Dimension scaling of posterior parameters for Bayesian logistic regression in TBI

Determine how the Bayesian logistic regression posterior mean μ_{θ|y} and covariance Σ_{θ|y}, and the scalar M_{θ|y} = μ_{θ|y}ᵀΣ_{θ|y}μ_{θ|y}, scale with the parameter dimension d under the proposed thermodynamic Langevin sampling protocol, so as to derive rigorous time-complexity bounds that do not rely on ad-hoc assumptions.

Background

For Bayesian logistic regression, the derived runtime bound depends on a scalar quantity M_{θ|y} involving the posterior mean and covariance (M_{θ|y} = μ{θ|y}ᵀΣ{θ|y}μ{θ|y}). The authors’ analysis shows logarithmic dependence on d provided that M{θ|y} grows at most linearly with dimension.

The paper explicitly states that there are currently no results constraining the dimensional scaling of μ{θ|y} and Σ{θ|y}, making the overall time bound contingent on an ad-hoc assumption. Establishing these scaling properties would make the complexity guarantees rigorous.