Characterizing the randomness of effective dimension under shrinkage priors

Provide a precise characterization of the randomness of the Bayesian effective dimension induced by shrinkage priors, particularly heavy-tailed global–local priors, and ascertain whether effective dimension concentrates in high-dimensional regimes or retains intrinsic variability as the sample size increases.

Background

The paper shows that global–local (shrinkage) priors randomize local signal-to-noise ratios, making the effective dimension data-dependent and random. While inequalities and bounds are provided, a full characterization is open.

Heavy tails (e.g., horseshoe) complicate analysis via mutual information decompositions, raising questions about concentration versus persistent variability of effective dimension as n grows.

References

A precise characterization of this randomness remains open, particularly for heavy-tailed global--local priors. It is unclear whether effective dimension under shrinkage concentrates in high-dimensional regimes or retains intrinsic variability even as the sample size grows.

— Bayesian Effective Dimension: A Mutual Information Perspective (2512.23047 - Banerjee, 28 Dec 2025) in Section 7 (Open problems and future directions)