Inv-Entropy: A Fully Probabilistic Framework for Uncertainty Quantification in Language Models

Thank you for your candid feedback.

Regarding Clarity & Mathematical Rigor:

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.

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.

We 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.

Random walk

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.

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.

Based 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.

The results highlight that using the stationary distribution yields much worse performance. This outcome is further explained below.

[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.

[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.

The rigor of Section 2.1

[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.

We 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.

Notations about tokens and embeddings

[Q2,Q3] Semantic input $S_{x_0}$ is input token sequences to LLM. It is defined in line 105.

[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.

[Q5] $\psi(S_{x_i})$ is the process of converting token sequences to their embedding.

[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”.

[Q9] The input space and output space are $X_n$ and $Y_n$, respectively.

Definitions about marginal and conditional probability

[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.

[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.

[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).

[Q10] In line 140, the original question should be $S_{x_0}$, rather than $x_0$.

[Q16] The three step transition defined in line 170 is a natural application of the Bayes rule based on (3) and (4).

Motivation for (5)

[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.

[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 $i$.

As 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.

[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.

However, 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.

[Q19] $R_i$ is the set of all LLM response by asking LLM the same question $S_{x_i}$ multiple times.

Clarifications about our numerical results

[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.

[Q23] All methods are defined from line 233 to 241.

Reference

[1] Belkin, Mikhail, and Partha Niyogi. "Laplacian eigenmaps for dimensionality reduction and data representation." Neural computation 15.6 (2003): 1373-1396.

[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.

Dear Reviewer cePH,

We are glad to hear that many of your confusions have been clarified. Please allow us to address your remaining concerns:

Main Concern

We acknowledge that all existing UQ methods for LLMs are currently ad hoc, given the complexities of natural language and transformer architectures. Indeed, this is a field still in its infancy, and much remains to be explored. While we do not claim to have solved the explainability problem, we believe our work represents a significant step forward in quantifying LLM uncertainty inspired by a probabilistic framework. The strong empirical performance across all datasets and evaluation metrics highlights the potential of our approach.

Gaining Insights

As emphasized in both the literature and our paper, UQ in language cannot readily provide interpretable summaries like confidence intervals for continuous variables. Instead, the goal of this growing body of work is to derive metrics that are highly correlated with the LLM's underlying knowledge, so they can inform downstream decisions: such as when to stop a reasoning chain or flag potential hallucinations.

Theory

While the initial motivation uses tangent spaces and first-order Taylor expansions, it serves to build geometric intuition for why a perturb-then-estimate framework should relate to semantic shifts in the output space. The probabilistic approach that follows formalizes this framework by modeling transitions between these perturbed samples using Markov chains, offering practitioners an effective tool to analyze how well LLM outputs align with input variations.

Clarifying Some Misunderstandings

Throughout the paper, we define the capital letter $X$ as a discrete random variable whose values lie in the finite set $X_n$. Similarly, $Y$ is a random variable with values in $Y_n$. In your language, the distributions of $X$ and $Y$ are the mixture of Dirac distributions, and we are only defining the mixture coefficients by (3) and (4). These definitions are introduced to motivate (5). Could you please further clarify what mathematical or notational inconsistencies you found ?

The probabilistic framework in Sections 2.2 and 2.3 aims to justify the construction of (5) using $X_n$ and $Y_n$. We interpret $\text{P}_x$ and $\text{P}_y$ (defined in (2)) as transition probabilities of two Markov chains, where the states correspond to elements of $X_n$ and $Y_n$, respectively. Then, Eq. (5) naturally corresponds to the “entropy” of self-transition probabilities under the composite dynamics of $\text{P}_x$ and $\text{P}_y$.

Importantly, what we refer to as conditional sample entropy is not the entropy of a probability distribution. We use the term “conditional sample entropy” only because its form matches that of Shannon entropy. In fact, it is an unnormalized entropy, intentionally designed to serve as our uncertainty measure.

