) are smaller than condition generally satisfied for large n), then the Iny-Entropy function H in becomes also increasing in P(æ; Mr Ez. Cy) and hence in Cy.
dey
(i++1 (nttIne)]in-I--c.)
2.6 GAAP
In this section, we present genetic algorithm-basee based adversarial perturbation (GAAP) that progres- sively modifies the semantic input So, to generate controlled perturbations, Per(Sna) Below we highlight our overarching framework, while algorithmic details are relegated to Appendix [ As shown in Fig. 4 the process consists of an initialization step and multiple iterative procedures. In the initialization step, we construct population Pop (Sza) consisting of perturbed versions of Sav.
More specifically, each text Popo (S is derived from Szo by replacing one keyword with synonym, hypernym, hyponym from WordNet [Miller 1995]. or a deletion.
Initialization
Sxo gulla only hissey Briid charpioraiipo
Randomly replace synonyms, hyponyms, and hypernyms.
vipl crowe tnle
Exionns
Sample
Iterative evolution
Crossover
Mutation
Figure Illustration of GAAP on TriviaQA [Joshi et al. 2017] question.
With an initial (S..) GAAP updates the (S..) ) through subsequent steps of crossovers and mutations. In the crossover step, we first select a random subset of Pop, (Sx,)) based on (Similarty(zo:)) such that samples are chosen with higher probability if they closer 2o. Next, we randomly segment each selected sentence. These sentence segments are then randomly concatenated to generate new sentences, as illustrated in Crossover of Fig.
In the mutation step, we perturb the recombined sentences with further key word substitutions and deletions, which introduce additional variations to the cross-pollinated texts. We construct the next generation population P (Szz) as the union of the selected, crossovered, and mutated texts.
GAAP proceeds by iteratively updating Pop, (Szo) with crossovers and mutations for T' iterations or until all texts in (S.,) have similarity to Se, smaller than predefined constant 5. Finally, we construct Per(Sz)) by sampling from populations
different generations (Pop, (S2g) )==00 where T E Z is fixed gap. Since texts in Рop, (Sp, ($2.) tend to deviate from Sz, further with larger t, such construction of (Sr((2z)) ensures diverse representation of perturbed texts with different levels of similarity to Szg.
Experiments
Models and tasks We conducted experiments using two language models: GPT-3.5-Turte black-box model accessed via API, and -LLLMA-..-----Inntt a grey-box model. We evaluated our framework on datasets spanning three categories: question answering (TriviaQA [Joshi et al..
2017], SciQ WWebbl et al.) (017]. Natural Questions [Kwiatkowski et al. 2019 (NQ, long-answer questions with details in Appendix (D.3)), multiple choice (MMLU [Hendrycks et al. 2020) ) and
mathematical reasoning (GSM8K [Cobbe et al. 2021))

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Table 1: Comparison of AUROC, PRR, and Brier scores across all the 5 datasets. We use GPT-3.55 Turbo with ChatGPT-based paraphrasing, and xalag MNLI embedding function. Bold and underline denote the best and second-best performers, respectively.
Datasets
Metric
Method
TriviaQA
SciQ
NQ
MMLU
GSM8K
Semantic Entropy
0.579 =: 0.844
0.679 1: 0.045
0,521
0,004
0.518 = 0.045
0.589 4: 0.082
Kernel Entropy
0.687 0.062
0.685 0.033
0,556
0.055
0.653 = 0.099
0.560 ::
VU
0.695
0.480 = 0.060
0,533 0.056
0.523 = 0.054
0,557 #
P(True)
0.604 =
0,522 + 0.026
0.519 0.000
0.474 = 0.027
0.571 + 0,598
LexSim
0.649
0,681
0.518 0.055
0.643 0.054
DegMat
0.734
0,672 0,089
0.551
0.052
0.608
0.058
0.678 1: 0.059
AUROC (1)
LUQ
0.637
0.726 0.048
0.627 0.055
0.648 0.057
0.662 + 0.064
KLE
0,333 0,054
0,341 0.056
0.410 0,060
0.360
0.064
0,338 1. 0.061
unv-Entropy
0,788
0.740
0.661 0,052
0.780
0.695
NI-Entropy
0.786
0,68
0.066
0.637
0.710
0.052
0.650 :1 0.069
NR-Iny-Fntropy
0.743 0.061
0.720 0.049
0.627 0.054
0.604
0.059
0.677 + 0.064
WD-px-py
0,518 0,080
0,303 0.060
0.558 0,055
0.573
0.061
0,605 1: 0.069
MAX-py-x
0.723 0,054
0,674 + 0.054
0.547 0,051
0,585
0.059
0.618 :: 0.059
Semantic Entropy
0.517 =
0.763 =
0,505 0.049
0.690 =
0.335 +
Kernel Entropy
0.794
0,812
0.573
0.768
0,333 1:
VU
0.723
0.677 :
0.537 0.053
0.654 0,055
0.328 :: 0.057
P(True)
0.797 0,842
0.679 0.050
0.502 0.050
0.671 0.043
0.303 + 0.056
LexSim
0,810 0,045
0,770 0.051
0.563 0,064
0.767 0.053
0,356 1 0.076
DegMat
0.882 = 0.041
0,802
0406
0.549 0.009
0.771 0.058
0.462 + 0.091
PRR (1)
LUQ
0.854 =: 0.043
0,840 = 045
0.595
0,066
0.787 = 0.052
0,504 +: 0.094
KLE
0,704 0.845
0.592 0.059
0.449 0.009
0.612 0.163
0,224 + 0.043
Inv- ow--Etroppy
0,885
0.853 0.042
0.614 0.067
0.898 0.050
0.521 + 0.094
NI-Entropy
0.883 0,043
0,781 0,053
0.592 0,064
0.823
0.055
0,501 +: 0.088
NR-Inv-Enne
0.840 0,054
0,844
045
0.576 0,069
0.743
= 0.064
0,518 1: 0.087
WD-px-py
0.763 0.051
0.587
0,056
0,586 0.065
0.777
0.054
0.420 + 0.085
MAX-py-1 -X-x
0.875 1: 0,038
0,821 1 0.048
0.536 0,066
0.749
0.062
0,413 .. 0.081
Semantic Entropy
0.166 0,023
0,173
0.029
0.242 0.006
0.208
0.020
0,188 0.018
Kernel Entropy
0,160 0,025
0.153 0.022
0.221 0.011
0.179 1. 0.018
0,190 1 0.017
VU
0,160 0,022
0.196 0017
0.223 0.014
0.219 0.017
0,188 1. 0.020
P(True)
0.172 = 0.022
0,215 + 0017
0.244 0.005
0,215 = 0.015
0.1 0.021
LexSim
0.151 = 0.124
0,179 + 0.220
0.225 0,000
0.187 Q.020
0,174 0.019
DegMat
0.140 1: 0.021
0,164 0018
0.229 1 0.012
0.191 1: 0.018
0.156 11 0.019
Brier (U)
LUQ
0.148 + 0.020
0.159 + 0016
0.208 + 0.014
0.180 = 0.019
0.151 + 0.019
KLE
0,188 0,021
0,218 +: 0.016
0,244 0,006
0,213 = 0.018
0.193 4:: 0.021
Inv-Entropy
0.128 0.120
0.157 0.18
0.201 0.014
0.147 0.017
0,152 # 0.020
NI-Entropy
0.124 0,020
0,164 1.64_00 0.077
0.204 1 0.014
0.168
0.021
0,156 1 0.022
NR-tnvyFattropy
0.138 = 0.021
0.159 = 0015
0.208 0.013
0.188
0,021
0.165 : 0.021
WD-px-py 0.184 = 0.019
0.212 = 0016
0.225 1 0.010
0.888
= 0.018
0.169 0.021
MAX-py-x
0.148 0.119
0.177 + 0.017
0.229 + 0.011
0.189 0.120
0.169 + 0.018
Baselines We compared our method with various state-of-the-art benchmarks highlighted in Sec.
Appendix A and recent paper (Vashurin et al. 2024 identifying them top-performers. These
include: Semantic Entropy [Farquhar et al. 2024]. Kernel Entropy [Gruber and Buettner 2023]
Verbalized Uncertainty (VU) [Tian al. [023]] P(True) [Kadavath al. 2022]. Lexical Similarity
(LexSim) Fooicchva et 2020), Degree Matrix(eeMaat) [Lin et 22.44) Long-text Uncertainty
Quantification (LUQ) [Zhang et al. 20244], Kernel Language Entropy (KLE) [Nikitin et al.) 2024
We also include additional UQ measures based on our framework: (i) NI-Entropy: Non-inverse
entropy which uses P(Y X) derived in Sec. 2.3 instead of P(X Y); the rest remains the same.
(ii) NR-Inv-Entroop entropy in (5) without replications. (iii) WD-px-py: Wasserstein distance WD(P(X), P(Y)) (iv) MAX-py-x: max; P(y
Evaluation metrics We evaluate performance using four metrics grouped into correctness-based and uncertainty-basse categories. Correctness-baset metrics: AUROC, PRR Malinin and Gales 202T), and Brier Score [Brier T950), measure how well confidence aligns with correctness. For MMLU, correctness is defined via exact match, while for other datasets we use GPT-3.5-Ture to
8

---

assess whether generated response is semantically equivalent to the reference (ground-truu answer. Confidence is taken as the negative of the UQ measure (i.e. Inv-Entropy) for AUROC and PRR. For the Brier Score, we apply isotonic normalization [Zadrozny and Elkan 2002] to map uncertainty scores to the [0. I] range. Correctness- open-ended or weakly supervised settings. To address this, we propose the Temperature Sensitivity of Uncertainty (TSU), which quantifies how often uncertainty increases with temperature. Since higher temperatures flatten the softmax distribution, they should yield greater randomness and uncertainty [Hinton et al., 2015). Formally, given a sequence of temperature values T1 < T2 < < <T,, TSU defined
metrics, however, rely on ground truth and may fail in
TSU(T1,
=
Σ (UQ Ti) UQ(S T2) <.. UQ(S,, (AS,;TT)) Tw));
(7)
D
S,ED
where D is the dataset, S is question in this dataset, UQ, T) represents a UQ subroutine (such as Inv-Entropy) for input S,, at temperature T, and II(-) is the indicator functionA salient feature in the definition of TSU in 7 is that it only depends on S.. thus is agnostic to the "ground truth" output y. In addition, TSU extends beyond conventional correctness-based metrics by evaluating the granularity of uncertainty estimation. By leveraging temperature scaling probing mechanism, TSU assesses how effectively a method distinguishes between gradations of uncertainty.
Implementation details Our framework requires three inputs: v/ Per, and ('Similarity- For Per, we apply two strategies: (1) ChatGPT-based paraphrasing, generating nine perturbed versions per question, (2) GAAP introduced in Sec. 2.6 with a similarity threshold of = 0.7. For th, we employ three state-of-the-eeeeeeeo-
approaches: (i) SBERT-small (paraphrase-Minnila MiniLL-----22),, (ii) SBERT-large (all-mpnet-ban) base-vaas222) [Reimers and Gurevych 2019]. and (iii) DeBERTa-v2-xlale 1laae---x-aa-e---L. [He et al. 2021]. For (i) and (it), we use cosine similarity ASimiluity (r, x') (1 + cos(x, x222. While, (ii)) generates an entailment score; however, this score is not symmetric. To address this, we take the average (ASimilarity (z, 22) + aSimilarityy (a', x2)) All experiments were conducted with an NVIDIA A100 GPU. Detailed experimental set up including prompts and parameters used are detailed in Appendix [ Appendix D also includes additional simulation results; we present only the core findings in the main paper for clarity and focus.
Table Comparison of TSU across different temperature ranges for TriviaQA and MML.U.
TriviQQA
Method
(SL(1.)14)
TSU.0.711.)
TSU(077-1.4)
TSU(0.3-14)
TSU(L.),1.4)
TSU((.7.1.4
TSU30,7-14)
TSUR03-1.4)
Semantic Eatropy
17:35
20.64
5.35
3.94
33.20
39,80
4.93
2,09
Kemel hamooy
43.5 43.92
55.56
18.25
9.64
59.45
09:10
37.32
12,63
38.78
42,86
4.92
0.00
37.62
38,73
2.59
0.00
P'Tree)
3.49
LexSim
53. 14.54
12.38
8.16
$5.06
61.22
30.61
15.28
DegMaa
45.37
47.96
20.02
13.27
69.30
73.55
32.58
14.34
LUQ
48.06
50.00
27.63
10.20
61.22
62.24
27.55
10,80
KLE
6.42
0.00
26.53
2.67
0.00
Eny-Fatropy
77.55
H8.78
47.21
19,85
73.47
56.73
43,58
18.37
NI-Fatopy
61.22
60.20
19.32
11.73
50.000
59.38
18.37
7.14
WD-gx--p
69.39
36.75
12.06
22.54
MAX-gy-x
81.63
32.58
16.03
63.41
25.51
3.1 Results
Correctness-based As shown in Table Inv-Entropy achieves consistently strong and stable performance across all five datasets and all three metrics. It attains ofate-of-the-art results in both AUROC and PRR. The improvement is particularly clear on MMLU, where Inv-Entropy reaches 0.780 AUROC and 0.898 PRR, noticeably higher than all basclines, reflecting the advantage of our probabilistic framework with inverse design when the output information is limited. It also achieves leading performance on the long-answer dataset NQ, indicating that its effectiveness is not constrained by answer length (a detailed sensitivity analysis with respect to answer length is provided in Table of the Appendix). It also ranks among the top two in Brier score, indicating well-calibrated confidence estimates. We intentionally use ChatGPT-based paraphrasing to highlight the advantages of our method independent of GAAP.
TSU Table reports TSU results TriviaQA and MML.U. These results align with the correctness- based UQ metrics in Table methods defined by probabilistic framework consistently achieve

