MEC: Machine-Learning-Assisted Generalized Entropy Calibration for Semi-Supervised Mean Estimation

Se Yoon Lee; Jae Kwang Kim

stat.MLApr 7, 2026

MEC: Machine-Learning-Assisted Generalized Entropy Calibration for Semi-Supervised Mean Estimation

Authors: Se Yoon Lee, Jae Kwang Kim

Organizations: Texas A&M University · Iowa State University

Abstract

Obtaining high-quality labels is costly, whereas unlabeled covariates are often abundant, motivating semi-supervised inference methods with reliable uncertainty quantification. Prediction-powered inference (PPI) leverages a machine-learning predictor trained on a small labeled sample to improve efficiency, but it can lose efficiency under model misspecification and suffer from coverage distortions due to label reuse. We introduce Machine-Learning-Assisted Generalized Entropy Calibration (MEC), a cross-fitted, calibration-weighted variant of PPI. MEC improves efficiency by reweighting labeled samples to better align with the target population, using a principled calibration framework based on Bregman projections. This yields robustness to affine transformations of the predictor and relaxes requirements for validity by replacing conditions on raw prediction error with weaker projection-error conditions. As a result, MEC attains the semiparametric efficiency bound under weaker assumptions than existing PPI variants. Across simulations and a real-data application, MEC achieves near-nominal coverage and tighter confidence intervals than CF-PPI and vanilla PPI.

Figures & tables

Figure 1: Coverage and width ratios of 95% confidence intervals across label fractions

f

for four ML predictors. Each column corresponds to one predictor—KRR, RF, FNN, and kNN. MEC (quadratic generator) attains near-nominal coverage and the narrowest valid intervals, consistently improving efficiency over CF–PPI. Vanilla PPI undercovers, especially at small

f

. Classical and oracle baselines are shown for reference. (See Figure 5 for the results with MEC using other generators; MEC performance is robust to the choice of generator.)

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 2 — Figure 2: Conceptual diagram for constructing the MEC estimator. The notation $\omega$ denotes a generic set of weights in the weighted estimating function $U_{n}^{\omega}(\theta)$ . In the MEC construction, $\widehat{d}_{i}$ denotes the baseline weight and $\widehat{w}_{i}$ denotes the MEC-updated weight. When all weights in $\omega$ are updated, we continue to use $\omega$ , rather than introducing separate $w$ -notation, to avoid notational clutter. This is the case in the semi-supervised mean-estimation setting in the main paper, where all weights in the rectifier term are updated.

Entropy	$G(u)$	$g(u)=G^{\prime}(u)$	$D_{G}(u\\|v)$
Quadratic	$\frac{1}{2}u^{2}$	$u$	$\frac{1}{2}(u-v)^{2}$
Kullback–Leibler	$u\log u$	$\log u+1$	$u\log(u/v)-u+v$
Empirical likelihood	$-\log u$	$-u^{-1}$	$-\log(u/v)+u/v-1$
Squared Hellinger	$(\sqrt{u}-1)^{2}$	$1-u^{-1/2}$	$\sqrt{v}\left(1-\sqrt{u/v}\right)^{2}$
Inverse	$\frac{1}{2u}$	$-\frac{1}{2}u^{-2}$	$\frac{1}{2}\left(\frac{1}{u}-\frac{1}{v}\right)+\frac{u-v}{2v^{2}}$
Rényi $(\alpha>0)$	$\frac{u^{\alpha+1}}{\alpha+1}$	$u^{\alpha}$	$\frac{u^{\alpha+1}-v^{\alpha+1}}{\alpha+1}-v^{\alpha}(u-v)$

Appendix

Table 1: Representative Bregman generators.

Figure 3: Bregman divergences

D_{G}(u\|v)

v=10

for six representative entropy generators (Quadratic, Kullback–Leibler, Empirical likelihood, Squared Hellinger, Inverse, and Rényi with

\alpha=1/2

); see Table 1 .

Figure 4: Iteration procedure of dual Newton solver on a single realization of the synthetic data with

f=0.2

in Section 6 of the main document. Panel (a) shows the calibration residual

\lVert Z^{\top}\omega-\mu\rVert_{2}

(tolerance

\varepsilon=10^{-10}

); panel (b) shows the Bregman objective

D_{G}(\omega\|d)

by iteration. The quadratic case converges in one step; others converge within

\leq 10

steps, with objectives remaining bounded.

Figure 5: Supplemental plots corresponding to Figure 1 in the main document, showing MEC with four generators—quadratic, Kullback–Leibler (KL), empirical likelihood (EL), and squared Hellinger. MEC performance is robust to the generator choice. Unlabeled sample size

N=1000

; labeled size

n=fN

with

f\in[0.1,0.5]

; covariate dimension

d=10

with i.i.d.