In the initial rebuttal, we define $\zeta_i$ as a diagonal element of $\text{P}_y\text{P}_x$, where $\zeta_i$ corresponds to $p(x_i \mid y_i)$ as defined in our main paper. Let $\Xi = \sum_i \zeta_i$. We can normalize as:

$$\tilde{\zeta}_i = \frac{\zeta_i}{\Xi}.$$

$\tilde{\boldsymbol{\zeta}} = (\tilde{\zeta}_1, \ldots, \tilde{\zeta}_n)$ then forms a valid probability distribution over $X_n$.

The normalized Shannon entropy is defined as:

$$\text{Entropy}(X_n \mid Y_n) = -\sum_i \tilde{\zeta}_i\log \tilde{\zeta}_i$$

Hence, our Inv-Entropy becomes:

$$H(X_n \mid Y_n) = -\sum_i \zeta_i \log \zeta_i = \Xi \text{Entropy}(X_n \mid Y_n) - \Xi \log \Xi$$

This expression aligns with our goal of modeling two sources of uncertainty:

(1) Total mass $\Xi$: $\Xi$ reflects the overall input-output specificity. Higher values indicate more isolated mappings, which we treat as a factor contributing to uncertainty

(2) Dispersion of $\tilde{\boldsymbol{\zeta}}$: Entropy over $\tilde{\zeta}_i$ captures variability across perturbations. Uniform patterns imply stability, while peaked ones reflect inconsistency.

We will revise the main paper to emphasize this interpretive lens for improved clarity.

Inverse Perspective

Eq. (5) is referred to as "inverse" because it measures the diagonal entries of $\text{P}_y\text{P}_x$, which is dependent on the diversity of elements in $X_n$. On the contrary, ordinary entropy often focuses only on the distribution related to $\text{P}_y$. Based on the reasoning in Section 2.2, Inv-Entropy captures uncertainty through the lens of $P(X|Y)$. We would like to refer the reviewer to our response to Reviewer uG7q under “W4: Extreme-Case Analysis”, which provides a theoretical ablation for the case where perturbing the input yields the same output. This analysis demonstrates that as the diversity of the input increases, Inv-Entropy decreases; highlighting the sensitivity of the approach and its ability to reflect the diversity of the input space that could have produced a given response.

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.

W1: TSU

Your 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.

W2: Calibration-based Metrics

Thank 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.

[Minor Issues-line 268] Regarding our statement in line 268, we will revise the sentence to:

“Inv‑Entropy attains the lowest Brier score, indicating better calibration than the other baselines.”

W3: Theoretical references

We add the following discussion in Section 2.1:

"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.

The 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."

[6] "Laplacian eigenmaps for dimensionality reduction and data representation." Neural computation, 2003).

[7] "Latent common manifold learning with alternating diffusion: analysis and applications." Applied and Computational Harmonic Analysis, 2019.

[8] "Graph kernels." The Journal of Machine Learning Research, 2010.

W4: Bootstraping and Monte-Carlo

We 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:

$\mathbb{E}_{Y_n \sim \text{Replications}} [H(X_n \mid Y_n)]$

Why not use all $r$ samples directly? In 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.

Empirically, 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.

Why use Bootstrapping with Monte Carlo averaging? For 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 $Y_n^{(s)} = \lbrace y_0^{(s)}, \dots, y_n^{(s)} \rbrace$.

However, 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:

$\widehat{H}(X \mid Y) = \frac{1}{S} \sum_{s=1}^{S} H(X_n \mid Y_n^{(s)}).$

We find this Monte Carlo estimator to be numerically stable than "NR-Inv-Entropy".

W5: Including WD-px-py and MAX-py

We 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.

To 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.

We will add more detailed explanations in the revised manuscript.

Q1 & Q2: Hyperparameter Selection & Holdout Set Usage

We emphasize that we did not perform any tuning on the test sets.

All 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.

[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.

Additionally, 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.

Q3: Lemma 2.1 Relationship with the Algorithm

Section 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.

The 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.

We will add a detailed correspondence between Section 2.1 and realistic LLMs in the revised manuscript.

Minor Issues

We 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.

Line 233: Adding Kernel Entropy Baseline in Reference [2]

Thank 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:

We 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].

