Valid Inference with Imperfect Synthetic Data

Dear Reviewer B76K,

Thank you for your feedback! We address your concerns below:

Assumption that text generations = original distribution

The prompting methods used by the researchers to generate samples may not be as robust as the paper implies. The assumption that prompting with T will result in text generations that exactly match the original distribution is incorrect

We sincerely apologize for the potential confusion! To start, we must clarify: our method does not require this assumption. We have edited the Introduction (lines 45-48) to make this explicit (as we think the misunderstanding may be our fault due to some ambiguous wording). Changes are bolded:

“”” Naively…impossible to provide statistical guarantees for the resulting estimate if the generative model does not perfectly match the real distribution—which is expected in practice. We propose a specific sampling strategy…This correlation structure will prove critical for principled methods for integrating synthetic data, as it enables us to more effectively share information across them without requiring the distributions to be equal to each other. “””

More importantly, we would like to highlight that in the Methods section we do not impose any assumptions on the quality of proxy or synthetic data to derive our key theoretical results on consistency (Section 4.4) (see Appendix A for proof). In Section 4.5, we explicitly analyze why synthetic data improves performance even when they are imperfect and derive an explicit formula for the asymptotic variance in Appendix C, where we show, “in the worst case, where synthetic data is completely informative, including it does not hurt, asymptotically” (Lines 257-258).

The core idea behind how we construct our GMM estimator (see Section 4.2) is that we estimate separate parameters ($\theta, \eta$) for each data source, precisely how the method avoids assuming the samples (from different data sources) come from the same distribution. If we assumed all data came from the same distribution (as the original distribution) then we would constrain $\theta = \eta$. The fact that we keep these separate is precisely how we allow for distributional differences while still benefitting from cross correlations.

“because the authors make strong claims relating to the algorithm's ability to identify the true distribution from artificially generated samples, there is a risk that researchers may succumb to overreliance on the algorithm's outputs. For this reason, the authors should clearly identify the bounds of their algorithm's validity…”

We apologize for the potential confusion! We would like to clarify again that we do not claim to identify the true distribution from artificially generated samples. Our theoretical results guarantee valid confidence intervals regardless of whether the synthetic samples match the real distribution; the confidence intervals will just be tighter if the synthetic samples match more closely. We state explicitly in the paper, even if the LLM is not good, the performance will still be at least as good as if the researcher had only used the human-labeled data: “When the [real and synthetic] residuals are independent, the formula reduces to the optimal variance based only on the real data” (Lines 256-257).

To further demonstrate this, we have added new experiments where the synthetic samples from the LLM are replaced by random noise with a set probability $p$ to further demonstrate how our method would empirically perform in even worse cases. More concretely, when p=0.5, we default to complete random noise for 50% of the samples and use the LLM predictions for the rest. We observe that even though the synthetic data is complete random noise for half of the samples, we don’t perform worse than not having used them (approximates the Human-Only baseline). We theoretically provide an explanation and proof in Section 4.5 (Lines 257-258) and Appendix C (Lines 831-851).

Ethical Concerns: Data quality and representativeness

Again, we would like to clarify that we do not make claims on data representativeness and one of the main strengths of our method is specifically that it does not make assumptions on the quality of the synthetic data.

Different LLMs

how the selection of language model might impact the reliability of results

We thank the reviewer for their valuable suggestion! We have added additional experiments using small, open-source models with worse performance (Llama-3-8b, Qwen-3-8b), where we observe that even with worse quality LLMs the same conclusions hold.

We report performance (MSE; lower is better) with Llama-3-8b, Qwen-3-8b:

Details about Baseline Comparisons

baseline comparisons do not seem to be the correct way to evaluate this work. PPI++ is used as a baseline, but it differs from the authors' methods along multiple dimensions (e.g. power tuning, the sampling method used to generate artificial data)

We would like to clarify that PPI++ uses the same synthetic data samples. The synthetic data samples used across all baselines are fixed to be the same across all methods.

