Conformal predictive systems allow forecasters to issue predictive distributions for real-valued future outcomes that have out-of-sample calibration guarantees. On a more abstract level, conformal prediction makes use of in-sample calibration guarantees to construct bands of predictions with out-of-sample guarantees under exchangeability. The calibration guarantees are typically that prediction intervals derived from the predictive distributions have the correct marginal coverage. We extend this line of reasoning to stronger notions of calibration that are common in statistical forecasting theory.

We take two prediction methods that are calibrated in-sample, and conformalize them to obtain conformal predictive systems with stronger out-of-sample calibration guarantees than existing approaches. The first method corresponds to a binning of the data, while the second leverages isotonic distributional regression (IDR), a non-parametric distributional regression method under order constraints. We study the theoretical properties of these new conformal predictive systems, and compare their performance in a simulation experiment. They are then applied to two case studies on European temperature forecasts and on predictions for the length of patient stay in Swiss intensive care units. Both approaches are found to outperform existing conformal predictive systems, while conformal IDR additionally provides a natural method for quantifying epistemic uncertainty of the predictions.