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.

翻译：我们提出保序条件法则（ICL），该法则通过在经典条件法则基础上增加一个额外要求——所关注的随机变量与条件随机对象之间存在保序关系——扩展了传统概念。我们基于给定σ-格的数学期望构建了ICL的存在性与唯一性。ICL与经典条件法则等价当且仅当后者本身具有保序性。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值来阐明其动机，其中评分规则可取自包含连续排序概率评分（CRPS）在内的一大类函数。此外，ICL具有校准性质，即对特定条件运算保持不变，且相应的事件概率与分位数在所有相关评分函数下同时达到最优。我们将发展一种基于σ-格的广义条件泛函新概念，该概念具有独立的研究价值。