---

PAU
OuaPetAAibee COVEK
340
Figure 5: AUROC of Inv-Entropy under different perturbation methods (GAAP or ChatGPT-based paraphrasing) and embedding functions, on both ChatGPT and LLaMA models.
top performance, while LexSim, DegMat and LUQ exhibit highly inconsistent results, occasionally leading in specific cases but underperforming in others. Methods like P(True) perform poorly in TSU due to their binary nature, which fundamentally lacks granularity. Similarly, Semantic Entropy shows limited diseriminatory power as its values often cluster around few discrete points. In contrast to these coarse- coarse-grained approaches, Inv-Entropy is fine- geeigradied with superior ability to reflect uncertainty variations, demonstrated by consistent top TSU performance across all datasets and temperature settings. This result highlights the probabilistic framework's ability to capture uncertainty trends beyond labeled datasets. The complete TSU results are presented in Table in the Appendix.
Performance breakdown (i) Impact of replications: The comparison between NR-Inv-Entre and Inv-Entropy in Tables and 2 Thighlights the impact of replication and bootstrapping. Notably, UR-Inv-Entropy often performs competitively, showcasing the strength of our framework even without replication. This suggests that one can trade off a small loss in accuracy for fewer queries. Appendix D includes an ablation over S and r, further confirming this finding. (ii) Impact of inversion: The comparison between NI-Entropy and Inv-Entropy further confirms the advantages of our inverse approach, which examines the diversity of inputs that could have led to a specific output. Although NI-Entropy consistently underperforms Inv-Entropy, still often ranks second, highlighting the strength of our defined probabilistic framework even without the inversion. (iii) Framework generality: The often competitive performance of WD-px-py -y and MAX-py-x highlights the robustness of our framework and its flexibility in defining wide range of UQ measures. (iv) Impact of perturbations: Fig. 5 shows that GAAP consistently improves AUROC across all three datasets and both LLMs compared to ChatGPT-based paraphrasing. By enhancing input diversity in a principled way, GAAP better tests model robustness and improves uncertainty quantification. These results underscore the importance of meaningful perturbations. (v) Impact of embedding function: Our framework delivers consistently strong uncertainty estimates with every encoder we tested, including SBERT (paraphrase- [puraphrase-MiniL--LLE MiniLM-L6--2), SBERT cil---ppptt--e-eete
and DeBERTa. Although the larger encoders provide slightly higher scores on several tasks, the overall gap is modest, showing that even lightweight models can support reliable UQ when paired with our method. The close alignment between results from SBERT was-mppteeeb-e22 and DeBERTa further suggests that entailment and similarity signals extracted by the two architectures contain overlapping information.
4 Conclusion
We present a fully probabilistic framework for uncertainty quantification that models the conditional distribution of inputs given outputs through dual random walk formulation. This inverse modeling perspective enables a principled characterization of uncertainty by capturing the semantic diversity of inputs associated with a given output. key strength of our framework its flexibility, allowing researchers to freely combine embedding functions, perturbation strategies, and similarity metrics to define customized uncertainty measures. As an instantiation of this idea, we introduce Inv-Entropy, novel uncertainty metric derived from the framework. Together with the proposed perturbation algorithm GAAP and evaluation metric TSU, our method achieves statc-of-the-ar performance across multiple datasets. We believe this framework opens up broad opportunities for future research, providing a general foundation upon which new uncertainty measures, tailored to different purposes, can be systematically developed. We acknowledge that, like other perturbation and replication- based UQ methods, our approach may face practical limitations due to computational cost. A promising direction is adaptively determine when further perturbation is unnecessary. We hope GAAP's sequential design can provide a step towards this goal.
10

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Appendix Outline
Appendix Related Work
Appendix Proof of Lemma
Appendix
GAAP 2.6 Implementation Details
Appendix
Settings & Additional Experimental Results:
Experimental setting D.I
Computational cost comparison
D.2
Additional experimental results D.3
* Table 5
Complete TSU results.
* Table 6
Sensitivity Analysis to answer lengths NQ.
Table
Abalation study on the number of bootstrapping iterations S.
* Table
Abalation study the number of replications
* Table
Abalation study on the embedding functions th.
* Table
10
Abalation study on temperatures T on TriviaQA.
* Table
Abalation study on temperatures T on SciQ.
A Related Work
UQ in LLMs has been gaining increasing interest recently (Huang al. 2024]. The existing, albeit limited, literature can generally be categorized into three perspectives.
Self-evaluation-based UQ: Self-evaluation techniques for LLMs [Chen et al. 2024 and the verbal expression of uncertainty (Lin et al. [2022] have recently been explored to enhance interpretability and reliability. For instance, recent approaches for evaluating frree-fomm generation tasks frequently utilize LLMs as evaluators [Zheng et al. 2023]. J. Additionally, for gray-box models, where the internal
workings are partially known, perplexity [Chen et al., [998] and entropy [Shannon 1948
can be
directly computed from the output logits, providing a natural UQ measure.
Replication- UQ:
These methods generate multiple outputs for a given input and measure
the deviation between them to estimate uncertainty [Grewal et al. [0024, Wagner et al., 2024 Kuhn al. 2023). Perhaps most prevalent semantic entropy [Farquhar et al. 2024], J, which computes uncertainty by clustering semantically equivalent answers from multiple responses and calculating the entropy of the resulting clusters. While effective in capturing aleatoric uncertainty, these approaches struggle with confidently wrong predictions, as resampling often yields similar incorrect results, leading to overconfidence and poor calibration. This issue exacerbates the challenges of handling hallucinations in L.MSs [Grewal et al. 2024).
Perturbation- based UQ: This a more recent approach that involves systematic perturbations of inputs latent representations to evaluate output variability [Zhang et al. 2023). While Dong et al. 2023) systematically evaluates LLM robustness for noisy slot-filling tasks under diverse input perturbations, SPUQ [Gao et al.) 2024] provides UQ metric by analyzing response variations to perturbed imputs. Notably, SPUQ achieves significant improvements in model uncertainty calibration, reducing Expected Calibration Error (ECE) by average of 50%. Indeed, our theoretical argument in Sec. 2.1 provides a theoretical justification for the need for perturbation in effective UQ.
In light of existing literature, our method unifies replication- based and perturbation based UQ while introducing a Bayesian perspective LLM uncertainty estimation by modeling the posterior distribution of inputs conditioned on outputs. This Bayesian inverse design approach provides new framework and perspective for UQ semantic models.
15

---

B Proof of Lemma 2.1
In this section, we present the proof of Lemma 2.1 the proof based on Taylor expansion of f(').
The order-2 Taylor expansion for f(1r (1 -1-1/v- (=) is,
f(zo (1 -)--)))))) f(=0)) (xro) -1- R(=),
(8)
where R(z) is the remainder term defined as,
1
R(z) - 2² - (I-nvy f(zo + E(z)(I - - (1) n
(9)
2
where {(z) E [0, 1] is constant dependent on z.
By assumption, there exists constant G > 0 such that || [>>llo S SG E Rd. Therefore,
R(=) S G ([[--1///]) -/. )= G||=||2/2
(10)
Since = admits a dddimensional isotropic Gaussian distribution (N o²), we can provide an upper bound for B(R(=)) and E (=))
E(R(=)) S Gdo?/2 /2,
(11)
and
E(R(2))) G
(12)
G s d' d'E
G²
d3
We can also calculate the expectation of f(r + - (1-1 )=) as
(200 + E(((()). Elf(20 + -1- -1 )=)]
(13)
Then, the variance of f(r0 + (I - --- is,
Vis(so + (1-9111 ((-1)1))=)) (f (I- EIi(E0 + +I m.
(14)
(U 27
+ 21E [(Vf(zo)] nz)))=) ((>)] + [{
where ((=) is defined as
<(=) = R(z) B(R(z)).
(15)
Notice that the first term in 14p is,
v f((o)) ((l (I-
E vf(20) T(I [I , )22 - (
)'V)(z)) T (v/.))
(16)
0² |Vf(20)]1 (z)) (v)(o))
[xo]]
0² (m))))) sin" ((n'0(()((a)) v)(o))).
The third term in 14) is upper bounded by
E [<(=)²]
$ 2
(17)
< 20 =
16

---

where we used and 12 in the third inequality.
The second term in 14
is upper bounded by the Holder inequality,
E ( ( (200) (1- s(=)
v [[()] - E (Vf(z)) -1)-)))))) \x)
(18)
- a V ( sin sin (V) (((o)
= =(()))
where we used the inequality 177 in the second inequality, and the assumption that |/ (x)))) is upper bounded in the last theorem.
We complete the proof by combining 16 18 and
C GAAP Implementation Details
In this section, we first introduce an example of the perturbation set (Per(So), then introduce the details of our GAAP algorithm that is used for the perturbation function Per(-).
C.1 An example
We show an example of (S in Table 3 The original input is a question from TriviaQA Sm. Whhhich golfer became only the fifth in history get both the British and US Open champi- onships in the same year, in 1982?
Sap
Which golfer
the fitth history
the British and US Open championships
Se1
year, Which driver became only the fifth in history to get both the British and US Open taple_crown the same year, in 1982?
Which
the fifth history to the British and US Open
Sag
823
Which modaliat hecame only the fift's histrey get both the British and US Open high. stamss in the same yesr, 19827
S24
gotter become only nitth in history
both
Open championshipi
same year.
825
Which polfer bocame only the fifth in history to win toch the Brilish and US Open in the same year, in 19829
Sz6
Which driver became only rifth
both medalist became only the Aihh history to
and US Open cipple_crown the year, 1982?
S27
British
Opm
same year.
S=8
Which driver become only the fifth in history win both the Britih and US Open championshipe the same year, in 1982?
529
Which
fifth history
both
Open the your. 1982
8z10
Which linksmon become only the rithh in history to win horh the British and US Open in the same year, in 19827
Table 3: question Szo and 10 perturbed versions of Szo as the output GAAP.
C.2
Algorithm details
We first introduce some notations. For an input sentence Szo, we can denote it in the form of a token (word) series.
S.p = (t) .t. tp))
where f1, .2, tp denote the sequence of tokens that constitute the input text Sz,, arranged in their original order. We will then elaborate on each step GAAP.
C.2.1 Key words selection
For better sampling efficiency, GAAP does not perturb all tokens equally, Instead, we identify the key tokens in S2, first and only perturb these tokens. As a result, we could explore the semantic space more efficiently under the perturbation budget constraint.
We define a function k(-,)) that identifies key tokens within a given text. The function takes two inputs: the first is a text and the second is ratio indicating the proportion of key tokens to all tokens in this text. k(Szg r) returns a subset of tokens:
ke (<S r) {t ty........................ t
where the indices f21:22,.... : C (1.....P) p). and the number of selected key tokens are: g = int|r- pl. In GAAP, we use KeyBERT Grroteenorstt 2024 to implement k(:,))
17

---

C.2.2 Initial population generation
Next, we define replacement function re(s,))) that substitutes specific token in the sequence. The function is defined follows:
re(Saor i, (1) (41 (11,-1., 4/ Go f, (p)
(19)
In GAAP, we choose 1's from substitution set SUB((),.. And the substitution set is defined the union of all possible hypernyms, hyponyms, synonyms, and an empty set denoting word deletion, SUB((,,) hypernyms(4;)
hyponyms((z,) synonyms(4,)) (0).
The initial population of GAAP for the input Sa, is defined as the union of the outcomes of all possible single-key token perturbations,
Popo (S2.))
U
U
re(S 41: 4.)
4, (r( eSUB(t,))
C.2.3 Iterative population update
Then, we introduce an iterative scheme for the perturbed population to evolve. In each iteration, we must follow the next three steps sequence.
1. Selection: The selection step aims choose subset of individuals from the population as parents for subsequent procedures. In GAAP, we design a random selection mechanism where individuals whose semantic meanings closer those of the original text S., will be selected with higher probability.
In the terminology of genetic algorithms, we define our fitness function the semantic similarity to Hо, (SSimilartty (zo. -). Then, we compute the fitness value 4similarity (zi., 24 for vS,. E (S2.) and use roulette wheel selection [Lipowski and Lipowska 2012 to choose parents. More specifically, the probability of selecting Sz, is:
ASimilarity (T0, x3s)
P(S..) Es., (Sop) 4Similarity (Zo; Xz)
The set of all selected parent individuals is denoted as Par (S
2. Crossover:
The crossover step aims to generate new offspring by recombining the segments (i.e., token sub-sequences) of parent individuals.
The inputs of the crossover operation are two randomly selected parent individuals SAA and Sen from Pa, (Sz,). Then, we uniformly randomly sample crossover point h from {1,2, P- 1}),, where p is the length of the shorter one between S,A and San. Next, we generate two offspring individuals Sny and S. as
Szy² = (111 .
B
p):
B Sz (t 1
h
where 1/ and 1 represent the i-th token of parents Sr, and Sag- respectively. The set of all generated offspring individuals is denoted Off, (Sz,).
3. Mutation:
The mutation operation aims to augment population diversity again by randomly replacing certain tokens the offspring individuals. For each offspring individual Sz, e (S ). randomly select token t, for replacement. Then, uniformly randomly choose new token t, from the substitution set SUB(t,,):
', E SUB(L,,) hypemyms((,,) U hyponyms((,)) U
Finally, we generate the mutated individual as
Sgy re(S 1,, ty)):
where re(s))) the replacement function defined in 19 individuals is denoted as Mu,
The set of all mutated offspring
18

---

The new population is formed by combining all the 3 sets above,
(S = (S.)) Off, U Mu, (Sao)-
(20)
Equation (20) defines an iterative algorithm update the population for 2, .... The iterative process terminates when either of the following conditions is met: (1) The number of generations t exceeds predefined maximum value N umm t 2 N Num, (2) the maximum of fitness values in the population is smaller than threshold & max; OSmmllriyy (Eo, Ri) < 8.
C.2.4 Perturbation set construction
Unlike standard genetic algorithms that aim to optimize the fitness function to extreme value, the objective for GAAP is to use populations Pop, at different generations f to construct a perturbation set (er(Sza). The ideal perturbation set should be diverse and contain texts with varying degrees of similarity to S. Since earlier generations contain fewer perturbations and later generations involve more perturbations, the generation index t a natural indicator similarity. As a result, we generate Per(S by random sampling from populations different generations.
More specifically, Per(S2a) consists of random samples from populations at generations 0, 2T. where T E Z is the sample interval:
int
Per(Szo) = Uniform(Pop,r- (Sa.))
(21)
q=0
where Uniform(.) is function that uniformly randomly selects subset of individuals from the population. The construction rule (2I) ensures that Per(Sz,) contains texts with progressively decreasing similarity to S.g as the genetic algorithm evolves toward lower fitness values (i.c., lower similarity).
After the termination condition is met, the perturbation set Per(S..)) is returned the final out- put. This set represents a collection of perturbed versions of Sng each with a different degree of perturbation.
D Settings & Additional Experimental Results
D.1 Experimental setting
Dataset Preprocessing No preprocessing required except for MMLU, due its heterogeneous and often non-standard question formats, which are incompatible with our perturbation- -based frame- unsuitable for
work. We excluded items that lacked self-contained semantic meaning and thus perturbation (e.g., "Which of the following statements is true?"), purely mathematical expressions
without contextual meaning (c.g., "If A 21, 2, 4). let B {1,, 2), (1, 3), (4. 2)). Then B is"), or other irregular or ambiguous phrasing. Only questions with clearly worded, self-contained statements
retained.
The following are all the prompts used in our experiments.
ChatGPT-based paraphrasing
Please Provide {number of perturbations) paraphrases for this sentence: (sentence)
Generating Responses
For TriviaQA, SciQ, and NQ:
{question) Answer concisely and return only the name.
For MMLU:
{question choices} Answer concisely and return only the name. For GSM8K:
19

