Bayesian Adaptive Sparse Copula: Tackling the Curse of Dimensionality in Multivariate Data

The Journal of Computational and Graphical Statistics has published an article by Martin Burda and Artem Prokhorov titled «Bayesian Adaptive Sparse Copula».

The paper addresses Bayesian nonparametric methods for estimating multivariate densities. Conventional approaches based on Dirichlet process mixtures employ single-scale priors and exhibit limited capacity to capture abrupt local variations in the data. Multiscale methods constructed from decision trees overcome this limitation; however, their multivariate implementation entails substantial computational burdens due to the exponential increase in the number of mixture components as dimensionality grows.

The authors propose an approach designed to mitigate this constraint. The method relies on a random Bernstein polynomial prior defined on the unit hypercube of arbitrary dimension. A spike-and-slab shrinkage structure is incorporated into the model, inducing automatic pruning of negligible branches in the decision tree at the posterior stage. This mechanism preserves the advantages of a multiscale framework while alleviating the curse of dimensionality. The proposed specification is embedded within a broader model via a copula link function and nonparametric marginal distributions.

The theoretical component of the study establishes conditions for posterior consistency under the weak topology. Simulation results confirm satisfactory finite-sample performance of the method. The practical applicability of the approach is illustrated through an application to forecasting Value at Risk and Expected Shortfall for a financial portfolio, in a setting where sampling from a non-sparse posterior is computationally infeasible.