Hierarchical Bayesian Bootstrap for Heterogeneous Treatment Effect Estimation
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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. This involves estimating a stratum-specific regression and integrating it over the distribution of confounders in that stratum - which itself must be estimated. Standard practice in the Bayesian causal literature is to use Rubin's Bayesian bootstrap to estimate these stratum-specific confounder distributions independently. However, this becomes problematic for sparsely populated strata with few unique observed confounder vectors. By construction, the Bayesian bootstrap allocates no prior mass on confounder values unobserved within each stratum - even if these values are observed in other strata and we think they are a priori plausible. We propose causal estimation via a hierarchical Bayesian bootstrap (HBB) prior over the stratum-specific confounder distributions. Based on the Hierarchical Dirichlet Process, the HBB partially pools the stratum-specific confounder distributions by assuming all confounder vectors seen in the overall sample are plausible. In large strata, estimates allocate much of the mass to values seen within the strata, while placing small non-zero mass on unseen values. However, for sparse strata, more weight is given to values unseen in that stratum but seen elsewhere - thus shrinking the distribution towards the marginal. This allows us to borrow information across strata when estimating HTEs - leading to efficiency gains over standard marginalization approaches while avoiding strong parametric modeling assumptions about the confounder distribution when estimating HTEs. Moreover, the HBB is computationally efficient (due to conjugacy) and compatible with arbitrary outcome models.
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