Isotonic Conditional Laws

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 will develop a new notion of general conditional functionals given $\sigma$-lattices which is of independent interest.