Abstract
Conformal prediction provides distribution-free prediction intervals with finite-sample coverage guarantees, and recent work by Snell \& Griffiths reframes it as Bayesian Quadrature (BQ-CP), yielding powerful data-conditional guarantees via Dirichlet posteriors over thresholds. However, BQ-CP fundamentally requires the i.i.d. assumption -- a limitation the authors themselves identify. Meanwhile, weighted conformal prediction handles distribution shift via importance weights but remains frequentist, producing only point-estimate thresholds. We propose \textbf{Weighted Bayesian Conformal Prediction (WBCP)}, which generalizes BQ-CP to arbitrary importance-weighted settings by replacing the uniform Dirichlet \Dir(1,\ldots,1) with a weighted Dirichlet \Dir(
eff \cdot \tilde{w}_1, \ldots,
eff \cdot \tilde{w}_n), where
eff is Kish's effective sample size. We prove four theoretical results: (1)~
eff is the unique concentration parameter matching frequentist and Bayesian variances; (2)~posterior standard deviation decays as O(1/\sqrt{
eff}); (3)~BQ-CP's stochastic dominance guarantee extends to per-weight-profile data-conditional guarantees; (4)~the HPD threshold provides O(1/\sqrt{
eff}) improvement in conditional coverage. We instantiate WBCP for spatial prediction as \emph{Geographical BQ-CP}, where kernel-based spatial weights yield per-location posteriors with interpretable diagnostics. Experiments on synthetic and real-world spatial datasets demonstrate that WBCP maintains coverage guarantees while providing substantially richer uncertainty information.