X\sim\mathcal{N}(0,1)

; outcome noise standard deviation

\sigma_{y}=5

Method	Bias	RMSE	Coverage	Mean CI Len	ESS	CV
Classical	0.008	0.561	0.948	2.237	–	–
PPI (oracle)	0.019	0.501	0.946	1.983	–	–
PPI (no cross-fitting)	-0.023	0.583	0.770	1.420	–	–
CF-PPI	0.007	0.597	0.948	2.361	–	–
Calibration via Moment Basis (Quadratic)	0.013	0.608	0.958	2.507	89.004	0.348
Calibration via Moment Basis (KL)	0.013	0.618	0.953	2.524	88.396	0.358

Appendix

Table 2: Additional simulation results including PPI++ as a baseline and weight calibration using an 11-dimensional moment-feature basis. ESS and CV are reported only for the calibration methods.

Figure 6: Additional Simulation Experiment 1: effect of covariate dimension

d

. We vary

d

from

5

15

while fixing

N=1000

n=200

(

f=0.2

\rho=0

, and

\sigma_{y}=5

. For MEC, we display only the quadratic generator for visual clarity, since MEC’s results are robust to the choice of generator.

Figure 7: Additional Simulation Experiment 2: effect of standard deviation

\sigma_{y}

. We vary

\sigma_{y}

from

1

10

while fixing

N=1000

n=200

(

f=0.2

\rho=0

, and

d=10

Figure 8: Additional Simulation Experiment 3: effect of covariate correlation

\rho

. We vary

\rho

from

0

0.8

while fixing

N=1000

n=200

(

f=0.2

\sigma_{y}=5

, and

d=10

Figure 9: Additional Simulation Experiment 4: effect of fold number

K

for sample-splitting. We vary

K

from

5

10

while fixing

N=1000

n=200

(

f=0.2

\rho=0

\sigma_{y}=5

, and

d=10

Figure 10: Real-data application (aligned case). Point estimates and 95% confidence intervals for the classical, PPI, CF–PPI, and MEC estimators across four learners (KRR, RF, FNN, and kNN) using the Energy Efficiency dataset. The labeled mean

\bar{Y}_{n}

is close to the reference mean

\bar{Y}_{\mathrm{full}}=22.307

. All debiasing methods produce estimates consistent with the reference and yield tighter intervals than the classical labeled-only estimator, demonstrating improved efficiency by leveraging unlabeled covariates.

Figure 11: Real-data application (off-aligned case). In this scenario,

\bar{Y}_{n}

differs noticeably from

\bar{Y}_{\mathrm{full}}

. The bias-correction property of PPI, CF–PPI, and MEC is evident: point estimates are pulled toward the reference mean, and the resulting 95% confidence intervals are tighter than those of the classical labeled-only estimator.

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Calibeating Prediction-Powered Inference

Lars van der Laan, Mark Van Der Laan

Department of Statistics, University of Washington · Department of Statistics and Division of Biostatistics, Center for Targeted Machine Learning and Causal Inference, University of California, Berkeley

Inverse Probability Weighting Parameter Estimation Post-Hoc Calibration Semi-Supervised Learning Prediction-Powered Inference Isotonic Regression

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We study semisupervised mean estimation with a small labeled sample, a large unlabeled sample, and a black-box prediction model whose output may be miscalibrated. A standard approach in this setting is augmented inverse-probability weighting (AIPW) [Robins et al., 1994], which protects against prediction-model misspecification but can be inefficient when the prediction score is poorly aligned with the outcome scale. We introduce Calibrated Prediction-Powered Inference, which post-hoc calibrates the prediction score on the labeled sample before using it for semisupervised estimation. This simple step requires no retraining and can improve the original score both as a predictor of the outcome and as a regression adjustment for semisupervised inference. We study both linear and isotonic calibration. For isotonic calibration, we establish first-order optimality guarantees: isotonic post-processing can improve predictive accuracy and estimator efficiency relative to the original score and simpler post-processing rules, while no further post-processing of the fitted isotonic score yields additional first-order gains. For linear calibration, we show first-order equivalence to PPI++. We also clarify the relationship among existing estimators, showing that the original PPI estimator is a special case of AIPW and can be inefficient when the prediction model is accurate, while PPI++ is AIPW with empirical efficiency maximization [Rubin et al., 2008]. In simulations and real-data experiments, our calibrated estimators often outperform PPI and are competitive with, or outperform, AIPW and PPI++. We provide an accompanying Python package, ppi_aipw, at https://larsvanderlaan.github.io/ppi-aipw/.

Abstract

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