---

{question) Answer concisely and return only the result itself.
We design prompts align closely with the highly concise reference answers in the datasets. The same prompts are used for both our method and all baseline models, across both GPT-3.5-Turbo and -L.MMA-311-8B-nstter
Correctness Evaluation
Are the following two answers to my question Q semantically equivalent?
Q: {question)
A1: (standard answer)
A2:
(answer)
Please answer with single word, either Yes or No.
VU Derivation (used of our benchmarks)
After the previous prompt of generating responses, we append the following prompt to elicit verbalized uncertainty:
And use percentage tell me your confidence in your answer.
The parameters used in our experiments are listed below.
Perturbation Configuration
As detailed in the main manuscript, generate nine perturbations per question using ChatGPTT-bsse paraphrasing, resulting in ten variants per question including the original. For GAAP, we set the threshold 0.7 and also fix the number of perturbations at nine. This uniformity is crucial for our probabilistic framework, where each question induces a distribution over variants. To ensure that these distributions are comparable and defined on the same scale, require the same number of perturbations per question across the dataset. When fewer than nine are generated, we randomly duplicate existing perturbations; when more than nine are produced, we randomly sample nine.
Replication and Bootstrapping
Our model incorporates bootstrapping to utilize replicated responses. Unless otherwise specified (notably the ablation studies analyzing the impact of and r), all reported results are based on experiments with $ = 30 bootstrapping iterations and replications.
Calculation of Mean and Variance
All reported evaluation metrics represent means with associated standard deviations computed via (i) Calculate the target
bootstrapping. Using 40 bootstrap samples generated with replacement, metric for each sample, (ii) Aggregate results by taking the mean of sample-level metrics as the final estimate, (iii) Compute the standard deviation across the 40 values as a dispersion measure.
LLM Configuration
For ChatGPT-bassdd paraphrasing, we set the temperature to 0.7. For correctness evaluation, a temperature of used. For LLaMA, due LLaMA's lack of automatic response termination and occasional output corruption, this may lead to incomplete or malformed answers. To mitigate this, adopt multi-attempt generation protocol with the following cleaning steps to ensure concise and valid outputs: (i) remove the echoed question if present, (ii) delete formatting tokens (e.g., 
, 
, #) and any trailing text, and (iii) retain only the first non non-empty line after trimming whitespace. The following is an example.
Question:
What capital of France?
Response before cleaning:
What is the capital of France? Paris 
 #
It is major European city and global Response after cleaning: Paris
20

---

D.2 Computational cost comparison
For non-locally hosted LLMs, uncertainty quantification methods generally involve two stages: obtaining responses from the model via API calls and computing the uncertainty scores based on those responses. This applies to both our method and all baselines; the only exception is Verbalized Uncertainty (VU), which directly returns score from the API without requiring post-processing. As discussed in the paper, any method that relies on perturbations and/or replications incurs additional computational cost. However, major advantage is that responses for perturbed inputs can be generated in parallel, making the process more scalable in practice.
With this in mind, we divide the computational cost into two components:
Computation time after responses are collected, and
(B) Total cost, which includes (i) along with API call time and perturbation generation.
Regarding API cost: Our method introduces n perturbations per question and generates 7 response replications for each version, resulting in total of (n 1) API calls per input. Thus, the cost scales linearly with both the number of perturbations and replications. That said, while our method introduces perturbations, it requires far fewer replications than several existing baselines. For instance, Semantic Entropy relies heavily on large r values (with n = 0); prior work recommends > 10 for stable performance [Farquhar al., 2024) In contrast, we demonstrate that even without replication (r = 1), our method remains effeettvve This is evidenced by the strong performance of NR-Inv-Entropy, which uses perturbation alone and still consistently outperforms many baselines.
Regarding computation after responses are collected: Below, we present the compute results in Table 4 without parallclization. The reported numbers are averaged over all sampled questions in Tri As shown, our method is highly efficient in the post-processing stage when computing Inv-Entropy scores, Furthermore, in settings where response generation time is negligible, such as when LLMs are deployed locally and perturbations are processed in parallel, our total computational cost often lower than that of existing baselines.
Table 4: Computation efficiency comparison across methods.
Method
Peak GPU Memory (MB)
Time A (s)
Time B (s)
Semantic Entropy
3575.31
13.985
20.183
VU
0
0
0,620
P(True)
0.001
2.201
LexSim
0
0.022
3.189
DegMat
1610.29
6.143
15.905
LUQ
1580.21
3,821
4.432
KLE
1608.52
1.323
6,725
Inv-Entropy (Ours)
86.65
1.990
10.323
NR-Inv NR-Inv-Entropy (Ours)
86.65
1.769
6,358
21

---

D.3 Additional experimental results
Table presents the complete TSU results across all five datasets.
Table 5: Comparison of average TSU values across all five datasets and temperatures
{0.3, 0.7, 1.0, 1.4], using TPrroo with ChatGPT-based paraphrasing and SBERT-small Saaiphrrae--iiilM--LG. TSU( 0.7, 1.0) becomes TSU(0.3-1.0)) All values are reported as percentages.
embeddings. TSU(a, b, .c) is abbreviated as (SU(a-c) (e.g..
TSU (0.3 0.7)
TSU 10)
T.U. 141
TSU (L.3 10)
1SU. 14)
TSU 10.. 14)
TSU (83-003-110
TSU (17-144)
ISU (03--44
TSU
TriviaQA
Semantic Entropy
11.57
14.26
17.35
21.56
25.51
20.64
5.18
5.35
3.94
VU
22.45
27.55
38.78
P(Truc)
1.02
2,13
385
1.38
341
0.00
0.00
0.00
LexSim
22.39
22.95
46.94
30.61
53.36
53.54
9.18
12.38
8.16
DegMat
33,21
45.37
20.02
LUQ
KLE
7.14
17.35
13.45
4.57
1.28
6.42
2.79
1.31
0.00
Iav Entropy
$2.42
66.33
77.55
76.53
92.86
$8.78
30.49
47.21
19.05
NI NI-Entrpy
55.14
27.96
61.22
59.35
67.35
60.20
21.43
19.32
11.23
WD WD
44.81
67.35
57.62
60.20
64.49
69.39
23.11
36.73
12.06
MAX-P5-x
50.83
Semantic Entropy
11.09
15.48
21.55
17.35
27.62
25.51
6.73
4.31
2.65
VU
21.43
33.67
28.57
35.71
31,42
32.65
1.55
6.12
0.00
P(True)
LexSim
31.73
29.19
47.96
40.82
61.22
59.18
19.39
13.27
11.22
DegMar
39.80
27.96
57.14
$2.04
64.28
64.45
23.47
23.99
13.04
LUQ
Entropy
62.24
22.45
Entropy
42.30
59.18
55.19
48.98
54.77
63.27
21.43
22.65
9.18
WD-px-py
50.00
59.16
63.25
59.37
71.40
69.39
22.13
28.59
11,22
MAX-p9-x
NQ
Semantic Entropy
VU
23.71
22.68
40.21
21.65
36.08
36.08
4.12
3.09
0.00
P(Tnue)
4.08
3,06
2.04
1,412
2.04
1.02
01.00
0.00
0.00
DegMat
$1.04
51.04
59.38
65.62
76.04
71.88
32.29
33.33
21.42
LUQ
43.16
29.47
67.37
$7.89
76.84
74.74
24.21
30.41
13.68
4.26
Imy- Entropy
61,86
59.79
78.35
74.23
91.75
85.57
30.93
41.24
22,43
NI-Entropy
WD
63.55
50.52
70.10
57.01
76.29
72.16
27.84
25.77
16.92
MAX-Py-x
55.67
62,89
73.20
66.30
84.54
82.47
28.87
39.18
18.56
MMLU
Semantic Entropy
21.43
17.35
33.20
21.83
42.08
39.80
7.14
4.93
2.09
VU
23.56
P(True)
1.92
4.56
5.87
4.92
5.02
5.79
0.00
0.00
0.00
LexSim
33.94
41.17
$5.06
50100)
68.33
61.22
24.78
30.61
15.28
DogMat
LUQ
$3.46
47.96
61.22
58.92
68.37
62.24
27.55
27.55
10.90
KLE
10.20
25.51
26.53
7.14
4.76
12:23
2.93
2.67
0.00
Iny-Entropy
60/88
67.35
73.47
79.59
90.82
86.73
34.31
43.88
18.37
NI-Eutropy
56.12
54.63
50.00
67.35
$2.45
59.38
19.39
18.37
7.14
py-
MAX-p9-X
62.24
62.35
65.41
23.42
81.23
80.61
31.62
25.51
14.42
GSM8K
Semantic Entropy
44.90
$6.12
35.71
71:43
77.55
62.24
20.41
13.27
4,08
VU
11:34
39.18
29.90
35.05
35.05
38.14
2.06
6.19
1.03
P(True)
17.35
3.32
LexSim
54.17
63.54
$4.17
65.62
64.58
30.31
23.96
10.42
DegMal
55.79
64.21
55.79
72.63
75.74
70.53
29.47
29.47
10.53
KLE
17.02
35.11
27.66
4.25
3.72
23.40
0.00
3.19
0.00
Inv-Entrogy
NI-Potropy
$3.61
58.76
$1.55
61.16
$7.73
59.79
25.77
20.62
6.19
WD-px-py
$4.08
$3.06
68.37
55.10
61.29
60.20
20.41
27.43
7.14
x
37.50
22

---

Table 6 shows the performance across different answer lengths in Natural Question (NQ) (wwiakkowsii et al. 2019 dataset. NQ dataset includes both short and long-form answers. We selected questions that contain only long-form answers. The reference answers in this subset range from
350 tokens in length. To analyze the sensitivity of different answer lengths, we further divided the dataset into three subsets based on the number of tokens in the reference answers: Short (< tokens), Medium (80-120 tokens) and Long (2 120 tokens). We evaluated both baseline methods and our proposed method each of these subsets as well as the full sample set.
Our method achieves state-of-the-ar art performance on the full dataset, ranking first across all three metrics. We also observe some sensitivity to answer length: it shows a clear and substantial advantage on the short-ane answer subset, highlighting the effectiveness the inverse-design mechanism when responses are more concise. On the medium-lenge subset, it continues to outperform all baselines, though with smaller margin. On the long-answer subset, while the top performer, our method remains competitive and yields reasonable results compared to strong baselines such as LUQ.
Intuitively, these results align with expectations under the inverse perspective: the shorter the answer, the more important it becomes to explore the diversity of the input space. Nevertheless, our approach consistently ranks among the top two methods even in the long-form QA setting. We end by noting that our probabilistic framework is also generic by nature and can accommodate a forward perspective or alternative uncertainty metrics beyond entropy computed over P(X Y). Exploring such metries across different answer lengths is promising direction we leave for future work.
Table 6: Comparison of AUROC, PRR, and Brier scores across Short, Medium, Long, and Full
subsets on NQ. Bold indicates the best performer. Underline indicates the second-best best.
Metric (1/4)
Method
Short
Medium
Long
Full
Semantic Entropy
0.509
0.461
0.584
0.521
VU
0.531
0.495
0.508
0.533
P(Truc)
0.529
0.473
0.548
0.519
LexSim
0,624
0,438
0,555
0.518
AUROC (1)
DegMat
0.547
0.621
0.484
0.551
LUQ
0.662
0.508
0.612
0.627
KLE
0.265
0.456
0.445
0.410
Inv-Entropy (Ours)
0.794
0.634
0,589
0.661
Semantic Entropy
0.420
0,584
0.507
0.505
VU
0.489
0.615
0.495
0.537
P(Truc)
0.427
0.584
0.478
0.502
LexSim
0.628
0.684
0.544
0.563
PRR (T)
DegMat
0,543
0.682
0.548
0.549
LUQ
0.649
0.655
0.523
0.595
KLE
0.354
0.649
0.444
0.449
Inv-Entropy (Ours)
0.747
0.742
0.508
0.614
Semantic Entropy
0.227
0.230
0.221
0.242
VU
0.209
0.218
0.216
0.223
P(True)
0.225
0.232
0.230
0.244
LexSim
0.169
0.199
0.207
0.225
Brier (L)
DegMat
0.187
0.181
0.213
0.229
LUQ
0.164
0.209
0.179
0.208
KLE
0.228
0.203
0.225
0.244
Inv Iny-Entropy (Ours)
0.125
0.175
0.193
0.201
23

---

Table[[]shone how the experimental results vary with the number of bootstrapping iterations S. We observe that increasing S initially enhances the performance of our method, Inv-Entropy. However, beyond certain threshold, further increases yield diminishing or significant returns. Therefore, using more than 10 iterations appears to provide limited additional benefit.
Table 7: Comparison of Inv-Entropy performance across all the 5 datasets with varying numbers of bootstrapping iterations S. We use GPT-3.5-Turbo with ChatGPT-based paraphrasing and DeBERTa- V2-xlarge-MNLI embedding function.
Dataset
Metric
S=1
S=5
S=10
S=30
S=50
S=100
AUROC
0.743 1:0007
0.785 0.785 10.056
0.788
0.780.10.054
0.788
TriviaQA
PRR
0.840=0005 +0.04
0.882
0.885=00044
0.882:10.044
0.885+:0.044
Brier
0.138=0.01
0.127=00.20
0.125=0.020
0. 0.128000020
0.131
0.128+0.000
AUROC
0.024
0..00
0.740
0.740
0.739.10.08
0.743
SciQ
PRR
0.821 0.840+0.016
0.04
0.843=00042
0.842:40046
0..45
Brier
0.163+00016
0.160+0.015
0.159+0.015
0.157=00018
0.160:10.015
0.159:+0.015
AUROC
0.659
0.686:10.000
0.699
0.703
0.702.1.058
0.705
NQ
PRR
0.709
0.730=t0.071
0.760=0.065
0.764=0.064
0.764
0.766:10.064
Brier
0.199
0. 0.0000
0.184=00019
0.182=00020
0.183++0.020
0.182++0.21
AUROC
0.054
0. 0.762.t:0042
2..022
0.780
0.790
0.789
MMLU
PRR
0.743
0.863
0.897
0.898
0.905
0.902
Brier
0.188##0021
0.152++0.017
0.148=00017
0.147
0.142:+00017
0.143
AUROC
0.674=0.054
0.684-10.056
0.685
0.695.20.88
0.690110.05
0.695
GSM8K
PRR
0.487-10.090
0.530-10.096
0.527
0.52 100094
0.520:10.004
0.527
Brier
0.159=0.021
0.101
0.150=0.021
0.152=
0.152
0.146:+0.021
Table shows the variation of several proposed UQ measures (Inv-Entropy, NI-Entropy, WD-px-py, MAX-py-x) the number of replications increases. Performance improves consistently with larger as additional replications help better capture aleatoric uncertainty. However, this improvement comes the expense of increased computational cost.
Table Comparison of UQ methods on the TriviaQA and SciQ datasets with varying numbers of replications r. We use GPT-3.5-Turke with ChatGPT-basse paraphrasing and SBERT-large (all- mpnet-base-v2) embedding function.
Dataset
TriviaQA r=
SciQ
Metric
==
=
7 2
y= 4
r == 6
AUROC
0.812+004
0.815
0.820#004
0.7944004
0,801
0.806
Inv-Entropy
PRR
0.916
0.915:20.09
0.9
0.888
0.892:10038 0.888
Brier
0.
0.
0. 0.
0.14
0.140
0.136:20019
AUROC
0.70240o7z
0.631
0.61720083
0.731 0.729
0.74622005
NI-Entropy
PRR
0,81340054
0.765
0.75140055
0.005
0,805
0.809
Brier
0.133
0.15220002
0.1
0.147
0. 12.4222022
0214
AUROC
0.762400s8
0.77 +00058
0.772+
0.684440022
0.688
0.685
WD-px-py
PRR
0,86610016 0.875
0,8762000
0,825
0.818:0.054
0.8192200511
Brier
0.13410021
0.131
0.130
0.17310017
0.169
0.172.10.099
AUROC
0.782+0047
0.794=20045
0.80422004
0.7542now
0.760=aa51
0.757=20049
MAX-py-x
PRR
0.90410.028
0.910#a.027
0.915
0.86411047
0.862+0.049
0.861400048
Brier
0.14
0.
0.124
0.1544200s
0.155
0.154
24

