Extension of sliced balancing to continuous treatments

Extend the sliced $L^p$ distributional balancing framework to continuous treatments by reformulating the balancing condition in terms of independence weights that make the weighted study population asymptotically independent between treatment and covariates.

Background

The developed SLDB framework is formulated for binary treatments, where balance is expressed by aligning treatment-specific covariate distributions with the marginal covariate distribution. For continuous treatments or dynamic treatment regimes, this two-sample formulation is not directly applicable. The paper identifies independence weights as a promising route for continuous treatments, but extending the sliced mechanism requires a new formulation of the balancing condition and is left unresolved.

References

A promising direction for extending our framework to continuous treatments is to pursue independence weights. Rather than balancing treatment-specific covariate distributions against the marginal distribution, this approach seeks weights $w$ such that the weighted study population satisfies, asymptotically, the independence of the treatment $A$ and covariates $X$ \citep{Huling2024independence}. Extending the sliced mechanism to such settings requires reformulating the balancing condition itself, which we leave as a topic for future work.

— Sliced $L^p$ Distributional Balancing  (2609.09600 - Zhang et al., 9 Sep 2026) in Section 7, Discussion