Bounding M_{θ|y} for Bayesian logistic regression in terms of prior and likelihood

Derive explicit bounds on M_{θ|y} = μ_{θ|y}ᵀΣ_{θ|y}^{-1}μ_{θ|y} in terms of the prior parameters (μπ, Σπ) and the logistic likelihood (data x and labels y), to obtain dimension-dependent runtime guarantees for the thermodynamic Bayesian logistic regression sampler.

Background

In Appendix F, the authors analyze Wasserstein-2 contraction for the Bayesian logistic regression sampler and identify the quantity M_{θ|y} = μ{θ|y}ᵀΣ{θ|y}^{-1}μ_{θ|y} as central to bounding convergence time.

They explicitly note that bounding M_{θ|y} in terms of prior and likelihood parameters is unclear and leave this as future work. Such bounds would enable removing ad-hoc assumptions and yielding rigorous runtime guarantees across dimensions.