---

Table 9 demonstrates the robustness of our framework across different embedding functions. On all three datasets (TriviaAA, SciQ, and MMLU), Inv-Entropy performs strongly with SBERT-small, SBERT--arge, and DeBERTa. This consistency allows practitioners to select an encoder based on available computational resources or domain-specifie requirements without compromising the quality of uncertainty estimation.
Table 9:
Comparison of AUROC, PRR, and Brier scores across the TriviaQA, SciQ, and MMLU datasets using different embedding functions. We use GPr-3.5-Turbo and CharGPT-based paraphras- ing.
Dataset
Metric
Embedding
Inv-Entropy
NI-Entropy
WD-px-py
MAX-py-x
SBERT-small
0.792
0.666:10.022
1..03
0,835
AUROC
SBERT-large
0.772
0.617:10083
0.816 0.004
DeBERTa
0.788
0.786+10005
0.518 000.00
0.723++0.044
SBERT-small
0.899
0.803-40.083
0.920=00.02
0,920+t0.029
TriviaQA
PRR
SBERT-large
0.876100.44
0.751:10.05
0.915 0.0022
0.915
DeBERTa 0.885
0.883-10.043
0.763
0.875
SBERT-small
0.127:0.021
0.151
0..06:20.021
0. 141:0.021
Brier
SBERT-large 0.130
0.150+1.024
0.001
0.124:10.020
DeBERTa 0.128:+0.00
0.1244000000
0..099
0. 0.148:+0019
SBERT-small
0.774110:499
0.750+0.054
0.655
0.742400000
AUROC
SBERT-large
0.806
0.706
D.685=20.057
0.757
DeBERTa 0.740
0.681
0.303
0.674
SBERT-small
0.874100044
0.820=1.059
0.796
0.852100049
SciQ
PRR
SBERT-large
0.89210.038
0.80940.054
0.8191100081
0.861 100088
DeBERTa
0.843
0.781
0.587 00006
0.821
SBERT-small
0. 0.
0.150
0.181 10.01
0. 0.1601
Brier
SBERT-large
0.136:100:099
0. 10.001
0.172-10.019
0. 0.1454100117
DeBERTa
0.157
0.164410017
0.212
0.177:10:01
SBERT-small
10.000
0.576-10.063
0..23:00.049
0.704:10.066
AUROC
SBERT-large
0.604-100058
0.532:+0.057
0.670
0.676
DeBERTa 0.780+00041
0.710++0055
0.573
0.585
SBERT-small 0.812+00064
0.719+0.069
0.869
0.832++0.057
MMLU
PRR
SBERT-large
0.790
0.695-10.068
0.834=10.049
0.820:10.06
DeBERTa 0..080
0.823-10.055
0.777
0.749
SBERT-small
0.170
0.192:+0.020
0.163
0.165:+0:019
Brier
SBERT-large
1.0.0000022
0.200
0.177 100019
DeBERTa
0. 0.147
0.168
0..08
O. 10.0020
25

---

Table 1O and Table 11 present how various uncertainty measures respond to temperature changes on the TriviaQA and SeiQ datasets, respectively. Our probabilistic methods consistently outperform baseline models across all temperature settings, highlighting their robustness to decoding variability.
Table 10: Comparison of AUROC, PRR, and Brier scores on the TriviaQA dataset under varying temperatures (T 0.3, 0.7, 1.0, 1.4). We GPT-o.5-Turbo with ChatGPT- paraphrasing Seaphrase-Miii.MM-LL..
and the SBERT-small the best and second-best performers, respectively.
embedding function. Bold and underline indicate
Temperature
Metric
Method
T=0.3
T=0.7
T=1.0
T=1.4
Semantic Entropy
0.579=0.044
0.679+00.52
0.700+0002
0.732.10.051
VU
0.695=0.000
0.625
0.629
0.578
P(True)
0.604=0.050
0.592
0.571 +0.03
0.583 :100038
LexSim
0.649=0.055
0.715
0.775
0.792:+0.051
DegMat
0.734=0.056
0.688=+0.067
0.794=10.049
0.730
AUROC(()
LUQ
0.637 2.067
0.712
0.748 L0.063
1..07100044
KLE
0.333 00.054
0.327 10.059
0.221 10.056
0. 0.155 10.042
Inv-Entropy
0.870=0.044
0.795+0.051
0.827+00043
0.810=0.042
NI-Entropy
0.762
0.66611.073
0.634.00083 100083
0.673 100055
WD-px-py
0.829
0.82710.0051
0.8 1816622 6:10.054
0.889910034
MAX-py-x
0.854
0.832
0.856110.00
0.846:10.00
Semantic Entropy
0.787
0.834-10.43 0.803
0.81 :000045
VU
0.836=0.044
0.798+00.53
0.727
P(True)
0.797=20044
0.760
0.776-10:042
0.726-10:046
LexSim
0.810
0.824
0.877
0.854
DegMat
0.882220.04
0.816.0.053
0.895
1..04
PRR(1)
LUQ
0.854
0.856.0.042
0.874
0.893 100099
KLE
0.704=0.048
0.646=+0050
0.623+00051
0.516
Inv-Entropy
0.939_..00
0.900
0.936110.22 10.022
0.903
NI-Entropy
0.802
0.799
0.777 10.00
0.765
WD-px-py
0.918+00029
0:
0.923
MAX-py-x
0.9
0.920
0.946
0.913
Semantic Entropy
0.166
n..20
00.02
+.0023
0. 0.150.000023
VU
0.120
0..08
0.175.0.027
0.203
P(True)
0.122
0.188 10.020
10.0010
0.198 100.020
LexSim
0.151
0.146+0.021
0.128+0.024
0.127=#0021
DegMat
0.140=0.021
0.149+00.22
0.115
0.145=#0.03
Brier({)
LUQ
0.148
0.142++000
0.121+00022
0.121#00018
KLE
0.088
0. 0.000000020
0.192++0.21
0.218
Inv-Entropy
0.085=0.019
0.127+00021
0.117++0019
0.132
NI-Entropy
0.104-0.021
0.151 +0.022
0.142++0024
0. 1644018
WD-px-py
0.115
0.102
0.121 0.117-10.02
MAX-py-x
0.103
0.1
0.108
0.114
26

---

Table 11: Comparison of AUROC, PRR, and Brier scores on the SciQ dataset under varying tem- peratures (T 0.3, 0.7, 1.0, 1.4). We use GPT-3.5-Turbo with CharGPT-based paraphrasing and the SBERT-large (all-mpnet-base-22 embedding function. Bold and underline indicate the best and second- best performers, respectively.
Temperature
Metric
Method
T=0.3
T=0.7
T=1.0
T=1.4
Semantic Entropy
0.548 00000
0.679100045
0.6991000M6
0.773 1:0003
VU
0.625
0.480+1.000
0.625-10.500
0.607=10006
P(True)
0.566=0.037
0.522+0.026
1.531
0.545 000002
LexSim
0..42
0.681 1:0046
0.713:10:049
0.770.10.051
DegMat
0.569
0.672
0.730++0.055
0.816
AUROC(1)
LUQ
0.565=0.062
0.740=0.050
0.650++0000
0.772::0.051
KLE
0.386=0.054
0.341=0.066
0.277++00559
0.184=+0.051
Inv-Entropy
0.767
0.800
0.754
0.795.10.00
NI-Entropy
0.682=0.062
0.755+00055
0.596++0.06
0.649
WD-px-py
0.701
0.689
0.795+00047
0.771
MAX-py-x
0.766=0.048
0.760+0.048
0.796++0.046
0.793=10:44
Semantic Entropy
0.705=0.045
0.763
0.780++0043
0.798
VU
0.739
0.677
0.736
0.678
P(True)
0.713=0.044
0.679
0.687+00045
0.634
LexSim
0.701=00052
0.770=t0001
0.794+10004
0.796=10.056
DegMat
0.739
0.802+0.046
0.834
0.864=10.040
PRR(1)
LUQ
0.709
0.843 10.042
0.734.10.06
0..0111.0061 1000661
KLE
0.632.100448
0.592
0.554
0.479
Inv-Entropy
0.900=0.029
0..09
0.864+0.04
0.870=10033
NI-Entropy
0.814=00047
0.820=00085
+0.9
0.693
WD-px-Py
0.858=0.0m
0.820
0.868 +0.000
0.851:10.00
MAX-py-x
0.893=0.029
0.862+0.048
0.887+00037
0.867=40033
Semantic Entropy
0.203
0..00
0,171 =0.021
0.159
VU
0.101
0. 0.000
0.202++0018
0.212-40016
P(True)
0.204=0.017
0.215
0.212++0016
0.225
LexSim
0,207 200000
0.179
0.169 10.022
0.156.10.019
DegMat
0..17
0.164
0.149.10.00
0.135 10.020
Brier(.)
LUQ
0.195 00.00
0.1577:0.000
0.173
1.159910.02
KLE
0.216=0.016
0.218+00016
0.219+0.016
0.233++0011
Inv Inv-Entropy
0.151 =00018
+0.020
0.153-+0.020
0.146=10.020
NI-Entropy
0.124=0.021
0.140
0.188
0.186:00.020
WD-px-Py
0.167=
0.17540.019
0.140 +0.000
0.156=20.015
MAX-py-x
0.151
0.155
0.040
0.153
27

