Isotonic conditional laws

TL;DR

This paper introduces isotonic conditional laws (ICL), a new concept extending classical conditional laws with an isotonic relationship, providing unique, calibrated, and optimal probabilistic predictions under certain scoring rules.

Contribution

The paper defines ICL, proves its existence and uniqueness, and demonstrates its properties and optimality in probabilistic forecasting within a new theoretical framework.

Findings

ICL exists uniquely under certain conditions.

ICL minimizes a broad class of scoring rules including CRPS.

ICL is invariant to specific conditioning operations, ensuring calibration.

Abstract

We introduce isotonic conditional laws (ICL) which extend the classical notion of conditional laws by the additional requirement that there exists an isotonic relationship between the random variable of interest and the conditioning random object. We show existence and uniqueness of ICL building on conditional expectations given $\sigma$-lattices. ICL corresponds to a classical conditional law if and only if the latter is already isotonic. ICL is motivated from a statistical point of view by showing that ICL emerges equivalently as the minimizer of an expected score where the scoring rule may be taken from a large class comprising the continuous ranked probability score (CRPS). Furthermore, ICL is calibrated in the sense that it is invariant to certain conditioning operations, and the corresponding event probabilities and quantiles are simultaneously optimal with respect to all relevant scoring functions. We develop a new notion of general conditional functionals given $\sigma$-lattices which is of independent interest.