We would also like to clarify that power tuning is a strategy the PPI++ authors use to improve their original weaker baseline PPI. Power tuning finds an optimal weighting parameter $\lambda$ that takes a combination between the classical estimator (using only real data) and the PPI estimator (using both real and synthetic data). $\lambda$ is chosen to minimize the width of the confidence interval. When the prediction model performance is poor, such that incorporating unlabeled data hurts performance, it sets $\lambda$ = 0 to recover the classical estimator. Therefore, we are actually making a comparison with a stronger method by reporting numbers that involve power tuning for that baseline. We also report numbers without power tuning for more context:

Title and Abstract

The authors' actual contributions are far more interesting, but neither the title nor abstract clearly convey the ideas behind the authors estimator algorithm, nor the principled sampling method used by the authors to generate synthetic data.

We appreciate that you find our contributions interesting! We have updated the title to “Valid Inference with Imperfect Synthetic Data via Generalized Method of Moments” to better reflect the core methodology behind the estimator algorithm and have updated the abstract to include further details about the principled sampling method, the estimator algorithm, and intuition behind why/when synthetic data improves performance.

Expand Motivation

Motivation is not clearly established

We are motivated by limited-labeled data settings where practitioners face constraints in obtaining sufficient human-labeled data due to budget constraints. This has led researchers to increasingly turn to LLMs as cheap but noisy alternatives for data generation and annotation (e.g., simulating human responses to surveys) [1][2][3].

The persistent challenge is that naively combining these imperfect synthetic data samples with human-annotated samples for downstream inference tasks can lead to biased estimates, compromising statistical validity of its conclusions. Current methods are only able to incorporate noisy predictions and not fully synthetic samples [4][5], which further motivates the purpose of this work. Our work addresses this fundamental gap by providing the first theoretically-grounded framework that can safely incorporate fully synthetic samples while maintaining statistical validity, regardless of the quality of the synthetic data.

Add Detailed Description of PPI++ in Main Text

PPI++ is referenced within the paper and used as a key baseline, but this algorithm is not clearly explained within the text of the paper

Thanks for the suggestion! Currently, we state the PPI++ objective in Section 5, Proposition 2 (Lines 274-277) and the pseudocode for the algorithm implementation in Appendix F, Algorithm 1 (Lines 894-895). To address this concern, we have (1) updated the draft to add the pseudocode in the main text (Section 5) and (2) add a line-by-line explanation of the algorithm in Section 5 to help provide a more direct explanation of the method.

We thank the reviewer again for their time and consideration! We are more than happy to address any other concerns they might have.

1. Hwang et al. Opportunities and challenges of applying llm-simulated data, 2025.
2. Geng et al. Are large language models chameleons? an attempt to simulate social surveys, 2024.
3. Rothschild et al. Opportunities and risks of llms in survey research, 2024.
4. Gligoric et al. Can Unconfident LLM Annotations Be Used for Confident Conclusions, 2024.
5. Egami et al. Using Imperfect Surrogates for Downstream Inference, 2023.

Dear Reviewer B76K,

Thank you for your response! Below, we respond explicitly to each point raised:

1. We would like to clarify that our paper does not make this claim and we acknowledge this explicitly in the Limitations section of our original submission

First, we would like to remark that our original submission acknowledges this explicitly in the Limitations section of our submission (Lines 348-352):

“A potential limitation of our framework is its reliance on the quality of the generative model (e.g., an LLM). As with other debiasing approaches in the literature, very poor-quality synthetic data would yield little-to-no benefits in statistical efficiency. Moreover, our theoretical guarantees, like those of debiasing methods, hold asymptotically and thus may fail to hold in extremely low-data regimes, potentially leading to undercoverage of the target parameter.”

2. We do not assert any guarantees on classification accuracy.

We apologize for the confusion. To clarify: we never claim to guarantee “classification accuracy” in the sense of the individual classification outputs from the LLM. Rather, our guarantees are on the “consistency and proper asymptotic coverage” of statistical inferences, regardless of how accurate the LLMs’ classifications are (see e.g. Lines 35-36 and 72-73 in the Intro for examples of this specific phrasing). We then prove that consistency and proper asymptotic coverage hold in Proposition 4.1 in the paper. Is there a specific concern about the proof of Proposition 4.1?

3. Current experiments in the paper actually already demonstrate the setting when LLM generations are very biased (see Figure 7)

We would like to clarify that our experiments in the paper explicitly include settings where the LLM generations are very biased (Please see Figure 7 where we plot bias). In Figure 7, we can clearly observe that using the LLM generations alone leads to largely biased estimates that are skewed in one direction. In Figure 3 (Main Results), we observe the performance of our method using these same LLM generations.

