# Fused Multinomial Logistic Regression Utilizing Summary-Level External Machine-learning Information

arXiv: https://arxiv.org/abs/2604.03939 · HTML: https://mlsanity.com/paper/2604.03939/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 23 references that are in the map.
- First posted on arXiv 5 Apr 2026.

## Builds on (most cited first, 3 of 3)

- [Generalized Random Forests](https://mlsanity.com/paper/1610.01271/) (2016)
- [A synthetic data integration framework to leverage external summary-level information from heterogeneous populations](https://mlsanity.com/paper/2106.06835/) (2021)
- [Risk Projection for Time-to-event Outcome Leveraging Summary Statistics With Source Individual-level Data](https://mlsanity.com/paper/2003.01850/) (2020)

## Abstract (arXiv)

In many modern applications, a carefully designed primary study provides individual-level data for interpretable modeling, while summary-level external information is available through black-box, efficient, and nonparametric machine-learning predictions. Although summary-level external information has been studied in the data integration literature, there is limited methodology for leveraging external nonparametric machine-learning predictions to improve statistical inference in the primary study. We propose a general empirical-likelihood framework that incorporates external predictions through moment constraints. An advantage of nonparametric machine-learning prediction is that it induces a rich class of valid moment restrictions that remain robust to covariate shift under a mild overlap condition without requiring explicit density-ratio modeling. We focus on multinomial logistic regression as the primary model and address common data-quality issues in external sources, including coarsened outcomes, partially observed covariates, covariate shift, and heterogeneity in generating mechanisms known as concept shift. We establish large-sample properties of the resulting fused estimator, including consistency and asymptotic normality under regularity conditions. Moreover, we provide mild sufficient conditions under which incorporating external predictions delivers a strict efficiency gain relative to the primary-only estimator. Simulation studies and an application to the National Health and Nutrition Examination Survey on multiclass blood-pressure classification.