Theory and computation for structured variational inference
This work addresses a core methodology in statistical applications, providing theoretical foundations and computational guarantees for structured variational inference, which is incremental but extends prior results to a more complex setting.
The paper tackled the problem of star-structured variational inference by proving existence, uniqueness, and self-consistency results, and derived quantitative error bounds for the variational approximation to the posterior, extending prior mean-field work to this structured setting.
Structured variational inference constitutes a core methodology in modern statistical applications. Unlike mean-field variational inference, the approximate posterior is assumed to have interdependent structure. We consider the natural setting of star-structured variational inference, where a root variable impacts all the other ones. We prove the first results for existence, uniqueness, and self-consistency of the variational approximation. In turn, we derive quantitative approximation error bounds for the variational approximation to the posterior, extending prior work from the mean-field setting to the star-structured setting. We also develop a gradient-based algorithm with provable guarantees for computing the variational approximation using ideas from optimal transport theory. We explore the implications of our results for Gaussian measures and hierarchical Bayesian models, including generalized linear models with location family priors and spike-and-slab priors with one-dimensional debiasing. As a by-product of our analysis, we develop new stability results for star-separable transport maps which might be of independent interest.