Using effective dimension as a diagnostic for approximate Bayesian inference

Determine whether the Bayesian effective dimension d_eff(n) = 2 I(Θ; X^{(n)}) / log n can serve as a general, model-agnostic diagnostic for information loss in approximate Bayesian procedures.

Background

Section 5 shows that, in Gaussian settings, covariance inflation typical of approximate posteriors reduces mutual information and hence effective dimension. This suggests d_eff(n) could quantify information loss caused by approximation.

The open question asks for a general validation of effective dimension as a diagnostic across models and approximation schemes, beyond Gaussian cases and specific algorithms.