Gradient Estimation for Mixture Variational Inference

Mixture distributions are expressive variational families for black-box VI, but their discrete component choices complicate gradient estimation. We systematize reparameterization-based estimators for mixtures in a common notation, giving self-contained derivations and extending several to new settings. In particular, we provide an elementary derivation of a single-sample post-stratified estimator---previously derived via transport equations---and prove a variance reduction relative to simple random sampling. We also broaden the applicability of implicit reparameterization and reduce its computational complexity. Across different benchmarks, we find that stratified estimators are consistently robust when feasible; among single-sample methods, the post-stratified estimator frequently perform best, while implicit reparameterization is the most computationally demanding. Our analysis clarifies when each method should be used and provides efficient algorithms that make mixture-based variational inference practical.