4. We add additional experiments encompassing (1) noise in one direction, (2) anticorrelation, (3) worse quality LLMs, (4) and worse quality prompting strategies

We want to thank Reviewer B76K for these insightful suggestions! First, we would like to highlight that during the rebuttal, we have also conducted experiments using worse quality, open-source LLMs (Llama-3-8b, Qwen-3-8b) to demonstrate our method’s performance with worse quality synthetic data, as requested (e.g., the accuracy of predictions from Llama-3-8b is only 0.15 and for Qwen-3-8b is 0.23 for the Hedging task demonstrating poor performance for the target task). Note that we employ all LLMs in a zero-shot manner with no domain-specific fine-tuning in all experiments.

To explicitly address the setting of when all "LLM generations are biased towards one direction" we provide an experiment where instead of random samples between {0, 1}, we assign towards a single class, which reflects your point about how misaligned LLM generations can be biased in one direction.

To further address your concern, we also add an experiment where the LLM label is anticorrelated with the real label.

To further address your concern, we also add experiments that use varying types of simpler prompt strategies: few-shot prompting and zero-shot prompting, which also result in worse quality synthetic data.

Results for few-shot prompting, zero-shot prompting (respectively):

We sincerely thank Reviewer B76K again for their constructive remarks and suggestions to further enhance the impact of our work. We believe the following changes will address the reviewer’s concerns:

• Add further explicit statements in the Introduction and Limitations section to clarify guarantees and guide practitioners for correct usage of the method.
• Add results on random noise, noise in one direction, anticorrelation, worse-quality LLMs, worse quality prompting strategies to Section 6 to empirically evaluate our method in settings with poor LLM performance with both simulations and real data.
• Add a discussion of Figure 7 into the main text, which demonstrates the setting where the LLM is very biased.

We hope these revisions address Reviewer B76K's concerns and improve our manuscript to meet their expectations for an acceptable article. We are more than happy to address any other concerns they might have.

Thank you for the additional information, and for the reference to Figure 7 in the Appendix. I have revised my score upward.

Thanks! We are delighted to hear that our rebuttal was helpful in addressing your concerns! Best wishes

Dear Reviewer WjYv,

Thank you for your feedback! We’re grateful for your acknowledgement of our contributions. We address your concerns below:

Clarity and Expansion of Related Work

The contribution of the paper is significant. The main weakness of the paper is that it is hard to follow; some basic concepts are overexplained, while the important part and the contribution section are left ambiguous. A very critical concept, the Generalized Method of Moments, is not explained in the related work.

We appreciate that you find our contributions to be significant and thank you for this important feedback. We currently discuss our GMM approach in detail in the Methods section (Section 4). To address this further, we add the following subsection to the Related Work section (after Line 104) to provide further context earlier on in the draft:

“”” Generalized Method of Moments (GMM). The GMM framework [Hansen, 1982] is a flexible estimation method that generalizes many common statistical approaches (e.g., maximum likelihood, instrumental variables, least squares, etc.). GMM estimators are defined by moment conditions—functions of the data and parameters whose expectations equal zero at the true parameter values. The key insight is that when the number of moment conditions exceeds the number of parameters to estimate, GMM optimally combines these conditions using a weighting matrix that accounts for their variance-covariance structure. This framework has been widely adopted in statistics and econometrics due to its robustness to distributional assumptions and ability to handle overidentified systems (more conditions than parameters). In this work, we leverage GMM's flexibility to incorporate human-labeled, proxy, and synthetic data through separate moment conditions. The two-step GMM estimation procedure [Newey and McFadden, 1994] (see Section 4.3 for details) learns optimal weights for combining these data sources through the moment covariance matrix, eliminating manual hyperparameter tuning required by existing debiasing methods. The key intuition is that the weighting matrix captures the interactions between the real moment residuals and the synthetic moment residuals; the interactions between these moment residuals improves estimation when they are predictive of one another (see Section 4.5 for further discussion). “””

Currently, we provide a description of our GMM method in Section 4 in the following manner: Introduce moment conditions (Section 4.1), estimator construction with proxy and synthetic data (Section 4.2), the two-step estimation procedure (Section 4.3), and theoretical results on consistency and asymptotic behavior of our GMM estimator (Section 4.4 with the full proof in Appendix A).

