A way to proper generalization of $\phi$-divergence based on Choquet derivatives

Abstract

In this article, we discuss an important concept of the generalized Radon–Nikodym derivatives using Choquet integral and its correct treatment. It is consequently used in the definition of generalized $\phi$-divergence for fuzzy measures with the emphasis on its proper use. We give several results concerning basic properties of $\phi$-divergence. In both concepts, we focus mainly on sufficient conditions for which standard properties copy the additive case. As it is shown, the key role in most of the relevant properties for both derivative as well as divergence is the subadditivity of fuzzy measure.

Appendix

Table 1 Specific divergences

Definition A.1

(Dominance and absolute continuity of measures¹⁴)

• (a) Measure $\nu$ is dominated by measure $\mu$, denoted as $\nu \ll \mu$, if for every set $E \in {\mathcal {A}}$

  $$\begin{aligned} \mu (E) = 0 \quad \Rightarrow \quad \nu (E) = 0. \end{aligned}$$

  (Loosely speaking the $\mu$-null sets are also $\nu$-null sets.)

• (b) Measure $\nu$ is absolutely continuous with respect to $\mu$, denoted as $\nu \ll _{\varepsilon } \mu$, if for every $E \in {\mathcal {A}}$

  $$\begin{aligned} (\forall \varepsilon> 0) \, (\exists \delta > 0) \,\, \mu (E)< \delta \quad \Rightarrow \quad \nu (E) < \varepsilon . \end{aligned}$$

  (Loosely speaking the sets of small measure $\mu$ are also sets of small measure $\nu$.)

Definition A.2

(Mutual singularity of measures) Two measures $\mu$ and $\nu$ are called mutually singular (orthogonal), denoted as $\mu \perp \nu$, if there exists a set $A \in {\mathcal {A}}$ such that $\mu$ is concentrated¹⁵ on A and $\nu$ is concentrated on $A^c.$

Theorem A.3

(Computation of Choquet integral on half-line (Sugeno 2013)) Let $\mu$ be a fuzzy measure on $\Omega \subseteq [0,\infty )$, $g: {\mathbb {R}}^+ \rightarrow {\mathbb {R}}^+$ be a nonnegative continuous function and $\mu ([t, \tau ])$ is differentiable function with respect to t. If

• (I) g is non-decreasing, the Choquet integral of g with respect to $\mu$ on $[0, \tau ]$ is represented as

  $$\begin{aligned} (C) \int _{[0, \tau ]} g \,{\mathrm {d}}\mu = - \int _{0}^{\tau } \frac{\partial }{\partial t} \mu ([t, \tau ]) g(t) \,{\mathrm {d}}t; \end{aligned}$$

• (II) g is non-increasing, the Choquet integral of g with respect to $\mu$ on $[0, \tau ]$ is represented as

  $$\begin{aligned} (C) \int _{[0, \tau ]} g \,{\mathrm {d}}\mu = \int _{0}^{\tau } \frac{\partial }{\partial t} \mu ([0, t)) g(t) \,{\mathrm {d}}t. \end{aligned}$$

Definition A.3

(Weak and strong decomposition property (Graf 1980)) Let $(\Omega , {\mathcal {A}})$ be a measurable space and let $\mu , \nu : {\mathcal {A}} \rightarrow {\mathbb {R}}^+$ be subadditive fuzzy measures.

• The pair $(\mu , \nu )$ is said to possess the WDP if for all $\alpha \in {\mathbb {R}}{^+}$ there exists a set $E_{\alpha } \in {\mathcal {A}}$ such that

  $$\begin{aligned} \alpha \nu _{E_{\alpha }} \le \mu _{E_{\alpha }} \qquad {and} \qquad \alpha \nu _{E_{\alpha }^{C}} \ge \mu _{E_{\alpha }^{C}}. \end{aligned}$$

• The pair $(\mu , \nu )$ is said to possess the SDP if for all $\alpha \in {\mathbb {R}}^+$ there exists a set $E_{\alpha } \in {\mathcal {A}}$ such that the following conditions hold

• $\forall E, F \in {\mathcal {A}}: \,\, F \subset E \subset E_{\alpha } \qquad \alpha [\nu (E) - \nu (F)] \le \mu (E) - \mu (F)$

• $\forall E \in {\mathcal {A}} \qquad \alpha [\nu (E) - \nu (E \cap E_{\alpha })] \ge \mu (E) - \mu (E \cap E_{\alpha })$.

Theorem A.4

(Characterization of SDP (Graf 1980)) Let $(\Omega , {\mathcal {A}})$ be a measurable space and let $\mu , \nu : {\mathcal {A}} \rightarrow {\mathbb {R}}^+$ be subadditive fuzzy measures. Then $(\mu , \nu )$ has the SDP if and only if the following holds

• (I) $(\mu , \nu )$ has the WDP.

• (II) For every $\alpha \in {\mathbb {R}}^+$, for every $E \in {\mathcal {A}}$ with $\alpha \nu (E) \le \mu (E)$, and for every $F \in {\mathcal {A}}$ with $F \subset E$ the inequality

  $$\begin{aligned} \alpha \left( \nu (E) - \nu (F) \right) \le \mu (E) - \mu (F) \end{aligned}$$

  is satisfied.

• (III) For every $\alpha \in {\mathbb {R}}^+$, for every $E \in {\mathcal {A}}$, and for every $F \in {\mathcal {A}}$ with  $F \subset E, \, \alpha \nu (F) \le \mu (F)$ and  $\alpha \nu (E/F) \ge \mu (E/F)$ the inequality

  $$\begin{aligned} \alpha \left( \nu (E) - \nu (F) \right) \ge \mu (E) - \mu (F) \end{aligned}$$

  is satisfied.