  A major focus of causal inference is the estimation of heterogeneous average
treatment effects (HTE) - average treatment effects within strata of another
variable of interest such as levels of a biomarker, education, or age strata.
Inference involves estimating a stratum-specific regression and integrating it
over the distribution of confounders in that stratum - which itself must be
estimated. Standard practice involves estimating these stratum-specific
confounder distributions independently (e.g. via the empirical distribution or
Rubin's Bayesian bootstrap), which becomes problematic for sparsely populated
strata with few observed confounder vectors. In this paper, we develop a
nonparametric hierarchical Bayesian bootstrap (HBB) prior over the
stratum-specific confounder distributions for HTE estimation. The HBB partially
pools the stratum-specific distributions, thereby allowing principled borrowing
of confounder information across strata when sparsity is a concern. We show
that posterior inference under the HBB can yield efficiency gains over standard
marginalization approaches while avoiding strong parametric assumptions about
the confounder distribution. We use our approach to estimate the adverse event
risk of proton versus photon chemoradiotherapy across various cancer types.
