Learning Prognostic Representations via Score-Ratio Dimension Reduction

High-dimensional covariate adjustment can preserve causal identification while worsening overlap. We propose a score-ratio method for learning low-dimensional prognostic representations for the average treatment effect on the treated. By subtracting the score ratio associated with treatment from that associated with treatment and outcome jointly, the method removes treatment-related information and isolates directions governing the control outcome. We show that the resulting spectral decomposition bounds the error of the low-dimensional conditional approximation and yields a valid prognostic representation when the spectral remainder vanishes. Under a Gaussian single-index model, the method recovers the overlap-optimal prognostic direction. In a nonlinear symmetric experiment where standard linear and dimension-reduction methods perform poorly, score-ratio refinement achieves a median squared cosine similarity of 0.967.