Dear Reviewer pA2S,

We are very happy to hear that the issues regarding hyperparameters, literature, and calibration have been fully resolved from your perspective. Thank you for your thoughtful engagement!

Regarding the revised manuscript

As you know, we have no control over the NeurIPS policy that disallows uploading a revised version during the rebuttal phase. We sincerely hope that this policy does not negatively affect your assessment of our work. As you know, the camera-ready version is intended to address such concerns, and we will move the details about TSU from the appendix to the main paper to fully address your concern.

To help address your concerns, we have included below (at the end of this comment) the wording we intend to add to the main text regarding TSU in the revised manuscript.

We would also like to kindly note that TSU is not the central contribution of the paper. This, in addition to its simplicity, is why we placed it in the appendix. Our central contribution lies in the proposed probabilistic framework and its state-of-the-art performance across multiple metrics and in comparison to various benchmarks.

Regarding clarity

With respect to clarity and broader reviewer feedback, we would like to respectfully highlight that several reviewers explicitly praised the presentation, theoretical contributions, and clarity of our paper:

• Reviewer Q6Vh called the paper “well written and easy to follow”, and emphasized the “highly valuable theoretical perspective.”

• Reviewer uG7q described the paper as “exceptionally clearly written” and “elegant,” awarding it top marks for both quality and originality.

• Reviewer tjtF stated that the paper was “well-written and easy to follow,” noted that the modeling section was “clear and understandable,” and praised both the motivation and empirical results.

Reviewer cePH was the only reviewer who expressed concerns about clarity. We believe these concerns may stem from field-specific terminology rather than a lack of rigor or explanation. For instance, the reviewer questioned terms such as “semantic input” and “semantic embedding”, which are standard in the representation learning and NLP literature. We fully understand, however, that these terms may not be immediately accessible to readers from other methodological communities.

We are very grateful for your engagement and sincerely hope our clarifications help strengthen your confidence in the revised work.

TSU discussion to be brought from appendix to main paper

Using correctness as a proxy for uncertainty quantification (UQ) has clear limitations in open-ended or weakly supervised settings where correctness is unavailable or ill-defined. To circumvent this, we introduce a new evaluation metric: Temperature Sensitivity of Uncertainty (TSU), which captures the proportion of instances where uncertainty increases consistently with temperature.

This metric is motivated by the role of temperature in controlling the randomness of sampling in LLMs. Specifically, higher temperatures flatten the token distribution, increasing entropy and randomness, while lower temperatures make generation more deterministic [Hinton et al. 2015].

Formally, let $\mathbb{T}_1 < \mathbb{T}_2 < \dots < \mathbb{T}_n$ be a sequence of temperature values. Then TSU is defined as:

Formally, TSU is defined as:
$TSU(\mathbb{T}_1, \mathbb{T}_2, \dots, \mathbb{T}_n) = \frac{1}{|\mathcal{D}|} \sum \mathbb{I}\left( \text{UQ}(S_x, \mathbb{T}_1) < \text{UQ}(S_x, \mathbb{T}_2) < \cdots < \text{UQ}(S_x, \mathbb{T}_n) \right)$

where $\mathcal{D}$ denotes the dataset, $S_x$ represents a question or input in the dataset, $\text{UQ}(S_x, \mathbb{T})$ is the uncertainty score (e.g., Inv-Entropy) evaluated at temperature $\mathbb{T}$, and $\mathbb{I}(\cdot)$ denotes the indicator function.

Importantly, TSU is agnostic to the output label $y$ and depends only on the input $S_x$, making it applicable even when "ground truth" is undefined or unavailable.

Moreover, TSU goes beyond binary correctness evaluation by assessing how finely a method can track the gradation of uncertainty in response to controlled perturbations in temperature. This provides a richer and more generalizable criterion for UQ methods.

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.

Below we provide an itemized response to your suggestions & concerns:

W1: Computational Needs

Great 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.

W2: Sensitivity Analysis on Hyperparameters

Thank you for the insightful suggestion. To fully address your comment, we have now run ablation studies on the parameters you highlighted.

The number of perturbations n:

The 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 $n$.