We further include a concrete example of our moment construction in Appendix B (Lines 821-831). However, we forgot to note a reference to Appendix B in the main text, which might have caused some ambiguity. To help improve clarity, we have updated the draft to move this concrete example from Appendix B into the main text before Section 4.3. This provides readers with a concrete illustration of our moment construction before turning to the estimation procedure (Section 4.3), making the method easier to follow. We hope these additions help step through the main contribution/method of the paper better. We would be happy to further improve the explanation if there are any additional remaining concerns.

My suggestion for the authors is to summarize the content in the early pages of the paper and expand on how the Generalized Method of Moments works and how it can help with this problem.

Thank you for this suggestion! We have updated the Introduction (Lines 61-70) with the following paragraph to summarize and further expand on how the GMM works and helps with the problem at hand more earlier in the draft:

""" Our primary methodological contribution is to propose a new estimator based on the generalized method of moments (GMM) framework that naturally incorporates this synthetic data. The construction of our GMM estimator defines separate parameters and moments for each data source. It is not initially obvious whether the incorporation of moments based exclusively on synthetic data should yield any benefits (or even affect) the estimation of the target parameter. However, we find that in the GMM estimation procedure, the off-diagonal terms in the weight matrix captures interactions between the real moment residuals and synthetic moment residuals, allowing the information from the synthetic data moments to influence the estimation of the target parameter. The key intuition is that synthetic data will improve performance when the synthetic data residuals are predictive of the real data residuals. More concretely, the asymptotic variance is the remaining variance after you regress the real residuals on the span of the synthetic residuals. In other words, when the residuals are independent of each other, the asymptotic variance reduces to the optimal variance based only on real data. This means that in the worst case, when synthetic data is completely uninformative, it does not hurt performance asymptotically (see Section 4.5 and Appendix C for further details). """

Clarification Question (Line 135)

Conditions under line 135 are not clear. Are the equations showing how T, X, and Y are generated, or are T and X when the text is labeled, and T and Y when the text is unlabeled? The current formulation is misleading.

Thank you for raising this clarification point! Yes, the equations are showing how T,X,Y are generated. We have updated the draft to list the equations in a list and not side-by-side to avoid this potential confusion:

$\tilde{T}_k \sim \mathbb{P} (\cdot \mid T_i, X_i)$ if labeled

$\tilde{T}_k \sim \mathbb{P} (\cdot \mid T_j, \hat{X}_j)$ if unlabeled

$\tilde{X}_k \sim \mathbb{P} (\cdot \mid \tilde{T}_k)$

$\tilde{Y}_k \sim \mathbb{P} (\cdot \mid \tilde{T}_k)$

We thank the reviewer again for their time and consideration! We are more than happy to address any other concerns they might have.

Dear Reviewer WjYv,

We thank the reviewer for their comments that they find our contributions to be significant and that our empirical evaluations support our theoretical results. In our response above, we have tried to address all your questions and comments. According to your suggestions, we have strengthened our explanation of Generalized Method of Moments by adding a new section to the draft, moved up a concrete example from Appendix B into the main text, and summarized the content on how the GMM works and how it helps with the problem in early pages of the paper, as suggested.

We once again thank you for your valuable time spent reviewing our work. We hope these revisions address Reviewer WjYv's concerns and improve our manuscript to meet their expectations for an acceptable article. We are more than happy to address any other concerns they might have.

Dear Reviewer psyf,

Thank you for your feedback! We address your concerns below:

Varying LLM quality/prompt design types

While the paper mentions that low-quality synthetic data won’t help, it doesn’t empirically explore the effect of varying LLM quality…The framework relies on the availability of strong LLMs like GPT-4o. This may not generalize well to low-resource settings, such as open-source models.

