Bayesian Adaptive Sparse Copula

Abstract

Bayesian nonparametric density estimation procedures typically utilize single-scale methods, such as Dirichlet process mixtures. In contrast to these, alternative multiscale estimators have a number of well-known advantages, including the ability to characterize abrupt local changes and to provide an estimate with a desired level of resolution. Despite their theoretical appeal, multiscale methods developed in the literature have been typically univariate. Their multivariate versions are in general very costly to use in practical applications, rendering such methods infeasible in many cases of interest. One of the key reasons is the rapidly increasing number of multiscale mixture components required to represent a nonparametric dependence structure in higher dimensions. In this paper, we propose a multivariate sparse multiscale Bernstein polynomial model for a copula dependence structure based on a Bayesian spike-and-slab prior. Its implementation has a flavor of multiscale adaptive importance sampling whereby important Bernstein polynomial components are preserved in the multivariate tree while components with small weights are omitted, yielding tree sparsity alleviating the curse of dimensionality. We combine the proposed copula model with nonparametric marginals for general density estimation. The resulting sparse posterior requires only a fraction of the implementation time and memory size relative to its non-sparse counterpart. This makes our approach feasible in  multivariate settings when other scenarios are inoperational. We further verify the conditions for posterior consistency and provide an application to forecasting the Value at Risk and Expected Shortfall of a financial portfolio.

Keywords: copulas, nonparametrics, multiscale, Bernstein polynomial, Value-at-Risk