  We develop scalable methods for producing conformal Bayesian predictive
intervals with finite sample calibration guarantees. Bayesian posterior
predictive distributions, $p(y \mid x)$, characterize subjective beliefs on
outcomes of interest, $y$, conditional on predictors, $x$. Bayesian prediction
is well-calibrated when the model is true, but the predictive intervals may
exhibit poor empirical coverage when the model is misspecified, under the so
called ${\cal{M}}$-open perspective. In contrast, conformal inference provides
finite sample frequentist guarantees on predictive confidence intervals without
the requirement of model fidelity. Using 'add-one-in' importance sampling, we
show that conformal Bayesian predictive intervals are efficiently obtained from
re-weighted posterior samples of model parameters. Our approach contrasts with
existing conformal methods that require expensive refitting of models or
data-splitting to achieve computational efficiency. We demonstrate the utility
on a range of examples including extensions to partially exchangeable settings
such as hierarchical models.

Methodology;Computation