Favorable calibration conditions beyond the three-point design

Establish broader, verifiable design and predictor conditions under which the covariance-calibrated PPI update based on the unlabeled-minus-labeled prediction-score covariance provably improves finite-sample parameter risk beyond the three-point regression design and its local perturbations.

Background

The paper proves a favorable finite-sample risk direction for CC-PPI in a specific scalar regression model with covariates in {0,−a,a}, Gaussian errors, a linear predictor, and an explicit condition on the labeled and unlabeled proportions. It also establishes stability under small perturbations of that favorable configuration. However, the sign of the risk change is design-dependent: the same three-point calculation produces both favorable and adverse regimes. Consequently, the authors do not claim that replacing the labeled prediction-score covariance with its unlabeled counterpart universally improves parameter risk.

A broader result would characterize practically checkable conditions on the design, prediction error, covariance structure, or residual metric that guarantee improvement, ideally beyond the scalar three-point setting and without relying on local perturbation arguments.