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Reviews:
[{"review_id": "7qgtFg6BKm", "reviewer": "Reviewer_Q6Vh", "summary": "This paper studies uncertainty quantification of LLMs through the lens of LLM input-output behavior as a dual random walk. This leads to quantifying uncertainty in terms of the diversity of the inputs which lead to a particular output, rather than the other way around. The proposed uncertainty metric, Inv-Entropy, is shown to empirically outperform existing approaches across a variety of benchmarks.", "questions": "* How does the method perform for questions which long-form answers? If this is not in-scope, that is fine, but it should be stated that this method works for short-form questions only.\n* Does the perturbation function always maintain semantic equivalence of the questions? The example in the Appendix shows \"golfer\" being replaced by \"driver\" which works in the shown question, but does not universally work as a replacement.", "limitations": "yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths:\n* Many papers in UQ for LLMs use perturbation-based approaches, but this paper makes a theoretical argument for why such an approach is reasonable. Their method then capitalizes on the perturbation approach by using both input and output perturbations, and it's effectiveness is supported by extensive empirical results comparing with many relevant baselines.\n* The paper is well written and easy to follow.\n* The theoretical perspective is highly valuable to this problem which is usually approached with empirically motivated methods.\n* The approach of using an inverse model of entropy appears highly effective in practice and provides space for further work on how the uncertainty of the two random walks in the input and output space are quantified.\n* The paper proposes a new metric for evaluating UQ for LLMs called TSU which does not use correctness as a proxy for uncertainty. This is useful for settings without ground truth.\n\nWeaknesses:\n* The method is not evaluated in a long-form answer setting. One of the baselines is specifically designed for such a setting, so if it is assumed that the method is only applicable to short-answer questions, then this should be stated.\n* The limitation of perturbations being expensive, and the method being overall computationally expensive is acknowledged, but I did not see any analysis of how this compares to the baselines.\n* The design of the perturbation function is based on replacing synonyms and other simple text modifications which seems like it could possibly lead to accidentally changing the semantics of the question.", "was_revised": true, "final_justification": "During the discussion, the authors provided additional results on a long form answer dataset which make me more confident in my assessment. I am also satisfied by their response regarding computation time and details of the perturbation function. Overall, after the discussion, I am very positive about the paper's acceptance.", "rebuttal": "Thank you for the thoughtful review and encouraging feedback. We appreciate your recognition of the inverse perspective, the theoretical grounding via Lemma 2.1, and the clarity of the paper. We are also glad you found TSU valuable for evaluating UQ without ground truth.\n\nBelow we provide an itemized response to your suggestions \\& concerns:\n\n>**W1 & Q1: New experiments on a long-form answer dataset**\n\nThis is a great suggestion. Indeed, our method is generic by nature and can be applied to both long and short answers. To fully address your concern, we have conducted additional experiments using the Natural Questions dataset [1], which includes both short and long-form answers. We randomly selected 1,000 questions that contain only long-form answers. The reference answers in this subset range from 34 to 350 tokens in length.\n\nTo analyze performance across different answer lengths, we further divided the dataset into three subsets based on the number of tokens in the reference answers:\n\n* **Short** (<80 tokens): 280 samples\n* **Medium** (80–120 tokens): 320 samples\n* **Long** (≥120 tokens): 400 samples\n\nWe evaluated both baseline methods and our proposed method on each of these subsets as well as the full 1,000-sample set. The results are shown below with the same experiment setting as that of Table 1 in our main paper:\n\n| **Metric (↑/↓)** | **Method**            | **Short** | **Medium** | **Long** | **Full** |\n|------------------|------------------------|-----------|------------|----------|---------|\n| **AUROC (↑)**    | Semantic Entropy       | 0.509     | 0.461      | 0.584    | 0.521   |\n|                  | VU                     | 0.531     | 0.495      | 0.508    | 0.533   |\n|                  | P(True)                | 0.529     | 0.473      | 0.548    | 0.519   |\n|                  | LexSim                 | 0.624     | 0.438      | 0.555    | 0.518   |\n|                  | DegMat                 | 0.547     | **0.621†** | 0.484    | 0.551   |\n|                  | LUQ                    | **0.662†** | 0.508      | **0.612** | **0.627†** |\n|                  | KLE                    | 0.265     | 0.456      | 0.445    | 0.410   |\n|                  | Inv-Entropy (Ours)     | **0.794** | **0.634**  | **0.589†** | **0.661** |\n| **PRR (↑)**      | Semantic Entropy       | 0.420     | 0.584      | 0.507    | 0.505   |\n|                  | VU                     | 0.489     | 0.615      | 0.495    | 0.537   |\n|                  | P(True)                | 0.427     | 0.584      | 0.478    | 0.502   |\n|                  | LexSim                 | 0.628     | **0.684†** | 0.544    | 0.563   |\n|                  | DegMat                 | 0.543     | 0.682      | **0.548** | 0.549   |\n|                  | LUQ                    | **0.649†** | 0.655      | **0.523†** | **0.595†** |\n|                  | KLE                    | 0.354     | 0.649      | 0.444    | 0.449   |\n|                  | Inv-Entropy (Ours)     | **0.747** | **0.742**  | 0.508    | **0.614** |\n| **Brier (↓)**    | Semantic Entropy       | 0.227     | 0.230      | 0.221    | 0.242   |\n|                  | VU                     | 0.209     | 0.218      | 0.216    | 0.223   |\n|                  | P(True)                | 0.225     | 0.232      | 0.230    | 0.244   |\n|                  | LexSim                 | 0.169     | 0.199**    | 0.207    | 0.225   |\n|                  | DegMat                 | 0.187     | **0.181†** | 0.213    | 0.229   |\n|                  | LUQ                    | **0.164†** | 0.209      | **0.179** | **0.208†** |\n|                  | KLE                    | 0.228     | 0.203      | 0.225    | 0.244   |\n|                  | Inv-Entropy (Ours)     | **0.125** | **0.175**  | **0.193†** | **0.201** |\n\n **Bold** indicates the best performer. † indicates the second-best.\n\nOur method achieves state-of-the-art performance on the full dataset, ranking first across all three metrics. We also observe some sensitivity to answer length: it shows a clear and substantial advantage on the short-answer subset, highlighting the effectiveness of the inverse-design mechanism when responses are more concise. On the medium-length subset, it continues to outperform all baselines, though with a smaller margin. On the long-answer subset, while not the top performer, our method remains competitive and yields reasonable results compared to strong baselines such as LUQ.\n\nIntuitively, these results align with expectations under the inverse perspective: the shorter the answer, the more important it becomes to explore the diversity of the input space. Nevertheless, our approach consistently ranks among the top two methods even in the long-form QA setting. We end by noting that our probabilistic framework is also generic by nature and can accommodate a forward perspective or alternative uncertainty metrics beyond entropy computed over $p(X|Y)$. Exploring such metrics across different answer lengths is a promising direction we leave for future work.\n\n>**W2: Computational Needs**\n\nFor non-locally hosted LLMs, uncertainty quantification methods generally involve two stages: Obtaining responses from the model via API calls and computing the uncertainty scores based on those responses. This applies to both our method and all baselines; the only exception is Verbalized Uncertainty (VU), which directly returns a score from the API without requiring post-processing. As discussed in the paper, any method that relies on perturbations and/or replications incurs additional computational cost. However, a major advantage is that responses for perturbed inputs can be generated in parallel, making the process more scalable in practice.\n\nWith this in mind, we divide the computational cost into two components:  \n(**A**) Computation time after responses are collected, and  \n(**B**) Total cost, which includes (i) along with API call time and perturbation generation.\n\n**Regarding API cost**:\n  \nOur method introduces $n$ perturbations per question and generates $r$ response replications for each version, resulting in a total of  $(n + 1) × r$ API calls per input. Thus, the cost scales linearly with both the number of perturbations and replications. That said, while our method introduces perturbations, it requires far fewer replications than several existing baselines. For instance, Semantic Entropy relies heavily on large $r$ values (with $n = 0$); prior work recommends $r > 10$ for stable performance (see [2]). In contrast, we demonstrate that even without replication ($r = 1$), our method remains effective. This is evidenced by the strong performance of **NR-Inv-Entropy**, which uses perturbation alone and still consistently outperforms many baselines.\n\n\n**Regarding computation after responses are collected**:\n\nBelow, we present the compute results, without parallelization. The reported numbers are averaged over all sampled questions in TriviaQA. As shown, **our method is highly efficient in the post-processing stage** when computing Inv-Entropy scores. Furthermore, in settings where response generation time is negligible, such as when LLMs are deployed locally and perturbations are processed in parallel, our total computation cost is often lower than that of existing baselines.\n\n| Method                | Peak GPU Memory (MB) | Time A (s) | Time B (s) |\n|----------------------|----------------------|----------------------|-----------------------|\n| Semantic Entropy      | 3575.31              |      13.985          |      20.183                 |\n| VU                    | 0                    | 0                    |       0.620                |\n| P(True)               | 0                    |    0.001             |       2.201                |\n| LexSim                | 0                    |      0.022           |         3.189               |\n| DegMat                | 1610.29              |      6.143          |              15.905         |\n| LUQ                   | 1580.21              |      3.821           |             4.432          |\n| KLE                   | 1608.52              |      1.323           |             6.725          |\n| Inv‑Entropy (Ours)    | 86.65                | 1.990                 |   10.323                    |\n| NR‑Inv‑Entropy (Ours) | 86.65                | 1.769           |        6.358               |\n\n>**W3 & Q2: Perturbation function \\& maintain semantic equivalence**\n\nWe would like to start by noting that our framework and the Inv-Entropy metric are compatible with any perturbation algorithm. Notably, Table 1 in the main paper demonstrates our strong performance even when using ChatGPT-based paraphrasing as the perturbation method. As a form of rewriting rather than arbitrary perturbation, ChatGPT-based paraphrasing can be reasonably assumed to preserve semantic equivalence.\n\nRegarding our proposed GAAP approach, while we cannot guarantee perfect semantic equivalence, we include a parameter $\\delta$ that controls the maximum allowable deviation from the original question to preserve closeness to the input. Interestingly, it is precisely this small deviation that allows GAAP to explore a broader input space, contributing to its superior performance compared to ChatGPT-based paraphrasing (see Figure 4 in the main paper).\n\nAdditionally, in more sensitive applications, one could incorporate an external filtering mechanism (e.g., an LLM) to remove GAAP-generated perturbations that deviate too far semantically. Such heuristics serve as a practical safeguard against rare but problematic cases. We will revise the discussion in the final version of the paper to clarify this aspect more thoroughly.\n\n>**References**\n\n[1] \"Natural questions: a benchmark for question answering research.\" Transactions of the Association for Computational Linguistics, 2019.\n\n[2] \"Semantic Uncertainty: Linguistic Invariances for Uncertainty Estimation in Natural Language Generation.\" The Eleventh International Conference on Learning Representations, 2023."}, {"review_id": "aBCVMETVWR", "reviewer": "Reviewer_uG7q", "summary": "The authors propose a new framework for disentangling aleatoric and epistemic uncertainty in settings where one can perturb input prompts. The method effectively unifies semantic entropy and input perturbation-based approaches, permitting more sophisticated judgements about a model's uncertainty (capturing both cases where one particular input results in many semantically related outputs and also those where one particular set of outputs is yielded by many semantically distinct questions). The authors conduct extensive experiments using one instantiation of their framework and show that it outperforms common baselines at the task of predicting correctness.", "questions": "Why is the TriviaQA AUROC for semantic entropy in Table 1 so low? In that paper they report TriviaQA AUROCs of > 0.8 (granted, using OPT).\n\nHow does the new method compare to a simple ensemble of existing semantic entropy and input perturbation based approaches?", "limitations": "As I mentioned, the paper needs to discuss limitations in more detail.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\n\nThe method is general, combines elements of several baselines from the literature (as I mention above), and is overall quite elegant. The paper is exceptionally clearly written and the experimental results also seem quite strong. I really like this paper.\n\nWeaknesses:\n\nSome small gripes:\n\n1. I think both figures are quite opaque. Figure 1 depicts level sets as concentric, which is not really accurate. Figure 2 is also pretty confusing; it would help to add indices to the caption.\n2. The paper should really include a comparison of the runtimes of different baselines. How much of the benefit of the new approach comes from simply spending more on compute?\n3. Related work and limitations sections are both pretty sparse (yes, there's related work in the appendix, but it would be nice to make room for more in the main paper). An obvious limitation is that this is only tested (and only really applicable at all) on short-form Q/A.\n4. Some of the evaluated tasks (e.g. TriviaQA) exhibit very little aleatoric uncertainty, and I imagine that similarity scores between different outputs on these tasks are systematically low. It would be nice to see this special case worked out theoretically.", "was_revised": true, "final_justification": "I wouldn't say this paper reaches the level of \"groundbreaking impact,\" but it's very strong and I hope it is accepted.", "rebuttal": "First, we would like to thank you for the encouraging and thoughtful review. We sincerely appreciate your positive feedback on the generality, clarity, and elegance of our framework, and we’re glad the experimental results resonated with you.\n\n\n>**W1: Figures 1 \\& 2**\n\nThank you for the helpful feedback. For Figure 1, we will revise the illustration to eliminate the misleading appearance of concentric level sets and clarify that it is meant as a conceptual sketch. For Figure 2, we will add a detailed explanation of the transitions involving the highlighted nodes in the final version, to help readers grasp our model framework more intuitively.\n\n>**W2: Computational Needs**\n\nGreat suggestion. We would also like to point you to our response (under \"W2: Computational Needs\") to **Reviewer Q6Vh**, who raised a similar question, where we also provide supporting experimental results. Overall, our method is highly efficient in the post-processing stage, after obtaining responses from the perturbations via multiple API calls. The main computational cost arises from the API calls themselves; however, responses for perturbed inputs can be generated in parallel, making the process significantly more scalable in practice.\n\n>**W3: Related Work in the Appendix**\n\nAs you suggested, we will bring the related work section of the appendix into the final version of the main paper.\n\n>**W3: Applicability to Long Form Q/A**\n\nYou raise a great point. Indeed, our inverse perspective for defining the final metric was designed with short answers in mind; since the shorter the answer, the more important it becomes to explore the diversity of the input space. That said, our probabilistic framework is general and can be applied to any answer format, with various metrics derivable from it. For instance, in the paper, we also present results using non-inverse entropy as well as the Wasserstein distance between $p(x)$ and $p(y)$. Exploring how different metrics perform across answer lengths is a promising direction we leave for future work.\n\nTo fully address your concern, we have added additional experiments on a long-form answer dataset, where we study the method’s sensitivity to answer length. **These results are provided in our rebuttal to Reviewer Q6Vh**. Our experiments show that the method remains effective in long-form QA settings. Due to space constraints, we refer you to our response to **Reviewer Q6Vh** (under \"W1 & Q1: New experiments on a long-form answer dataset\"), who raised a similar question.