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.

翻译：本文提出保序条件分布（ICL），该概念在经典条件分布的基础上附加了如下要求：目标随机变量与条件随机对象之间存在保序关系。我们基于给定σ-格的期望条件，证明了ICL的存在性与唯一性。当且仅当经典条件分布本身已是保序的时，ICL才与之等价。从统计学角度出发，我们通过证明ICL等价于某个期望评分的最小化问题来阐明其动机，其中评分规则可取自包含连续排序概率评分（CRPS）在内的广泛类别。此外，ICL具有校准性，即其对特定条件运算具有不变性，且相应的事件概率与分位数在所有相关评分函数意义下同时达到最优。我们还发展了给定σ-格的一般条件泛函的新概念，该概念本身具有独立的研究意义。