Thanks for the suggestion! We have added additional experiments using small, open-source models (Llama-3-8b, Qwen-3-8b) to encompass varying types of LLM quality. We observe that even with worse quality LLMs, the same conclusions hold. We have also added an experiment where the synthetic samples from the LLM are replaced by random noise with a set probability $p$ to further demonstrate how our method would empirically perform in even worse cases. More concretely, when p=0.5, we default to complete random noise for 50% of the samples and use the LLM predictions for the rest. We observe that even though the synthetic data is complete random noise for half of the samples, our method is able to safeguard against this (approximates performance you would get with human-only samples). This is important since practitioners can only know retrospectively if an LLM was not good for their task at hand. We would like to highlight that we theoretically provide an explanation and proof of this in Section 4.5 (Lines 257-258) and Appendix C (Lines 831-851).

We report performance (in terms of MSE; lower is better) with Llama-3-8b, Qwen-3-8b (respectively)

For 50% of the samples, we default to complete random noise.

We would also like to remark that our method offers a statistical framework for integrating information from LLMs, which provides value when LLMs offer useful signal (as you should only expect it to help if the LLM is “good”). In the case when the LLM is “bad”, what is important is that the method does not result in extremely biased estimates and can safeguard against this. The performance of models today and their rapidly improving accessibility (e.g., open models also keep improving) is what motivates practitioners to want to leverage predictions and generations from them, which in turn provides strong motivation for such integration frameworks.

Varying prompt design. It would be helpful to see a comparison between the proposed conditioned sampling vs. naive zero-shot generation to justify the design decision…Can your framework support generation strategies that are not conditioned on real examples? For instance, can it be extended to few-shot or zero-shot prompting?

We thank the reviewer for their valuable suggestion. We have added additional experiments encompassing varying types of prompt strategies: few-shot prompting and zero-shot prompting. We clearly see that conditional sampling leads to larger benefits downstream, justifying our design decision. However, even when the synthetic samples are of worse quality due to simpler prompt strategies, we still outperform baselines.

Results for few-shot prompting, zero-shot prompting (respectively):

More Tasks

The experiments are focused on regression tasks with structured covariates; it’s unclear how well the framework generalizes to classification, causal inference, or time series.

We have added additional experiments to now cover regression, classification, causal inference, and time series tasks. We report performance in terms of MSE (lower is better) averaged over 200 trials. We observe that our method outperforms baselines across the additional tasks, demonstrating our framework's flexibility beyond regression: any estimand expressible as a moment condition can leverage our approach.

First, in addition to the logistic regression results (classification task) in the paper, we run ordinary least squares (regression task). We clearly see that our method outperforms baselines and the same conclusions hold.

Next, we construct a semi-synthetic causal inference task to evaluate treatment effect estimation under missing covariates. Starting with real data containing features $X_i$ and outcomes $Y_i$, we synthetically generate treatment assignments $D_i$ via covariate-dependent randomization and simulate causal outcomes with heterogeneous treatment effects across covariate strata. We define moment conditions $m_i(\tau) = S_i · (\phi_i - \tau)$, where $\phi_i$ is the augmented inverse propensity weighted (AIPW) influence function combining outcome regression and propensity score components. Nuisance functions (propensity scores and outcome models) are estimated non-parametrically on labeled data, then we use these moments in our GMM estimator. This demonstrates our framework's flexibility beyond regression: any estimand expressible as a moment condition can leverage our approach.

We further demonstrate our applicability to time series by estimating autoregressive (AR) parameters under missing data. We generate $N$ independent AR(1) processes $Y_{it} - \mu = \phi(Y_{i,t-1} - \mu) + \varepsilon_{it}$ where $\phi$ is the target parameter, supplemented by noisy proxy and auxiliary time series data. The GMM formulation uses AR(1) moment conditions: $E[\varepsilon_{it}] = 0$ and $E[(Y_{i,t-1} - \mu)·\varepsilon_{it}] = 0$, where $\varepsilon_{it}(\phi,\mu)$ represents model residuals. Again, we observe that our method outperforms baselines.

Questions

What's the performance gap between open-source models and GPT-4o?

We report the performance gap between GPT-4o and open-source models (Llama-3-8b, Qwen-3-8b) in the table below.

What's the diversity of the generated samples, which should also be discussed and verified.

Thank you for this suggestion! We report Self-BLEU (lower is more diverse) and embedding-based metrics (average cosine distance) (higher is more diverse). We also report the metrics for samples generated by different prompt strategies (zero-shot, few-shot) for comparison.

We thank the reviewer again for their time and consideration! We are more than happy to address any other concerns they might have.

