Precise evaluation of effective dimension under non-Gaussian priors

Precisely evaluate the Bayesian effective dimension d_eff(n) = 2 I(Θ; X^{(n)}) / log n for models with non-Gaussian priors, including controlling mutual information components that arise in global–local scale mixtures.

Background

In shrinkage settings, mutual information decomposes through latent scales, and controlling terms like I(λ; Y) becomes technically challenging, especially under heavy tails. The authors emphasize that this difficulty persists beyond the Gaussian conjugate framework.

The statement identifies a broad gap: even though effective dimension unifies many behaviors, exact evaluation under non-Gaussian priors generally lacks tractable formulas or guarantees.