Role of effective dimension in posterior contraction

Determine whether posterior concentration can be characterized directly in terms of the Bayesian effective dimension d_eff(n) = 2 I(Θ; X^{(n)}) / log n, and ascertain whether small effective dimension is necessary for contraction under standard loss functions or whether contraction can occur despite large mutual information.

Background

The paper intentionally avoids contraction-rate theorems but proposes that effective dimension might unify understanding of posterior behavior across models. Establishing a direct link between contraction and d_eff(n) could replace ambient-dimension or sparsity proxies with an intrinsic, information-theoretic quantity.

The authors also question necessity: whether low effective dimension must hold for contraction, or whether there are regimes where contraction persists even with large information gain, highlighting gaps in current theory.