When outcome data are expensive or onerous to collect, scientists increasingly substitute predictions from machine learning and AI models for unlabeled cases, a process which has consequences for downstream statistical inference. While recent methods provide valid uncertainty quantification under independent sampling, real-world applications involve missing at random (MAR) labeling and spatial dependence. For inference in this setting, we propose a doubly robust estimator with cross-fit nuisances. We show that cross-fitting induces fold-level correlation that distorts spatial variance estimators, producing unstable or overly conservative confidence intervals. To address this, we propose a jackknife spatial heteroscedasticity and autocorrelation consistent (HAC) variance correction that separates spatial dependence from fold-induced noise. Under standard identification and dependence conditions, the resulting intervals are asymptotically valid. Simulations and benchmark datasets show substantial improvement in finite-sample calibration, particularly under MAR labeling and clustered sampling.
Figures & tables
Figure 1: Synthetic spatial-field simulation (superpopulation target): boxplots of empirical 90% coverage versus dependence level σ . Panels split by missingness arm (MCAR vs MAR) and sampling regime (iid vs soft-block). For each σ , each box summarizes the distribution across 100 population replicates of empirical coverage computed from 200 repeated sample draws of size n=600 from a single population replicate. Methods shown are Cross-PPI, PPI++, Bootstrap-PPI, and Spatial DR-JK-HAC. The dashed line marks nominal coverage. Under MAR, baseline models under-cover even under iid sampling; under MCAR, these models under-cover when soft-block sampling makes spatial dependence visible. Spatial DR-JK-HAC stays near nominal in all four cells, at the cost of modestly wider intervals (see Appendix D ).
Dataset
Cross-PPI
PPI++ (IID)
Bootstrap-PPI
DR-IID (CF-m)
Spatial DR-HAC
Spatial DR-JK-HAC (K=5, t )
Panel: MAR
Forest
0.613
0.619
0.575
0.825
0.825
0.925
Galaxies
0.775
0.875
0.825
0.825
0.825
0.875
Census income
0.631
0.831
0.613
0.725
0.744
0.806
Malaria
0.850
0.900
0.725
0.750
0.750
0.850
Health+
0.456
0.406
0.119
0.925
0.938
0.912
Table 1: Coverage for observed data auxiliary-split prediction design, averaged over IID and soft-block sampling.
Table 3: Observed-data auxiliary-split prediction design: mean confidence-interval width by dataset and arm (nominal 90%).
Figure 2: Synthetic simulation (superpopulation target): confidence-interval length boxplots by missingness arm and sampling regime. The x -axis is σ∈{0,40,80,120} . Each box summarizes mean interval length over 200 repeated sample draws across 100 replicate populations. Purple shades denote DR-based methods (DR-iid, spatial DR-HAC, spatial DR-JK-HAC). The figure is self-contained: the dependence-aware methods inflate width relative to iid post-prediction baselines, and spatial DR-JK-HAC narrows intervals relative to spatial DR-HAC while retaining dependence-aware coverage.
Figure 3: Synthetic simulation: coverage boxplots for the full method menu. Methods include Cross-PPI, PPI++ (iid), Bootstrap-PPI (Efron 2025), DR-iid, spatial DR-HAC, and spatial DR-JK-HAC. The figure is self-contained: DR-iid and spatial DR-HAC improve over iid post-prediction baselines under dependence and MAR but do not match spatial DR-JK-HAC in the most challenging soft-block MAR cells, motivating the jackknife-HAC correction in the main text.
Figure 4: Replicate-level interval visualization under MAR and soft-block sampling. Each interval is centered by subtracting its replicate target mean, so the dashed vertical line at 0 is the target in every panel. Cross-PPI (blue) shows systematic mis-centering in multiple panels, consistent with the MAR under-coverage seen in Figure 1 . Spatial DR-JK-HAC (purple) remains centered and attains near-nominal coverage.
Figure 5: Replicate-level interval visualization under MCAR and soft-block sampling, with the same centering convention as Figure 4 . Under MCAR, iid baselines are often closer to nominal than under MAR, but Cross-PPI still under-covers in the dependence-rich synthetic panel as dependence becomes visible under soft-block sampling. Spatial DR-JK-HAC remains near nominal across panels at the cost of wider intervals.
