Technical Summary: Supercharging Bayesian Inference with Reliable AI-Informed Priors

Problem Statement

Modern predictive systems, particularly machine learning (ML) and foundation models, can generate outputs informative about scientific quantities of interest. In data-limited settings, leveraging these outputs for Bayesian prior elicitation offers a pathway to significant statistical efficiency. However, a fundamental tension exists: while an informative prior derived from an ML system can concentrate posterior mass and reduce variance, it risks propagating systematic errors (miscalibration) from the ML model into the posterior distribution. If the ML system's beliefs are biased, a strong prior can shift the posterior away from the true parameter, rendering inference unreliable. The core challenge is to incorporate AI-generated information to improve efficiency without sacrificing inferential validity or introducing uncontrolled bias.

Methodology

The authors propose a framework for AI-informed prior elicitation that mitigates this tension by "rectifying" the AI-induced law before it is used to construct a prior. The methodology centers on three key components:

1. Rectified Base Measure Construction:
   Instead of using the raw distribution of AI-imputed data ($P_{AI}$) as the base measure for a Dirichlet Process (DP) prior, the authors introduce a rectifier ($T_\eta$). This is a data-dependent map that transforms $P_{AI}$ into a rectified base measure $P_{\hat{\eta}_m} = T_{\hat{\eta}_m}(P_{AI})$.

   • The rectifier is learned using a small labeled calibration sample ($m$).
   • The goal is to align the estimating equation induced by the rectified measure with that of the true data-generating process ($P_0$). Specifically, the method seeks to minimize the score discrepancy $\|(P_0 - P_{\hat{\eta}_m})g_{\theta_0}\|$, where $g_\theta$ is the gradient of the loss function.
   • Two primary rectification strategies are discussed:
     • Quantile Mapping: For univariate outcomes, this transports the marginal distribution of AI-imputed outcomes to match the calibration data while preserving rank ordering.
     • Targeted Rectification: This aligns specific moments relevant to the loss function (e.g., mean matching for squared error loss) to ensure the score equation matches under the true distribution.

2. Rectified AI Prior and Posterior:
   The rectified measure $P_{\hat{\eta}_m}$ serves as the base measure for a Dirichlet Process prior: $P_0 \sim \text{DP}(\alpha, P_{\hat{\eta}_m})$.

   • The concentration parameter $\alpha$ (scaled as $\gamma = \alpha/n$) controls the strength of the prior relative to the labeled data.
   • Posterior sampling is performed via the posterior bootstrap, which leverages the conjugacy of the DP prior. This allows for embarrassingly parallel sampling by minimizing a weighted sum of losses over the labeled data and the synthetic data generated from the rectified base measure.

3. Theoretical Framework:
   The authors establish asymptotic properties for the rectified AI posterior.

   • Gaussian Asymptotics: Under non-vanishing prior strength, the posterior distribution converges to a Gaussian distribution centered at a mixed oracle target.
   • Bias Characterization: The leading order centering bias is explicitly derived as a function of the score discrepancy $(P_0 - P_{\hat{\eta}_m})g_{\theta_0}$. The theory shows that if the rectifier reduces this discrepancy to $O_p(m^{-1/2})$, the resulting bias is controlled and comparable to standard posterior fluctuations, provided the calibration sample size $m$ is sufficient.

Key Contributions

• Rectification Framework: The paper introduces a general procedure to correct AI-induced laws using labeled calibration data, transforming them into reliable base measures for prior construction.
• Theoretical Guarantees: The authors derive the asymptotic distribution of the rectified AI posterior and provide a first-order expression for its centering bias. This clarifies the conditions under which AI-informed priors remain reliable even when the prior strength is non-vanishing.
• Practical Rectifiers: The paper provides concrete, implementable rectification strategies, including quantile mapping and moment matching, tailored to different problem structures (e.g., univariate outcomes vs. regression).
• Empirical Validation: The methodology is validated across three distinct domains:
  1. Gene Expression: Inference on quantiles of gene expression levels using transformer-imputed data.
  2. Demographic Regression: Estimation of age-income regression coefficients using census data and gradient-boosted tree predictions.
  3. Skin Disease Classification: A downstream classification task using an LLM (GPT-5.5) to impute class probabilities for erythemato-squamous diseases.

Results

The empirical results demonstrate that the rectified AI prior substantially outperforms both standard approaches (using only labeled data) and uncorrected AI priors (using raw AI base measures):

• Inference Validity: In the gene expression and age-income regression experiments, the raw AI prior suffered from severe bias, resulting in credible intervals with near-zero coverage (e.g., 0% coverage for the 0.25 quantile in gene expression) and high interval scores. In contrast, the rectified AI prior restored coverage to near-nominal levels (approx. 90%) and achieved interval scores competitive with Prediction-Powered Inference (PPI++) and classical methods.
• Predictive Performance: In the skin disease classification task, the rectified AI prior enabled a Bayesian deep learner to achieve significant gains in test accuracy across a wide range of prior strengths ($\gamma \in [0.01, 64]$). The raw AI prior only showed marginal improvements in a narrow window of $\gamma$ and was highly sensitive to calibration, whereas the rectified prior provided robust, additive performance improvements (approx. 4.9% accuracy gain over the labeled-only baseline).
• Robustness to Prior Strength: The rectified approach decouples the variance reduction benefits of informative priors from the risk of systematic bias, allowing practitioners to use stronger priors without fear of invalidating inference.

Significance and Claims

The paper claims that its primary contribution is making modern predictive systems usable as sources of reliable prior knowledge. By shifting the focus from calibrating the prior strength parameter to rectifying the base measure, the authors argue that one can achieve the "best of both worlds": the variance reduction of an informative prior without the systematic bias of a miscalibrated model.

The authors position this work as a solution to the "tension" between efficiency and reliability in AI-augmented statistics. They emphasize that their method allows for the incorporation of external information (like LLM outputs or complex ML models) in a way that is philosophically consistent with Bayesian updating (where data corrects beliefs) while being practically robust to model misspecification. The work suggests that with proper rectification, AI systems can be trusted to guide inference in data-scarce regimes, potentially accelerating discoveries in fields like genomics and clinical diagnostics where gold-standard labels are scarce.

The paper remains modest regarding limitations, noting that the choice of rectifier is problem-dependent and that the quantile mapping approach is specific to univariate outcomes. They suggest future work in rectifier model selection and fully Bayesian approaches to propagating uncertainty from the rectifier estimation.