Weighted Bayesian Conformal Prediction

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 $\mathbf{\text{Weighted}}\mathbf{\text{Bayesian}}\mathbf{\text{Conformal}}\mathbf{\text{Prediction}}\mathbf{\text{(WBCP)}}$, which generalizes BQ-CP to arbitrary importance-weighted settings by replacing the uniform Dirichlet $\text{Dir}(1,\dots ,1)$ with a weighted Dirichlet $\text{Dir}(\text{neff}\cdot {\tilde{w}}_1,\dots ,\text{neff}\cdot {\tilde{w}}_n)$, where $\text{neff}$ is Kish’s effective sample size. We prove four theoretical results: (1)~$\text{neff}$ is the unique concentration parameter matching frequentist and Bayesian variances; (2)~posterior standard deviation decays as $O(1\mathrm{/}\sqrt{\text{neff}})$; (3)~BQ-CP’s stochastic dominance guarantee extends to per-weight-profile data-conditional guarantees; (4)~the HPD threshold provides $O(1\mathrm{/}\sqrt{\text{neff}})$ 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.