\n\n>**W4: Extreme-Case Analysis: Small or no Aleatoric Uncertainty**\n\nIf we understand your question correctly, you are asking us to consider what happens in cases with very little aleatoric uncertainty; for instance, when perturbations of an input yield the same output each time. To address your concern, we attempt to understand this scenario mathematically by analyzing an extreme case in which perturbing an input always yields exactly the same output. This case may not be uncommon in tasks involving very short-form answers.\n\nMathematically, suppose all outputs are identical, even after perturbing the inputs: $y_0 = y_1 = \\cdots = y_n$. As a result, all pairwise output similarities are maximized: $a_{\\text{similarity}}(y_i, y_j) = 1$ for all $i, j$. \n\n**1. Identical Inputs.** Suppose all perturbed inputs are identical: $x_0 = x_1 = \\cdots = x_n$, and all outputs are also identical: $y_0 = y_1 = \\cdots = y_n$. Then $a_{\\text{similarity}}(x_i, x_j) = a_{\\text{similarity}}(y_i, y_j) = 1$ for all $i, j$, and both $\\text{P}_x$ and $\\text{P}_y$ become uniform matrices with entries $1/(n+1)$. The conditional distribution becomes $P(X = x_i \\mid Y = y_j) = 1/(n+1)$, and the Inv-Entropy is maximized: $H(X \\mid Y) = \\log(n+1)$.\n\n**2. Slightly Varying Inputs.** If inputs differ slightly, say $a_{\\text{similarity}}(x_i, x_j) = 1 - \\epsilon$ for small $\\epsilon$, then $P_x$ becomes nearly uniform, with diagonal entries slightly higher. As a result, $P(X \\mid Y)$ deviates mildly from uniformity, and the entropy decreases slightly: $H(X \\mid Y) = \\log(n+1) - \\mathcal{O}(\\epsilon^2)$.\n\n**3. More Diverse Inputs.** When inputs are semantically diverse, $a_{\\text{similarity}}(x_i, x_j)$ spans a wide range. The matrix $\\text{P}_x$ becomes structured, and $P(X \\mid Y)$ becomes more peaked. Consequently, Inv-Entropy drops further, reflecting reduced uncertainty due to input-output specificity.\n\nThese regimes confirm that Inv-Entropy is sensitive to input variability even when outputs remain constant; precisely as intended by the inverse modeling perspective.\n\n**Thank you for raising this question. We have not thought of showing this analysis before and it largely enriches our paper**.  \n\n>**Q1: Difference in Reported AUROC for Semantic Entropy on TriviaQA**\n\nThank you for this insightful question. There are several key differences between our setup and that of the prior Semantic Entropy paper [1] that explain the discrepancy in AUROC values:\n\nFirst, the prior work used OPT, a white-box model with access to token-level logits. In contrast, we use ChatGPT-3.5-Turbo, a black-box model. While both implementations of Semantic Entropy cluster outputs by semantic meaning, the method for estimating cluster probabilities differs. The white-box version uses log-probabilities to weight clusters more precisely. The black-box version, which lacks access to logits, estimates cluster probabilities using relative frequency. This approximation is inherently less precise and tends to result in lower AUROC scores.\n\nSecond, we use ChatGPT to assess whether a generated answer is semantically equivalent to the gold reference. The original Semantic Entropy paper, however, used ROUGE-L scores, which rely on lexical overlap and tend to overestimate correctness for similar-looking outputs.\n\nThird, we use instruction-style prompting uniformly across all methods. The prior work adopted few-shot prompting via in-context learning, which is known to enhance both accuracy and calibration, especially for white-box models.\n\n\n>**Q2: Adding Semantic Entropy–Perturbation Ensemble Baselines**\n\nFollowing your suggestion, we investigate two ensemble strategies for combining **Semantic Entropy** with input perturbations to enhance uncertainty estimation. We adopt the notation and perturbation setup introduced in Section 2.5 of the main paper. \n\nLet $S_{x_0}$ denote the original question prompt, and let $\\text{Per}(S_{x_0}) = \\lbrace S_{x_0}, S_{x_1}, \\dots, S_{x_n} \\rbrace$ be the set of $n+1$ semantically perturbed versions (including the original). For each perturbed prompt $S_{x_i}$, we sample $r$ responses from the language model, yielding response sets $R_i = \\lbrace y_{i,1}, \\dots, y_{i,r}\\rbrace$.\n\nWe consider the following:\n\n* **Semantic Entropy.**  \n  The original method computes entropy solely over the $r$ responses to the original prompt $S_{x_0}$. No perturbation is used.\n\n* **Semantic Entropy (Joint).**  \n  This ensembled  Semantic Entropy computes entropy over all responses from the original and perturbed prompts $ \\mathcal{R}_{\\text{joint}} = R_0 \\cup R_1 \\cup \\dots \\cup R_n$.\n\n* **Semantic Entropy (Average).**  \n  An entropy score is computed independently for each response set $R_i$, and the final score is the average over all $i$.\n  \nThe results are shown below with the same experiment setting as that of Table 1 in our main paper:\n\n| **Metric (↑/↓)**         | **Method**                     | **TriviaQA**          | **SciQ**              |\n|--------------------------|--------------------------------|------------------------|------------------------|\n| **AUROC (↑)**            | Semantic Entropy      | 0.698 ± 0.048          | 0.611 ± 0.040          |\n|                          | Semantic Entropy (Joint)       | 0.783 ± 0.048          | 0.709 ± 0.050          |\n|                          | Semantic Entropy (Average)     | 0.784 ± 0.046          | 0.698 ± 0.053          |\n|                          | Inv-Entropy (Ours)             | **0.810 ± 0.051**      | **0.771 ± 0.051**      |\n| **PRR (↑)**              | Semantic Entropy     | 0.813 ± 0.040          | 0.725 ± 0.047          |\n|                          | Semantic Entropy (Joint)       | 0.877 ± 0.039          | 0.820 ± 0.028          |\n|                          | Semantic Entropy (Average)     | 0.873 ± 0.039          | 0.805 ± 0.027          |\n|                          | Inv-Entropy (Ours)             | **0.920 ± 0.030**      | **0.843 ± 0.054**      |\n| **Brier (↓)**            | Semantic Entropy     | 0.154 ± 0.019          | 0.168 ± 0.023          |\n|                          | Semantic Entropy (Joint)       | 0.134 ± 0.023          | 0.154 ± 0.018          |\n|                          | Semantic Entropy (Average)     | 0.140 ± 0.021          | 0.158 ± 0.018          |\n|                          | Inv-Entropy (Ours)             | **0.117 ± 0.018**      | **0.149 ± 0.022**      |\n| **TSU (0.7, 1.4)**       | Semantic Entropy      | 17.35%                 | 25.51%                 |\n|                          | Semantic Entropy (Joint)       | 59.18%                 | 61.86%                 |\n|                          | Semantic Entropy (Average)     | 63.27%                 | 73.20%                 |\n|                          | Inv-Entropy (Ours)             | **88.78%**             | **80.61%**             |\n\nAdding perturbation-based ensemble variants of Semantic Entropy improves its performance. However, Inv-Entropy still consistently outperforms all variants across all metrics and both datasets, confirming the strength of our approach.\n\n\n>**References**\n\n[1] \"Semantic Uncertainty: Linguistic Invariances for Uncertainty Estimation in Natural Language Generation.\" The Eleventh International Conference on Learning Representations, 2023."}, {"review_id": "OgKXjkIGCA", "reviewer": "Reviewer_tjtF", "summary": "Although there are lots of uncertainty quantification methods which trying to measure the uncertainty of Large Language Model, they are usually heuristic. To address this issue, in this paper, they propose a new uncertainty quantification framework called Inv-Entropy which is based on the perturbation algorithm called GAAP. Before suggesting the method, the authors raise motivations and Lemma, utilizing a simple 1d case which helps readers to understand more clearly.", "questions": "See the Weaknesses section.", "limitations": "See the Weaknesses section.\n\nOverall I like the paper and I am willing to increase my rating after seeing the response from the authors.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n- The paper is well-written and easy to follow\n- The motivation is clear and give appropriate justification of their proposed method.\n- Description of their methods, especially for the modeling part of their conditional probability and multiple sampling parts, are clear and understandable.\n- Experimental results show that Inv-Entropy clearly outperforms other methods.\n\nWeaknesses\n- The most clear and straightforward concern about the proposed method is the computational cost. It would be helpful to include an analysis of the exact computational time and memory cost.\n- This method seems to rely quite heavily on certain hyperparameters, such as $n$ and $S$. It would be helpful to see how robust the method is through an ablation study on these parameters.\n- It would be helpful for understanding if there were practical entropy analyses or qualitative examples showing cases where the model outputs different $y$ values for similar $x$, or similar $x$ values for different $y$.\n- Including an ablation study on the augmentation used in the GAAP method would help readers better understand the appropriate level of augmentation needed to estimate $\\hat{H}$. It would also be beneficial to add ablation studies on the GAAP method's hyperparameters, $\\delta$ and $\\tau$, to analyze their impact. Additionally, incorporating ablations on cases without crossover or without mutation would further aid in understanding the individual contributions of these components.", "was_revised": true, "final_justification": "I am satisfied with the author's response.", "rebuttal": "First, we would like to thank you for your constructive review. We appreciate your positive remarks regarding the clarity of our writing, the strength of our motivation and theoretical foundation, and the performance of our proposed Inv-Entropy method. We also sincerely thank you for noting that you like the paper and are open to increasing your rating.\n\nBelow we provide an itemized response to your suggestions & concerns:\n\n>**W1: Computational Needs**\n\nGreat suggestion. We would like to point you to our response (under \"W2: Computational Needs\") to **Reviewer Q6Vh**, who raised a similar question, where we also provide supporting experimental results. Overall, our method is highly efficient in the post-processing stage, after obtaining responses from the perturbations via multiple API calls. The main computational cost arises from the API calls themselves; however, responses for perturbed inputs can be generated in parallel, making the process significantly more scalable in practice.\n\n>**W2: Sensitivity Analysis on Hyperparameters**\n\nThank you for the insightful suggestion. To fully address your comment, we have now run ablation studies on the parameters you highlighted. \n\n>>**The number of perturbations n**:\n\n| Metric         | n = 3 | n = 6 | n = 9 |\n|----------------|-------|-------|--------|\n| AUROC          |   0.725 ± 0.063    |   0.815 ± 0.052    |  0.837 ± 0.047      |\n| PRR            |   0.813 ± 0.057    |   0.884 ± 0.049    |    0.904 ± 0.042    |\n| Brier          |    0.145 ± 0.022   |   0.122 ± 0.021    |     0.115 ± 0.020   |\n\nThe table above shows the effect of perturbation times $n$ on Inv-Entropy using ChatGPT-3.5-Turbo on TriviaQA with $r=5$. We observe that as the number of perturbations $n$ increases, the performance of Inv-Entropy improves consistently across all metrics. Notably, even with as few as 3 perturbations, the method still achieves competitive results, indicating robustness to the choice of $n$.\n\n>>**The number of replications r**:\n\n| **Metric** | **TriviaQA**        |                   |                   | **SciQ**           |                   |                   |\n|------------|---------------------|-------------------|-------------------|--------------------|-------------------|-------------------|\n|            | *r = 2*           | *r = 4*           | *r = 6*           | *r = 2*            | *r = 4*           | *r = 6*           |\n| **AUROC**  | 0.812        | 0.815      | 0.820      | 0.794       | 0.801      | 0.806      |\n| **PRR**    | 0.916       | 0.915      | 0.915      | 0.888      | 0.888      | 0.892     |\n| **Brier**  | 0.128      | 0.121      | 0.123     | 0.143      | 0.140     | 0.136      |\n\nThis table presents the performance of Inv-Entropy on the TriviaQA and SciQ datasets using ChatGPT-3.5-Turbo as the number of replications $r$ increases. Although performance improves with larger $r$, the gains are relatively small. The results with $r = 2$ are already strong, confirming our claim in the \"Computational Needs\" that our framework does not rely heavily on a large number of replications. Compared to the ablation study on the number of perturbations $n$, we find that model performance is much more sensitive to changes in $n$ than in $r$. \n\n**These results also shed light on our theoretical findings, which highlight the importance of perturbation for uncertainty quantification**.\n\n>>**The temperature t**:\n\n| **Metric** | **TriviaQA**     |             |             | **SciQ**         |             |             |\n|------------|------------------------|-------------------|-------------------|------------------------|-------------------|-------------------|\n|            | *t = 0.3*        | *t = 1.0*   | *t = 1.4*   | *t = 0.3*        | *t = 1.0*   | *t = 1.4*   |\n| **AUROC**  | 0.870            | 0.827       | 0.810       | 0.767            | 0.754       | 0.795       |\n| **PRR**    | 0.939            | 0.936       | 0.903       | 0.900            | 0.864       | 0.870       |\n| **Brier**  | 0.085            | 0.117       | 0.132       | 0.151            | 0.153       | 0.146       |\n\nThe table above presents an ablation study on the effect of temperature $t$ in the response sampling process for Inv-Entropy, evaluated on the TriviaQA and SciQ datasets using ChatGPT-3.5-Turbo. We observe that lower temperatures lead to better performance across all metrics. This suggests that reducing randomness in the model’s outputs helps stabilize the uncertainty estimation and improve reliability. Nonetheless, Inv-Entropy remains effective across a range of temperature settings, demonstrating its robustness.\n\n>>**The bootstrapping iterations S**:\n\nPlease refer to Table 6 in the appendix. We observe that increasing S initially enhances the performance of our method. However, beyond a certain threshold $S=10$, further increases yield diminishing or no significant returns. Therefore,\nusing more than 10 iterations appears to provide limited additional benefit.\n\n\n>**W3: Qualitative examples for responses with similar x and different y**\n\nThe following table presents a representative case from the TriviaQA dataset, where semantically similar questions yield highly diverse answers. \n\n|          | original question                                                  | perturbed question1                                                     | perturbed question2                                                               | perturbed question 3                                                      |\n|--------------------|--------------------------------------------------------------------|---------------------------------------------------------------------------|--------------------------------------------------------------------------------------|------------------------------------------------------------------------------|\n|          | In 1999 Anna Kournikova signed a lucrative contract to model what? | What did Anna Kournikova sign a profitable agreement to showcase in 1999? | In 1999, Anna Kournikova agreed to a well-paying contract to promote which product? | Anna Kournikova inked a financially rewarding deal in 1999 to endorse what? |\n| Answer 1 | Sports Illustrated Swimsuit Issue                                  | Omega Watches                                                             | Taco Bell                                                                           | Adidas                                                                      |\n| Answer 2 | Sports Illustrated Swimsuit Issue                                  | Adidas                                                                    | Berlei Bras                                                                         | Adidas                                                                      |\n| Answer 3 | Sports Illustrated Swimsuit Issue                                  | K-Swiss                                                                   | Tennis shoes                                                                        | Adidas                                                                      |\n| Answer 4 | Sports Illustrated Swimsuit Issue                                  | Adidas                                                                    | Tennis shoes                                                                        | Adidas                                                                      |\n| Answer 5 | Sports Illustrated Swimsuit Edition                                | Omega Watches                                                             | K-Swiss                                                                             | Adidas                                                                      |\n\nAnswers 1 through 5 correspond to independent replications for each question. This table clearly shows the value of our **perturbation-based approach**. If one only relies on replications and compares multiple outputs directly, as in other methods, the underlying uncertainty in this case would not be captured. By introducing semantically meaningful perturbations, our method is able to reveal this uncertainty and provide a more informative uncertainty estimate.\n\n\n\n>**W4: Sensitivity analysis on GAAP**\n\n| Metric   | $\\delta = 0.3$ | $\\delta = 0.5$ | $\\delta = 0.7$ | $\\delta = 0.9$ |\n|----------|----------------|----------------|----------------|----------------|\n| AUROC    | 0.723          | 0.752          | **0.843**      | 0.740          |\n| PRR      | 0.792          | 0.843          | **0.917**      | 0.801          |\n| Brier    | 0.165          | 0.143          | **0.103**      | 0.161          |\n\nThis table reports the performance on the TriviaQA dataset using ChatGPT-3.5-turbo. As the similarity threshold $\\delta$ decreases, more loosely related perturbated questions are included. Among the settings, $\\delta = 0.7$ achieves the best overall performance, striking a balance between semantic diversity and fidelity to the original question. In contrast, overly high or low thresholds result in worse performance.