Thank you for confirming that our response has addressed all your concerns! We sincerely appreciate your valuable time and feedback. Best wishes

Dear Reviewer 7qA1,

Thank you for your feedback! We address your concerns below:

Method details

The method is rather involved, with ML model giving proxies, the LLM giving synthetic data, and then a GMM doing the statistical inference. which ML model for proxy data? Which LLM for synthetic data? Which LLM prompts?

We apologize for the confusion! The same model (same LLM) is used to give proxy data (predictions) and synthetic data (generations). We have added the following clarification in the Methods section (Line 126) to clarify this point: “For simplicity, we use the same machine learning $f$ used above for proxy data also for synthetic data generation.” For the LLM, we use GPT-4o (Line 303). For the LLM prompts, we include the full prompts used for all prediction and generation tasks in Appendix D.1 (Lines 853-860).

Ablations and more baselines

There is just one baseline

We would like to highlight that PPI++ is the current state-of-the-art method for these tasks. This line of work termed prediction-powered inference (also referred to as debiasing-based methods) has been the predominant strategy for incorporating model predictions in the literature [1][2] and represents the current most competitive method in incorporating imperfect LLM predictions [3][4]. Therefore, we focused on comparisons with this method rather than including multiple, weaker baselines from earlier iterations of this line of research, since they resulted in worse performance.

We would like to further highlight that we also implement an “oracle” version of PPI++ that “cheats” in hyperparameter selection (this is the baseline shown in the main plots). PPI++ requires selecting a hyperparameter alpha via cross-fitting, which cuts the sample size and leads to worse performance. To obtain an upper bound on the performance of this family of methods even if the hyperparameter could be chosen perfectly, the oracle method runs PPI++ on a large grid of hyperparameters and then reports the result closest to the ground truth (which isn’t knowable in practice). Our method outperforms even this strong baseline, while Appendix E (Fig 6) shows that the real, cross-fitting version of PPI++ performs worse than the oracle, as expected.

To better address your concern, we have also added an additional baseline Ji et. al [5]. We report the MSE (lower is better) averaged over 200 trials in the following Table. We clearly see that our method outperforms this method and the same conclusions hold.

Just a single LLM, and unclear how accurate GPT-o is for these tasks. It would have been interesting to test [worse synthetic data] in practice, either with more difficult tasks or worse LLMs.

We thank the reviewer for their valuable suggestion! We have added additional experiments using small, open-source models (Llama-3-8b, Qwen-3-8b), where we observe that even with worse quality LLMs the same conclusions hold. We have also added an experiment where the synthetic samples from the LLM are replaced by random noise with a set probability $p$ to further demonstrate how our method would empirically perform in even worse cases. More concretely, when p=0.5, we default to complete random noise for 50% of the samples and use the LLM predictions for the rest. We observe that even though the synthetic data is complete random noise for half of the samples, we don’t perform worse than not having used them (outperforms the Human-Only baseline). We would like to highlight that we theoretically provide an explanation and proof of this in Section 4.5 (Lines 257-258) and Appendix C (Lines 831-851).

We report performance (MSE; lower is better) with Llama-3-8b, Qwen-3-8b (respectively)

For 50% of the samples, we default to complete random noise.

We also report the accuracy of GPT-4o, Llama-3-8b, Qwen-3-8b for these tasks to more comprehensively understand performance.

Tasks are simple

We would like to remark that we run our experiments on standard benchmarks used in recent “LLM for statistical inference” works [3][4]. In fact, the inference tasks we use are actually more elaborate than previous work in this area, since we include a nonlinear task (logistic regression) as opposed to more simple linear inference tasks (such as simple mean estimation, quantile estimation, etc.) used in past works [1][2]. To further address your concern, we have also included new results on ordinary least squares (OLS) – another common statistical inference task— in the following Table. We see that even for linear tasks and even when Y is not binary but continuous, the same conclusions hold. OLS and logistic regression are perhaps the most widely used statistical models in computational social science practice and strong performance on these tasks is widely applicable.

Organization of figures

Key observations in Experiments should be structured better. Right now, jumps from Figures 3 and 4, to Figure 6 in the Appendix, to finally get to Figure 2, but then going back to Figure 3.

Thank you for this suggestion! We have grouped Figure 3, 4 and Figure 6 together. Followed by Figure 2, for better flow.