Figure 6: Residual dependence diagnostics by dataset under the exact coordinates used in inference. Left panel reports Moran’s I with permutation intervals and right panel reports nearest-neighbor residual correlation with the same permutation reference. The figure is self-contained: forest and malaria are clearly dependence-rich under observed coordinates, while galaxies and census are weak under proxy embeddings.
Figure 7: Residual semivariograms for each empirical dataset under analysis coordinates. Rising semivariograms indicate stronger local dependence. Forest and malaria show the clearest distance-gradient pattern.
Figure 8: Overlap diagnostics under the auxiliary-split observed-data design: design and estimated propensity lower-tail quantiles ( q05 ) by dataset and arm, averaged over sampling regimes. The dashed line at 0.10 marks the declared overlap floor.
Figure 9: Inverse-propensity effective-sample-size ratio (ESS / labeled count) under the auxiliary-split observed-data design, averaged over sampling regimes. MAR arms have lower ESS ratios, quantifying finite-sample information loss from nonuniform labeling probabilities.
Method
MCAR cov.
MCAR width
MAR cov.
MAR width
Cross-PPI
0.813
0.283
0.463
0.273
PPI++ (iid)
0.800
0.273
0.541
0.269
Bootstrap-PPI (Efron 2025)
0.652
0.215
0.345
0.288
DR-iid
0.805
0.274
0.823
0.315
Spatial DR-HAC
0.864
0.308
0.869
0.346
Spatial DR-JK-HAC
0.908
0.385
0.903
0.430
Appendix
Table 5: Mechanism attribution in the hardest synthetic cell ( σ=120 , soft-block sampling, 20% labels, nominal 90% superpopulation target). Each entry reports empirical coverage and mean interval length over 100 populations and 200 draws per population (20,000 intervals per arm).
Figure 10: Modular extension simulation with overlapping two-way dependence at nominal 90% . Cross-PPI remains anti-conservative in MAR. The jackknife two-way DR interval is less conservative than the direct two-way plug-in at weak dependence, while both methods are highly conservative in strong-overlap regimes.
Figure 11: Interval width in the modular extension simulation at nominal 90% . Jackknife and direct two-way plug-in widths are nearly identical throughout, and both increase sharply as overlap dependence strengthens.
Figure 12: Modular extension coverage at nominal 80% .
Figure 13: Modular extension coverage at nominal 95% .
The following motif is common in spatiotemporal settings: we have a sequence of covariate and label pairs observed for a relatively short, recent time period. We have access to unlabeled covariates over a longer time period. Data is observed over many spatial locations. For instance, crop yield might be observed over a large geographical area for recent years, but weather data (which is informative about crop yield) is available for a much longer period. The goal is to estimate, at each spatial location, the expected label (e.g., crop yield) in the future and provide a valid confidence interval for this value. The observed time period alone is too short for reliable estimates. Imputing missing labels with machine learning can cause substantial bias. Prediction-powered inference (PPI) can correct for this bias, but it relies on an i.i.d. assumption that breaks under our expected temporal dependencies. Heteroskedasticity and autocorrelation consistent (HAC) procedures account for temporal correlation, but have not been adapted to cases where some labels are imputed. We provide reliable point estimates and confidence intervals given: short labeled time series (across spatial locations), a longer unlabeled time series, and an imperfect predictor of labels given covariates. We show our method outperforms natural alternatives.
We study prediction-powered conditional inference in the setting where labeled data are scarce, unlabeled covariates are abundant, and a black-box machine-learning predictor is available. The goal is to perform statistical inference on conditional functionals evaluated at a fixed target point, such as conditional means, without imposing a parametric model for the conditional relationship. Our approach combines localization with prediction-based variance reduction. First, we introduce an RKHS localization method that learns a data-adaptive weight from covariates and reformulates the target conditional moment at the target point as a weighted unconditional moment. Second, we incorporate machine-learning predictions through a correction-based decomposition of this localized moment, yielding a prediction-powered estimator and confidence interval that reduce variance when the predictor is informative while preserving validity regardless of predictor accuracy. We establish nonasymptotic error bounds and, in the abundant-unlabeled regime, minimax-optimal convergence rates for the resulting estimator, prove pointwise asymptotic normality with consistent variance estimation, and provide an explicit variance decomposition that characterizes how machine-learning predictions and unlabeled covariates improve statistical efficiency. Numerical experiments on simulated and real datasets demonstrate valid conditional coverage and substantially sharper confidence intervals than alternative methods.
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.