Copula Active Subspaces II: Error Decomposition, A Posteriori Estimation, and Sharpness of the Bounds

Abstract

The Copula Active Subspaces (CAS) method of Part I estimates a non-Gaussian observation-noise density $π_{\boldsymbol{n}}$ from residual samples. Its KL divergence $\mathcal{D}_{\mathrm{KL}}(π_{\boldsymbol{n}}\,\|\,\widehatπ_{\boldsymbol{n}})$ admits an exact decomposition into marginal, projection and reduced-density terms and bounds the posterior KL divergence averaged over the true predictive distribution. We derive a priori bounds for these terms, together with estimates or bounds from residual samples. Weighted nearest-neighbour entropy estimates are used for the marginal and reduced-density terms, with asymptotic standard errors in exact rank coordinates. For the projection error, the score is fitted on one subsample, and its mean squared magnitude outside the retained subspace is estimated on the other. Corrections for score and rank-transform errors then give a theoretical upper bound with probability at least $1-δ$. Tests on three known noise laws in dimension $d=20$ use the estimated rank transform. In $11$ of $12$ combinations of example and sample size, the sum of the two estimated terms and the trace bound exceeds the reference divergence after averaging over the estimated models. In these tests, score-estimation error is the largest contribution to looseness. The overestimate from the Gaussian log-Sobolev inequality alone is at most a factor of $1.98$.