Questions

In L.135, do you mean $\tilde{X}_k = f( \tilde{T}_k)$ idem ditto for $Y$?

Thanks for raising this clarification point! Yes, that is exactly what we mean. We have added this to the draft for better clarity.

If we have enough texts , what is the intuition for why modelling synthetic helps?

Thank you for this great question! Even when the practitioner possesses a large collection of raw texts, the bottleneck for inference is the labels $(X,Y)$, because hand-labeling texts is expensive. Intuitively, prompting the LLM to generate a synthetic sample $(\tilde{T}, \tilde{X}, \tilde{Y})$ for every real document $T$ can be viewed as obtaining a second noisy view of the same latent signal.

To further answer your question, we plot the relative improvement in variance (Variance_human_only - Variance_with_synthetic) / Variance_human_only, where the number of raw text samples (real T) grows. This metric represents the percentage reduction in variance achieved by adding synthetic data. We fix the labeled portion of the raw text samples (real T) to be fixed p=0.05. We run this on the Congressional Bills dataset, as it has the largest number of raw text samples. We observe that the relative improvement is almost constant, indicating that the improvement from adding synthetic data persists even when the number of real text grows large.

We thank the reviewer again for their time and consideration! We are more than happy to address any other concerns they might have.

1. Angelopoulos et. al. PPI++: Efficient Prediction Powered Inference.
2. Angelopoulos et. al. Prediction-Powered Inference.
3. Gligoric et. al. Can Unconfident LLM Annotations Be Used for Confident Conclusions.
4. Egami et. al. Using Imperfect Surrogates for Downstream Inference.
5. Ji et. al. Predictions as Surrogates: Revisiting Surrogate Outcomes in the Age of AI.

Dear Reviewer 7qA1,

We thank the reviewer for their appreciation of the new results. And yes, we will definitely include those in the final paper.

Regarding Q2, great point! The reviewer is correct that there is less motivation for employing synthetic data if the practitioner already has a large, sufficient number of real texts for the downstream inference task. We would like to clarify that the main motivation for this work is settings where this is not the case—where practitioners have an insufficient amount of real texts for the downstream inference task (due to reasons such as budget or accessibility). We have explicitly clarified this in Section 1 (Lines 27-34) with the following clarifications (changes are bolded):

""" In these settings, where manually labeling large text corpora can be prohibitively expensive, practitioners have leveraged large language models (LLMs) as cheap but noisy labelers [Egami et al., 2023, Gligoric ́ et al., 2024].

More recently, in settings where real texts are insufficient, practitioners have started to explore the possibility of also leveraging LLMs not only as labelers but also to output entirely new synthetic samples, e.g., simulating human responses to surveys or human participants in early pilot studies [Argyle et al., 2023, Brand et al., 2023, Dominguez-Olmedo et al., 2024, Anthis et al., 2025, Hwang et al., 2025b]. Such synthetic simulations aim to overcome the financial limitations and time costs of collecting sufficient real text samples, to improve both efficiency and accessibility [Alemayehu et al., 2018]. """

We thank the reviewer again for their appreciation of the new results. We hope these revisions and additional results address Reviewer 7qA1’s concerns. We sincerely thank the reviewer again for their valuable time and consideration and are more than happy to address any other concerns they might have.

Dear Reviewer 7qA1,

We thank the reviewer for their appreciation of the new results. In our response above, we have tried to address all your questions and comments. To summarize, we have:

1. Added results with more baselines.
2. Strengthened empirical evaluations to include ablations with worse quality synthetic data: We have added results using worse quality LLMs: small, open-source models (Llama3-8b and Qwen-3-8b) and have demonstrated the same conclusions hold. We have also added an experiment where we default to complete random noise for 50% of the samples to further demonstrate a setting where the synthetic data is not reflective of the underlying task at hand and reflect predictions of a very poor model.
3. Added new results on OLS tasks—another common statistical inference task in the literature.
4. Added clarifications of method details and organization of figures.

We once again thank you for your valuable time spent reviewing our work. We hope this response addresses your concerns. If so, we would be extremely grateful if you would consider updating your score. If you have any more questions or feedback, please let us know.

We thank the reviewer for their appreciation of our responses and new results. We sincerely appreciate your valuable time and feedback. Best wishes