The number of replications r:

This 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 $r$.

These results also shed light on our theoretical findings, which highlight the importance of perturbation for uncertainty quantification.

The temperature t:

The 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.

The bootstrapping iterations S:

Please 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, using more than 10 iterations appears to provide limited additional benefit.

W3: Qualitative examples for responses with similar x and different y

The following table presents a representative case from the TriviaQA dataset, where semantically similar questions yield highly diverse answers.

Answers 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.

W4: Sensitivity analysis on GAAP

This 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.

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.

Dear Reviewer tjtF,

Thank you for your engagement. We sincerely appreciate your positive assessment of our work.

We hope our rebuttal addressed your suggestions and concerns, and we would be happy to provide any additional clarification or simulation results if needed.

Warm regards

The Authors

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.

W1: Figures 1 & 2

Thank 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.

W2: Computational Needs

Great 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.

W3: Related Work in the Appendix

As you suggested, we will bring the related work section of the appendix into the final version of the main paper.

W3: Applicability to Long Form Q/A

You 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.

To 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.

W4: Extreme-Case Analysis: Small or no Aleatoric Uncertainty

If 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.

Mathematically, 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 $i, j$.

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: $H(X \mid Y) = \log(n+1)$.

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: $H(X \mid Y) = \log(n+1) - \mathcal{O}(\epsilon^2)$.

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.

These regimes confirm that Inv-Entropy is sensitive to input variability even when outputs remain constant; precisely as intended by the inverse modeling perspective.

Thank you for raising this question. We have not thought of showing this analysis before and it largely enriches our paper.

Q1: Difference in Reported AUROC for Semantic Entropy on TriviaQA

Thank 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:

First, 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.

Second, 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.

Third, 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.

Q2: Adding Semantic Entropy–Perturbation Ensemble Baselines

Following 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.

Let $S_{x_0}$ 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 $R_i = \lbrace y_{i,1}, \dots, y_{i,r}\rbrace$.

We consider the following:

• Semantic Entropy.
  The original method computes entropy solely over the $r$ responses to the original prompt $S_{x_0}$. No perturbation is used.

• Semantic Entropy (Joint).
  This ensembled Semantic Entropy computes entropy over all responses from the original and perturbed prompts $\mathcal{R}_{\text{joint}} = R_0 \cup R_1 \cup \dots \cup R_n$.

• Semantic Entropy (Average).
  An entropy score is computed independently for each response set $R_i$, and the final score is the average over all $i$.

The results are shown below with the same experiment setting as that of Table 1 in our main paper:

Adding 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.

References

[1] "Semantic Uncertainty: Linguistic Invariances for Uncertainty Estimation in Natural Language Generation." The Eleventh International Conference on Learning Representations, 2023.

Dear Reviewer uG7q,

Thank you once again for your thoughtful and supportive review. We sincerely appreciate your positive assessment of our work.

We hope our rebuttal addressed your suggestions and concerns, and we would be happy to provide any additional clarification or simulation results if needed.

Warm regards

The Authors

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.

Below we provide an itemized response to your suggestions & concerns:

W1 & Q1: New experiments on a long-form answer dataset

This 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.

To analyze performance across different answer lengths, we further divided the dataset into three subsets based on the number of tokens in the reference answers:

• Short (<80 tokens): 280 samples
• Medium (80–120 tokens): 320 samples
• Long (≥120 tokens): 400 samples

We 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:

Bold indicates the best performer. † indicates the second-best.

Our 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.

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 metrics across different answer lengths is a promising direction we leave for future work.

W2: Computational Needs

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 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.

With this in mind, we divide the computational cost into two components:
(A) 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 $r$ 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.

Regarding computation after responses are collected:

Below, 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.

W3 & Q2: Perturbation function & maintain semantic equivalence

We 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.

Regarding 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).

Additionally, 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.

References

[1] "Natural questions: a benchmark for question answering research." Transactions of the Association for Computational Linguistics, 2019.

[2] "Semantic Uncertainty: Linguistic Invariances for Uncertainty Estimation in Natural Language Generation." The Eleventh International Conference on Learning Representations, 2023.