From a Hierarchical Dirichlet-Type Construction to the Informative Bayesian Double Bootstrap

Efron’s double bootstrap and hierarchical Bayesian nonparametric methods have largely developed along separate paths. This paper connects them and uses that connection to motivate a new resampling procedure. We show that a suitable two-level Dirichlet construction, with base measures matched to the data, can approach the classical double bootstrap when its concentration parameters become large, while the hierarchical Dirichlet process of Teh et al. does not share this limit. This distinction identifies the double bootstrap as the endpoint of a construction of hierarchical Dirichlet type—though not of the hierarchical Dirichlet process itself—and as the boundary of a broader family of two-level resampling methods. Moving away from that boundary leads to the Informative Bayesian Double Bootstrap (IBDB). The method introduces prior information at the first stage while keeping the second stage focused on calibration, as in the classical double bootstrap. We also establish finite-concentration bounds describing how the proposed construction differs from its classical counterpart and when it can move beyond the support of the observed data. In simulations against four competing methods, the IBDB performs best for tail-sensitive quantities and heavy-tailed settings, while it tends to over-cover simple location parameters. Similar patterns appear in the Danish fire-insurance and Siemens equity-loss examples. Its main advantage is improved calibration through interval repositioning rather than simply wider intervals. The gains are most relevant when sample information is limited.