\n\n\n**Regarding the comment on isolating the effects of crossover and mutation**, we note that our GAAP algorithm integrates these operations in a tightly coupled manner, making it non-trivial to disable one without disrupting the overall perturbation quality or validity. We will clarify this more explicitly in the final version."}, {"review_id": "4HajKVr1C0", "reviewer": "Reviewer_pA2S", "summary": "This paper presents Inv-Entropy, a novel fully probabilistic framework for uncertainty quantification (UQ) in large language models (LLMs). It introduces an inverse modeling perspective grounded in random walk theory to characterize the diversity of inputs that could yield a specific output. Key contributions include a dual Markov chain formulation over input–output semantic embeddings and a new UQ metric (Inv-Entropy), supported by Monte Carlo bootstrapping. Further proposals are a GAAP perturbation algorithm utilizing genetic operations to enhance input diversity and an evaluation metric called Temperature Sensitivity of Uncertainty (TSU) that does not rely on correctness labels. The method is evaluated across various datasets and compared to state-of-the-art uncertainty quantification (UQ) baselines, demonstrating consistent improvements.", "questions": "- Past literature [2,4,5] did not use a separate holdout set for their evaluation. Their approaches have likely been optimized on the test set. Did the authors follow this practice, or was a holdout set used? And if not, what is the justification?\n- How are the hyperparameters chosen? Over what range of hyperparameters was optimized?\n- Lemma 2.1 uses the variance of a distribution, which depends on the tangent of $f^*$. However, $f^*$ is not known in practice. How does this theory and its discussion relate to the algorithm used in the experiments?\n\n**References (continued)**\n\n- [4] Kuhn, L., Gal, Y., & Farquhar, S. (2023). Semantic uncertainty: Linguistic invariances for uncertainty estimation in natural language generation. ICLR.\n\n- [5] Nikitin, A., Kossen, J., Gal, Y., & Marttinen, P. (2024). Kernel language entropy: Fine-grained uncertainty quantification for llms from semantic similarities. NeurIPS.", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 4, "originality": 3, "strengths_and_weaknesses": "## Overall\n\nIn summary, the paper shows significant advancements in uncertainty quantification for large language models, an important research area, but it also comes short in the clarity of the presentation. Even though the core contributions seem to be strong enough, a plethora of minor issues make the work appear unfinished. Especially the almost non-existent discussion of TSU in the main part, which is sold as a contribution and used to motivate the proposed algorithm, seems to be problematic. In its current state, I cannot recommend this work to the NeurIPS audience, but I am looking forward to an improved version.\n\n## Strengths\n- Flexible and novel probabilistic framework applicable to natural language processing\n- Important and relevant application\n- Simple algorithm\n- Extensive evaluation and comparison of baselines and datasets\n- Very strong empirical performance\n\n## Weaknesses\n- Temperature Sensitivity of Uncertainty (TSU) is highlighted as a contribution and takes up a large part of the evaluations, but is never clearly motivated and explained beyond a few sentences in the main paper. The authors should decide whether they want to properly include TSU in the main part (which comes with a proper explanation) or if its results are moved to the appendix, where they have enough space for the explanation.\n- There seems to be confusion about what calibration-based metrics are. A calibration-based metric quantifies in some way how close predicted probabilities are to ground-truth probabilities (c.f. [1]). The Brier score partly quantifies this; however, the AUROC and PRR do not. These are rejection-based metrics, which assess how informative an uncertainty metric is with respect to a correctness measure. Based on my understanding, calibration is not part of this research direction and should not be mentioned in this work (without a very extensive discussion).\n- Discussion of theoretical foundations misses references. There is a large corpus of literature on random walks, semantic similarities, and tangent spaces; however, almost nothing is cited here. The theoretical framework is labelled as a major contribution, but it is not possible to put it into relation with other theoretical frameworks without actively searching for related theory outside of this work.\n- Some of the algorithmic decisions seem ad hoc without theoretical connections, e.g., it is not made clear what theoretical quantities Monte Carlo and bootstrapping aim to estimate; why use bootstrap sampling with S samples when we generate r samples from the model? Why not use the r samples directly?\n- I am not sure including evaluations with WD-px-py and MAX-py in the main paper makes sense, since they are not properly discussed, and considering there are more fundamental discussions (like TSU) missing\n\n### Minor issues\n- Line 30: It would be nice to spell out UQ the first time it is mentioned (besides the abstract)\n- Line 52-65: Contributions could be more concise and compressed\n- Line 74-79: Missing references for input perturbations and equivalence classes\n- Line 82: ';' is an atypical set notation (use ':' or '|')\n- Line 80-87: Missing reference for tangent space and missing justification for gaussian distribution\n- Eq 3: Add bracket for matrix/vector multiplication to highlight order of operators when indexing (you did it right in Eq 4)\n- Line 160: that -> there\n- Line 169-172: This paragraph is very important. Maybe emphasize more? Further, the notation seems strange: First, we are not defining; we are making statements (so no delta). Second, it seems inconsistent to write $\\mathbb{P} ( Y = y_j \\mid X = x_i ) = \\mathbb{P} ( y_j \\mid x_i )$. Either $x_i$ and $y_j$ are constants or random variables, but using them as both seems confusing.\n- Eq 5: Flip the side of the $=$, since $H(Y|X) := - \\sum_i P(Y=y_i|X=x_i) \\log P(Y=y_i|X=x_i)$ is the known definition (and provide citation)\n- Line 176: Citation for wasserstein distance would be nice since it is used later on\n- Eq 6: $S$ is already used as variable (although with a subscript). A different variable would be nice\n- Algorithm 1: The following inputs (as far as I have understood) are missing: r, S, $\\delta$, $\\tau$, T\n- Fig. 3: text in fig too small. Maybe the fig margins are also too small? \n- Line 233: There are already plenty of baselines in use, which is immaculate; however, a work [2] from last year's ICML clearly outperformed semantic entropy and P(True) on almost similar benchmarks. Considering that KLE is more recent and used as a baseline, the method of [2] should be at least cited.\n- Line 239-240: Please correct \"P(Y |X) derived in Sec. 2.3 instead of P(Y |X)\"\n- Line 241: Add an explanation of why comparing the marginal distributions of input and output makes sense\n- Line 243: AUROC and PRR are not calibration metrics\n- Line 248: \"map map\"\n- Line 268: The Brier score allows to compare models, so we can say that model A is likely better calibrated than model B (even though the Brier score is not only about calibration but also about sharpness, c.f. [3]). But we cannot say under what threshold of the Brier score a model is well-calibrated. Please reformulate.\n- Table 1: Please explain how temperature 0.7 was chosen\n- Table 2: Too bloated for main part, might fit better in appendix\n- Figure 4: The figure presents important insights, but the font is too small to read\n\n\n**References**\n\n[1] Vaicenavicius, J., Widmann, D., Andersson, C., Lindsten, F., Roll, J., & Schön, T. (2019). Evaluating model calibration in classification. In The 22nd international conference on artificial intelligence and statistics (pp. 3459-3467). PMLR.\n\n[2] Gruber, S. G., & Buettner, F. (2024). A Bias-Variance-Covariance Decomposition of Kernel Scores for Generative Models. In International Conference on Machine Learning (pp. 16460-16501). PMLR.\n\n[3] Bröcker, J. (2009). Reliability, sufficiency, and the decomposition of proper scores. Quarterly Journal of the Royal Meteorological Society: A journal of the atmospheric sciences, applied meteorology and physical oceanography, 135(643), 1512-1519.", "was_revised": true, "final_justification": "Most of my concerns have been solved, which warrants an increase in the score.\nHowever, my major concerns (and this is shared by another reviewer) are the clarity and the presentation, which would require me to reread the revised manuscript.\nThis is important since the theoretical framework is introduced without citing the relevant literature it is based on, and TSU is presented as a contribution but not introduced and discussed in the main paper.\nThe authors promise to change this in the camera-ready version, and I trust them to a certain degree.\nHowever, the extent of the changes is big enough to keep me from giving a clear acceptance without being able to read the revision.", "rebuttal": "First, we would like to thank you for your detailed and thoughtful review. We appreciate your recognition of the significance and originality of our work, as well as your positive assessment of our empirical results and the flexibility of our proposed framework. \n\n>**W1: TSU**\n\nYour suggestion regarding the placement of the TSU definition is very well received. Due to space constraints and our decision to prioritize the modeling framework, we initially placed TSU in the appendix rather than the main paper. Its simplicity and generality are precisely why we introduced it. We will bring the definition and motivation into the beginning of section 3 of the main paper as you recommend. \n\n>**W2: Calibration-based Metrics**\n\nThank you for pointing out the naming issue. In the final version, we will revise all mentions of “calibration-based metrics” to the more appropriate term “correctness-based metrics” when referring collectively to AUROC, PRR, and Brier score. We include the Brier score not only because it partially reflects calibration ability, but more importantly because it enables a more fine-grained assessment of the alignment between predicted confidence (the negative of the uncertainty measure) and correctness labels. While AUROC and PRR are ranking-based metrics, Brier also accounts for the magnitude of the predicted confidence, offering a richer evaluation of how well uncertainty scores correspond to actual correctness.\n\n[Minor Issues-line 268] Regarding our statement in line 268, we will revise the sentence to:\n\n“Inv‑Entropy attains the lowest Brier score, indicating better calibration than the other baselines.”\n\n>**W3: Theoretical references**\n\nWe add the following discussion in Section 2.1:\n\n\"Similarity-based graphs are widely used in structure discovery [6] and feature alignment [7]. Recent work [4,5] leverages graph statistics to quantify LLM uncertainty using only model outputs. Our proposed Inv-Entropy further introduces perturbations in the input space and evaluates LLM uncertainty by assessing the alignment between the input similarity graph and the output similarity graph.\n\nThe use of random-walk-based metrics for graph alignment was first introduced in [8], and we are, to the best of our knowledge, the first to apply such techniques to UQ in LLMs.\"\n\n[6] \"Laplacian eigenmaps for dimensionality reduction and data representation.\" Neural computation, 2003).\n\n[7]  \"Latent common manifold learning with alternating diffusion: analysis and applications.\" Applied and Computational Harmonic Analysis, 2019.\n\n[8] \"Graph kernels.\" The Journal of Machine Learning Research, 2010.\n\n>**W4: Bootstraping and Monte-Carlo**\n\nWe clarify that bootstrapping and Monte Carlo estimation are not ad hoc, but are both theoretically motivated components designed to estimate a core quantity in our framework that integrates perturbations and replications:\n\n$$\\mathbb{E}_{Y_n \\sim \\text{Replications}} [H(X_n \\mid Y_n)]$$\n\n**Why not use all $r$ samples directly?**\nIn our dual random walk formulation, each Markov chain operates over a state space of paired inputs and outputs, $(x_i, y_i)$, with the critical assumption that these pairs are in one-to-one correspondence. However, LLM output $y_i$ is stochastic and such randomness may significantly affect the estimate of $\\text{P}_y$, which is a nonlinear function of $Y_n$. Therefore, we use $S$ replicates of LLM outputs to stabilize Inv-Entropy. \n\nEmpirically, we explored an alternative approach during development, where all $r$ outputs are used simultaneously. This is called \"NR-Inv-Entropy\" in our Table 1 and 2. This method consistently underperforms our proposed bootstrapping strategy, Inv-Entropy, across all evaluation metrics, confirming the concerns about output randomness and estimate stability.\n\n**Why use Bootstrapping with Monte Carlo averaging?**\nFor each bootstrapped set indexed by $s$, we sample one output $y_i^{(s)}$ from the $r$ replications of each $x_i$, forming a valid output set $Y_n^{(s)} = \\lbrace y_0^{(s)}, \\dots, y_n^{(s)} \\rbrace$.\n\nHowever, since each $Y_n^{(s)}$ reflects only one possible realization of outputs, its associated conditional entropy $H(X_n \\mid Y_n^{(s)})$ captures uncertainty only for that particular bootstrapped set  $Y_n^{(s)}$, which is highly stochastic due to the inherent randomness of LLM. To comprehensively estimate the expectation over all possible LLM output realizations, we apply Monte Carlo averaging:\n\n$$\\widehat{H}(X \\mid Y) = \\frac{1}{S} \\sum_{s=1}^{S} H(X_n \\mid Y_n^{(s)}).$$\n\nWe find this Monte Carlo estimator to be numerically stable than \"NR-Inv-Entropy\".\n\n\n>**W5: Including WD-px-py and MAX-py**\n\nWe included WD-px-py and MAX-py to highlight the flexibility of our probabilistic framework, which naturally supports alternative uncertainty metrics beyond Inv-Entropy; such as those based on comparing input and output marginals. Both variants perform competitively, outperforming most standard baselines, which supports their inclusion as valuable design alternatives.\n\nTo also address your later question: comparing the marginal distributions of inputs and outputs is meaningful in our context as they offer alternative approaches to capture the alignment between the diversity of semantic inputs and the spread of model outputs. If semantically distinct inputs lead to highly similar outputs, this mismatch can signal reduced epistemic uncertainty. Metrics like WD-px-py can capture this behavior within our dual random walk construction.\n\nWe will add more detailed explanations in the revised manuscript.\n\n>**Q1 & Q2: Hyperparameter Selection & Holdout Set Usage**\n\nWe emphasize that we did not perform any tuning on the test sets.\n\nAll hyperparameters except for temperature were chosen using a held-out set of 100 samples from the TriviaQA dataset. These samples were excluded from all subsequent evaluations and are not included in any of the reported results. All experimental results in the paper, across all datasets, are computed using these fixed hyperparameters, ensuring that evaluations are conducted on proper hold-out sets. We also note that when perturbing using ChatGPT, the GAAP-specific hyperparameters ($\\delta$, $\\tau$, $T$) are not used.\n\n[Minor Issues-Table 1] The temperature was set to 0.7 because it is the default configuration for many publicly available LLMs. We adopted this setting to reflect realistic usage and ensure that our evaluation aligns with typical deployment scenarios.\n\nAdditionally, we refer to our response to **Reviewer tjtF** (under \"W2 & W4 Sensitivity Analysis\"), where we conducted an ablation study showing that Inv‑Entropy remains robust across different hyperparameter settings.\n\n>**Q3: Lemma 2.1 Relationship with the Algorithm**\n\nSection 2.1 introduces a simplified theoretical prototype to justify the perturb-then-estimate foundation of Inv-Entropy. Lemma 2.1 shows that the variance of inputs mapping to a fixed output under a function $f$ can illustrate how sensitivity to input perturbations reflects predictive inexactness. \n\nThe tangent plane is a mathematical abstraction of the class of semantic invariant inputs to LLMs. In practice, we perturb the input question to different mutants which should have similar responses. Vaguely speaking, Inv-Entropy operationalizes the notion of \"tangent space\" of LLM by perturbing prompts. This is the nature of \"input semantic invariance\". However, generating diverse yet exactly semantic equivalent input perturbations is almost impossible. That is why Inv-Entropy also introduces an input similarity graph and uses a random walk on such graphs to characterize small input semantic variations.\n\nWe will add a detailed correspondence between Section 2.1 and realistic LLMs in the revised manuscript.\n\n>**Minor Issues**\n\nWe appreciate the detailed list of minor issues and will address all of them in the final version, including clarifying notation, adding missing references, fixing typos, improving figures, and streamlining contributions and tables as suggested.