Hierarchical Copula-based Conformal Prediction and Exact Validity via Nested Prediction Regions

Empirical, Archimedean and vine copulas have been repeatedly investigated and leveraged to infer conformal prediction regions for multivariate predictions, but they do not provide finite-size guarantees when the estimated copula is biased or misspecified. To address this limitation, we start with copula-based conformal prediction regions that are always nested and we leverage this property to counteract this copula-estimation bias, via an additional conformal re-calibration step. Furthermore, we introduce a simpler class of semi-parametric copulas (i.e., hierarchical Archimedean copulas) as an alternative to the more complex vine copulas for which incorporating prior knowledge is difficult. Using synthetic data sets, we compare biased and debiased copula-based conformal prediction methods, and we report the impact of the data size and the impact of the number of output dimensions. Using real data, we leverage prior knowledge via this simpler class of copulas. In these experiments, we observe that this additional re-calibration step effectively eliminates the estimation bias of empirical and semi-parametric copulas when its computations are precise (enough) and the data size is large enough. The debiased hierarchical Archimedean copulas yield performances that are comparable to the results of debiased vine copulas.