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Isotonic conditional laws

Sebastian Arnold, Johanna Ziegel

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 $σ$-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 $σ$-lattices which is of independent interest.

Isotonic conditional laws

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 -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 -lattices which is of independent interest.
Paper Structure (19 sections, 23 theorems, 96 equations, 2 figures)

This paper contains 19 sections, 23 theorems, 96 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Existence and uniqueness of ICL
  3. Factorization and connection to classical conditional laws
  4. Generated $\sigma$-lattices and measurability for random elements
  5. Factorization and connection to the classical conditional law
  6. Universality of ICL
  7. Identifiable functionals, scoring functions and scoring rules
  8. ICL as a scoring rule minimizer
  9. Calibration properties of ICL
  10. Conditional functionals given $\sigma$-lattices
  11. General construction
  12. Strictly increasing identification functions and conditional expectations
  13. Conditional quantiles and quantile calibration of ICL
  14. Proof of Theorem \ref{['thm:existence_and_uniqueness_ICL']}
  15. Ordered metric spaces
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $Y$ be a random variable and $\mathcal{A} \subseteq \mathcal{F}$ be a $\sigma$-lattice. Then there exists a conditional law of $Y$ given $\mathcal{A}$, denoted by $P_{Y\mid \mathcal{A}}$. More precisely, there exists a Markov kernel $P_{Y \mid \mathcal{A}}$ from $(\Omega, \mathcal{F})$ into $(\m

Figures (2)

  • Figure 1: Linearity and the tower property are violated for conditional expectations given $\sigma$-lattices. Left panel: The blue cone indicates the family $L_2(\mathcal{A})$ for $\sigma$-lattice $\mathcal{A}\subseteq \mathcal{F}$. Here, $\mathbb{E}(Y_1 \mid \mathcal{A})+ \mathbb{E}(Y_2 \mid \mathcal{A}) =X \neq Y=\mathbb{E}(Y_1+Y_2 \mid \mathcal{A})$. Right panel: For $\sigma$-lattices $\mathcal{A}_1\subseteq A_2 \subseteq \mathcal{F}$, the families $L_2(\mathcal{A}_1)$ and $L_2(\mathcal{A}_2)$ are denoted by the blue and orange cone, respectively. Clearly, $\mathbb{E}(\mathbb{E}(Y \mid \mathcal{A}_2) \mid \mathcal{A}_1) =X_3\neq X_1=\mathbb{E}(Y \mid \mathcal{A}_1)$.
  • Figure 2: The implications between auto-calibration (AC), isotonic calibration (IC), threshold calibration (TC) and quantile calibration (QC) as given in Proposition \ref{['prop:implications_calibration']}.

Theorems & Definitions (63)

  • Definition 2.1
  • Example 2.1
  • Theorem 2.1: Existence and uniqueness of ICL
  • Remark 1
  • Example 3.1: Euclidean space
  • Example 3.2: The space $\mathcal{P}(\mathbb{R})$
  • Definition 3.1
  • Lemma 3.1
  • Proposition 3.2
  • Theorem 3.3
  • ...and 53 more