\n\n>>**Line 233: Adding Kernel Entropy Baseline in Reference [2]**\n\nThank you for pointing out the work in [2]. To fully address your concern, we added Kernel Entropy in [2] with 5 and 50 replications per input. The results are shown below with the same experiment setting as that of Table 1 in our main paper:\n\n| **Metric (↑/↓)**   | **Method**                  | **TriviaQA** | **SciQ**    |\n|--------------------|-----------------------------|--------------|-------------|\n| **AUROC (↑)**      | Kernel Entropy (5)  | 0.687        | 0.685       |\n|                    | Kernel Entropy (50) | 0.797        | 0.706       |\n|                    | Inv-Entropy (Ours)          | **0.810**    | **0.771**   |\n| **PRR (↑)**        | Kernel Entropy (5)  | 0.794        | 0.812       |\n|                    | Kernel Entropy (50) | 0.884        | 0.837       |\n|                    | Inv-Entropy (Ours)          | **0.920**    | **0.843**   |\n| **Brier (↓)**      | Kernel Entropy (5)  | 0.160        | 0.153       |\n|                    | Kernel Entropy (50) | 0.124        | **0.131**   |\n|                    | Inv-Entropy (Ours)          | **0.117**    | 0.149       |\n| **TSU (0.7, 1.4)** | Kernel Entropy (5)  | 55.56%       | 62.89%      |\n|                    | Kernel Entropy (50) | 84.62%       | 76.73%      |\n|                    | Inv-Entropy (Ours)          | **88.78%**   | **80.61%**  |\n\nWe observe that while 50 replications improve performance compared to 5, our Inv-Entropy approach still outperforms across all metrics. In our updated version, we will cite [2]."}, {"review_id": "hXb1n7Elo0", "reviewer": "Reviewer_cePH", "summary": "The paper considers the uncertainty quantification of LLM question-answering tasks. They propose to a set of prompt variations, and perform random walks between their embeddings, and derive some entropy measures from them. I think the work is measuring how much the prompt variations change the embeddings, but I’m not totally sure. The results show excellent performance on UQ benchmarks.", "questions": "- I overall struggled to understand section 2. I couldn’t connect the presentation to the overall LLM setting (ie. prompt, LLM, next-tokens, responses) or the underlying problem. I also struggled to follow the presentation mathematically due to lack of rigor (eg. not defining variables, functions or their domains). It would be very useful to give a real-world LLM example and denote how all the presented variables and functions relate to it.\n- What is “semantic input”? Are there other types of inputs as well? What does S_x0 mean? What does x0 mean? What are their sizes or domains? Is S a function?\n- What does “LLM” f actually mean? LLMs are quite big models: is f the input-response mapping, or the model architecture, or some abstract model concept? What are their inputs and outputs?\n- What does “semantic embedding” psi mean? Is this related to transformers and next-token predictions, or is this some external model? What are their inputs and outputs?\n- We have x_i = psi(S(x_i)): it seems that nothing happens. Why do we do this, and what does this mean?\n- Xn and Yn contain “samples”. Why are elements of Xn samples? I see nothing stochastic here. Are the samples from some distribution? If so, how is it defined? Is f stochastic? In what way?\n- What does X,Y mean? Are they related to Xn,Yn or xi,yi?\n- What are the two markov chains? How are they defined? Where is the sequential (time) indexing in the markov chain?\n- What are the input and output spaces?\n- In 140 the x0 is described as “question”, but earlier x0 was embedding. Is it both?\n- Eq 3 defines a “marginal” probability that clearly is not marginal probability. Marginal probability describes chance of being in state j no matter where we are in the process without knowing past or future. Eq 3 is some kind of transition probability instead.\n- I can’t understand the \\pi P_y[j] notation. P_y is not defined. \\pi as operator is not defined.\n- Eq 4 is a conditional probability of moving from j to i through some intermediate k. I don’t understand why we do this. The notation should be explicit about how many steps there are. I don’t understand how can we mix the Px and Py matrices here, or what it means. I don’t understand why we move y from x to j, but x from k to i.\n- It’s quite odd that the markov presentation does not contain temporal indexing. Markov chain is a sequence of random variables indexed by time, ie. X_t \\in {0,1,…,n}. Here the time is not present, but instead it seems that X and Y are time-independent (but also a chain…), or maybe they are just some kind of 1 or 2 step chains that never proceed further [but that’s surely not a markov chain]. I find this quite confusing. Can you discuss?\n- I don’t understand why line 170 introduces a 3-step probability, or why it’s called “marginal”.\n- Why do you only consider few-step transition probabilities, and not true marginals of the stationary distribution of the markov chain?\n- I don’t understand what conditional sample entropy means here. Entropy is only defined for random variables with known densities. Samples don’t come with densities. Furthermore, conditional entropy and entropy of a conditional are different things. I think the latter is being used here, but wording implies former. Can you clarify?\n- Can you try to clarify what the entropy eq 5 measures conceptually?\n- What is R_i?\n- After reading the sec 2, I’m a bit confused what did we achieve? It seems that bunch of variations on some prompt are generated. We then quantify how aligned the embeddings of the prompt and its answer are, and compute an entropy of some transitions between variations. I don’t see how this now connects with the tangent space / invariance discussion of sec 2.1, or what this now measures or quantifies. If the \\hat{H} is now high or low, is that good or bad? What do we want to see or achieve here? Do we want to see the the correct pairings (xi,yi) being most probable [why or why not]? What does all of this tell us about LLMs?\n- What do you actually do in the experiments? Do you train something? Do you change the LLM prediction logits at test time in some way? The alg1 tells us that you compute hatH: how is the hatH used to improve the produced logits?\n- What is the nature of uncertainty you quantify? What is “uncertainty” in this paper?\n- The results seem to be contain methods that are not described in the paper", "limitations": "No issues.", "rating": 2, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 2, "originality": 3, "strengths_and_weaknesses": "The idea of using embeddings of prompt and output variations is generally sensible, but I did not fully understand what happens in this paper, so I’m not totally sure of this. The results are excellent, which indicates the method works and does something. \n\nThe clarify of the paper is subpar. The method description is conceptually vague, and I could not concretise the presented ideas to existing LLM ideas. The method description also lacks rigor, which makes the math difficult to follow or verify. The paper requires a major revision on both counts.\n\nI’m not totally sure if the method is well-defined or well-motivated. There is lot of sloppyness in the math, and I don’t understand why the paper considers random walks of just length 2 or 3. Surely one should look at the markov chain stationary distribution. Moreover, I could not understand why random walks are considered in the first place: couldn’t we just measure the concentration of the embeddings more directly?\n\nDespite the great results, the paper is not yet ready for neurips and requires a major overhaul of the presentation to verify the method and its math are sound.", "was_revised": true, "final_justification": "The authors have engaged in informative discussion during response period and the contributions have become clearer. I retain my reject rating due to presentational issues. The paper has some disconnect between the tangent space and random walk arguments, and the random walk presentation is not mathematically sufficiently rigorous. The method is a black-box, and there is little attempt to try to give insights behind the produced uncertainty summary metrics.", "rebuttal": "Thank you for your candid feedback.\n\n>**Regarding Clarity & Mathematical Rigor**: \n\n**Clarity**: We respectfully note that the paper was consistently praised by other reviewers for its readability and accessibility. Reviewer Q6Vh described it as “well written and easy to follow,” Reviewer uG7q called it “exceptionally clearly written,” and Reviewer tjtF similarly stated that it is “well-written and easy to follow.” We appreciate that you noted difficulty in connecting the concepts to existing LLM ideas. However, we would have greatly appreciated more specific pointers to where the exposition could be improved so that we could make targeted revisions.\n\n**Mathematical rigor**: We respectfully disagree with the claim that the paper lacks mathematical rigor. Section 2.1 is built on foundational tools from functional analysis and probabilistic modeling, and all core quantities are well-defined. We thoroughly reviewed your comments and found no substantive issues in the mathematics. In fact, other reviewers highlighted the strength of our theoretical contribution; for example, Reviewer Q6Vh emphasized the “highly valuable theoretical perspective,” and Reviewer uG7q described the method as “elegant,” awarding the paper top marks for both quality and originality. If there are particular definitions or results you believe are incorrect or unclear, we would be eager to clarify them.\n\nWe sincerely appreciate your questions and respond to each of your 23 points in detail below, and hope our clarifications help improve clarity and reinforce our contributions.\n\n>**Random walk** \n\n*Why do we consider random walk*: In its core, Inv-entropy examines the alignment between LLM inputs and output embedding. Random walk is a tool we introduce to calculate such alignments. Similarity-based random walk is extremely popular in structure identification [1] and data alignment [2]. Since it is intuitive and flexible, we design our uncertainty measure with the help from random walk.\n\n*Why do we consider random walk with short steps*: Indeed, there are multiple ways to characterize the similarity between the two transition probability. In our paper, we examined the performance of $5$ metrics, Inv-Entropy, NI-Entropy, NR-Inv-Entropy, WD-px-py, MAX-py.  Table 1 shows that Inv-Entropy yields the best performance. \n\nBased on your suggestion, we replace $P(Y)$ in (3) by the stationary distribution of $\\text{P}_y$ and replace the conditional distribution $P(X_i|Y_i)$ by the stationary distribution of $\\text{P}_y\\text{P}_x$. The comparison in performance of Inv-entropy in the TriviaQA dataset (other settings are the same as in the main paper) is provided in the table below. \n\n| Metric         | Stationary | Non-Stationary  |\n|----------------|-------|-------|\n| PRR            |    0.697 ± 0.043    |    0.902 ± 0.042      |\n| Brier          |  0.209 ± 0.017    |    0.117 ± 0.020     |\n\nThe results highlight that using the stationary distribution yields much worse performance. This outcome is further explained below.\n\n[Q15,Q17] We only use two steps of random walk in Inv-Entropy is optimal for two reasons to introduce an implicit bias. The defined marginal distribution $P(Y)$ is the marginal distribution of $n+1$ states after one step of random walk from uniform distribution under $\\text{P}_y$. This provides an implicit bias over $P(Y)$ that it should not differ too much from uniform distribution $\\pi^{\\text{Uniform}}$. Numerically, such an inductive bias provides stability in our estimate. If we instead use the stationary distribution of $\\text{P}_y$ as $P(Y)$, the distribution could significantly deviate from the initial $\\pi^{\\text{Uniform}}$, which results in worse performance in practice.\n\n[Q8,Q14] Since we only consider the Markov chains that mix for at most two steps, we do not introduce a time subscript to simplify notations.\n\n>**The rigor of Section 2.1**\n\n[Q1] Section 2.1 is based on rigorous functional analysis. The domain of $f$ is $\\mathbb{R}^d$ and the output is a scalar in $\\mathbb{R}$. The definitions are elaborated in line $74$ to $87$. We are glad to clarify any potential misunderstanding.\n\nWe also acknowledge that the rigorous analysis in section 2.1 is only a simplification of realistic LLMs, whose inputs are token sequences and outputs are also probabilities in the token space. Nevertheless, we can use $f$ to denote the mapping from LLM input token sequences to one embedding coordinate of LLM output tokens. The purpose of Lemma 2.1 is explained in line 97 to 99.\n\n>**Notations about tokens and embeddings**\n\n[Q2,Q3] Semantic input $S_{x_0}$ is input token sequences to LLM. It is defined in line 105.\n\n[Q4] Input embedding $x_0$ is the embedding of the input sequence $S_{x_0}$ extracted by an external model. We intentionally leave the choice of this external model unspecified to provide practitioners with flexibility in implementation.\n\n[Q5] $\\psi(S_{x_i})$ is the process of converting token sequences to their embedding.\n\n[Q6] $X_n$ is a set of input sequences and $Y_n$ is the set of corresponding LLM outputs. A more appropriate name should be “elements”.\n\n[Q9] The input space and output space are $X_n$ and $Y_n$, respectively.\n\n>**Definitions about marginal and conditional probability**\n\n[Q7,Q11] In line 130, $X$ and $Y$ are defined as discrete random variables whose supports are $X_n$ and $Y_n$. In principle, there are infinitely multiple ways to define the marginal and conditional probability of $X$ and $Y$. This notation of marginal or conditional should not be confused with the dynamics related to Markov chains. [Q16] This is also the case for line 170.\n\n[Q8,Q12] We only use Markov chains as probabilistic tools to construct the marginal of $p(Y)$ and the conditional distribution $p(X|Y)$. The states, indices, and transition probabilities of the two Markov chains are defined from line 116 to line 123. $\\text{P}_x$ and $\\text{P}_y$ are two matrices defined in (2). $\\pi^{\\text{Uniform}}$ is a vector defined in line 136 to 137. \n\n[Q13] In (4), the summation form of $P(X=x_i|Y=y_j)$ is the direct application of matrix multiplication of the two matrices defined in (2).\n\n[Q10] In line 140, the original question should be $S_{x_0}$, rather than $x_0$.\n\n[Q16] The three step transition defined in line 170 is a natural application of the Bayes rule based on (3) and (4).\n\n>**Motivation for (5)**\n\n[Q17] Thank you for pointing out your concerns about our naming. In fact, (5) is neither entropy or conditional entropy in the standard sense. It is the Shannon entropy of the diagonal entries of $\\text{P}_y\\text{P}_x$, which is dependent on $X_n$ and $Y_n$. Since $X_n$ consists of samples from the input perturbation algorithm and $Y_n$ contains their corresponding outputs, we call (5) conditional sample entropy.\n\n[Q18] For a more intuitive explanation, suppose $\\zeta_0,\\zeta_2,\\cdots,\\zeta_{n}$ are the diagonal entries of matrix $\\text{P}_y\\text{P}_x$. Then $\\zeta_i$ measures how likely is it that, starting at point $i$, a two-step walk through $\\text{P}_y$ then through $\\text{P}_x$ will return to $i$.\n\nAs a result, $H(X_n|Y_n)$ in (5) is equal to $H(X_n|Y_n)=\\sum_{i=0}^{n} - \\zeta_i\\log\\zeta_i$, which decreases with larger $\\zeta_i$. It measures the uncertainty or diversity in self-transition behavior across all states under the combined dynamics of $\\text{P}_y\\text{P}_x$. Low entropy often correspond to high $s_i$'s, which indicate $\\text{P}_y$ and $\\text{P}_x$ are more aligned, which is considered to be good. \n\n[Q21] Section 2.1 introduces a simplified theoretical prototype to justify the perturb-then-estimate foundation of Inv-Entropy. The tangent plane is a mathematical abstraction of the class of semantic invariant inputs to LLMs. Section 2.1 says if $f(\\cdot)$ generates very different outputs to inputs that should have similar outputs, then $f$ is probably inaccurate, which is bad. \n\nHowever, in practice, generating diverse yet exactly semantic equivalent input variants is almost impossible. That is why Inv-Entropy also introduces an input similarity graph and uses a random walk on such graphs to characterize small input semantic variations. Our defined $H$ is a natural statistics that describes the local alignment between input and outputs. This is the nature of our measure of uncertainty.\n\n[Q19] $R_i$ is the set of all LLM response by asking LLM the same question $S_{x_i}$ multiple times.\n\n\n>**Clarifications about our numerical results**\n\n[Q22] In our experiments, we did not train nor improve the logits of LLMs: that's not the task of Inv-Entropy. Instead, we take existing LLMs and evaluate their uncertainty. $\\hat{H}$ is a metric of how uncertain the LLM output is. We cross-validate $\\hat{H}$ with other established measures of uncertainty, including correctness or uncertainty distribution, using AUROC, PRR, and Brier score. Our experiment setup is explained from line 242 to line 252.\n\n[Q23] All methods are defined from line 233 to 241.\n\n**Reference**\n\n[1] Belkin, Mikhail, and Partha Niyogi. \"Laplacian eigenmaps for dimensionality reduction and data representation.\" Neural computation 15.6 (2003): 1373-1396.\n\n[2] Talmon, Ronen, and Hau-Tieng Wu. \"Latent common manifold learning with alternating diffusion: analysis and applications.\" Applied and Computational Harmonic Analysis 47.3 (2019): 848-892."}]

Author rebuttal:

Meta-review: This paper proposes a fully probabilistic framework for uncertainty quantification in LLMs based on an inverse entropy perspective. The reviews for this paper were largely positive, though a clear consensus was not reached, making this a borderline case. Three reviewers (Q6Vh, uG7q, tjtF) recommended acceptance, noting the submission's theoretical grounding, originality, and strong empirical evaluation. Initial concerns from these reviewers regarding the scope (long-form QA), computational cost, and hyperparameter sensitivity were were discussed and reviewers appeared satisfied with the authors' responses. However, reviewers pA2S and cePH raised concerns about the clarity of the presentation and the rigor of the mathematical framework. Reviewer cePH remained unconvinced post-rebuttal, maintaining a recommendation for rejection due to what they perceived as a lack of rigor and motivation. While I sympathize with the points made by Reviewer cePH regarding the clarity of the mathematical presentation, I found the authors' explanations in the rebuttal to be clear. The level of rigor presented in the manuscript appears suitable for the UQ in LLMs community, which straddles theoretical motivations and strong empirical validation. In my assessment, the overwhelmingly positive feedback on the paper's novelty and performance from the other reviewers outweighs this dissenting opinion. I (weakly) recommend acceptance.
Decision: Accept (poster)