Conjugate and MCMC Bayesian Chain-Rule Prediction- Powered Inference for Binary Prevalence Estimation

TMLR Paper9119 Authors

Abstract:

Prediction-Powered Inference (PPI) combines abundant machine predictions with scarce labels to estimate population functionals with improved efficiency. Recent Bayesian treatments of PPI have introduced general conjugate and Monte Carlo formulations. We focus on a narrower but practically important setting: binary prevalence estimation through the chain-rule functional

$$
g=P(H=1\mid A=1)P(A=1)+P(H=1\mid A=0)P(A=0).
$$

For the base Beta-Bernoulli model, we show that the posterior factorizes into independent Beta distributions, so uncertainty in $g$ can be propagated by direct sampling without MCMC. We reserve NUTS only for genuinely non-conjugate extensions, including hierarchical partial pooling, logit-normal priors, $K$-bin score models, and joint threshold uncertainty.

We benchmark this conjugate Bayesian chain-rule estimator (CRE) against three baselines: a labeled-only Bayesian estimator, the classical difference estimator, and a prior-free analytic PPI estimator based on continuous probabilities with small-sample $t$ critical values. Empirically, we report (i) simulation-based calibration for the conjugate engine, (ii) repeated-labeling resampling studies with $M=500$ replications on ADNI-derived out-of-fold prediction tables, and (iii) an Alzheimer's disease MRI case study with out-of-fold threshold selection, bootstrap cut-point dispersion, and propensity-overlap diagnostics. In the full cohort, CRE achieves near-nominal coverage and narrower intervals than the labeled-only Bayesian baseline, while substantially improving stability over the difference estimator at small label budgets; gains are smaller but remain competitive in a fixed 65-70 subset. These results position conjugate Bayesian chain-rule PPI as a lightweight and auditable option for deployment-time prevalence monitoring.

Submission Type: Regular submission (no more than 12 pages of main content)

Assigned Action Editor: ~Ozan_Sener1

Submission Number: 9119