Entropy Scope of Relevance

Proposed answer to the following question(s):

Let \(\operatorname{dom}{ f}\) denote the domain of a function \(f\). Given \(\pi\) a set of sets, let \( {\cup{ \pi}}\) denote the union of all member sets in \(\pi\). When \(\pi\) is a partition, \(\pi\) covers \( {\cup{ \pi}}\).

In a scope of relevance, partitions represent questions with each member set (part) an answer. Each answer represents an event of a probability space.

Given a function over partitions of events of probability space \(\Omega\), define levels of the domain:

\[\begin{eqnarray*} \operatorname{dom}_{ 0}{ \theta} & := & \{ \pi \in \operatorname{dom}{ \theta} : {\cup{ \pi}} = \Omega \} \\ \operatorname{dom}_{ i+1}{ \theta} & := & \left\{ \pi \in \operatorname{dom}{ \theta} : {\cup{ \pi}} \in \rho, \rho \in \operatorname{dom}_{ i}{ \theta} , \pi \not\in \operatorname{dom}_{ i}{ \theta} \right\} \\ \end{eqnarray*}\]

A scope of relevance \(\theta\) is a non-negative real-valued function over partitions of events (subsets) of a probability space \(\Omega\) with the following conditions:

  1. At most one partition can cover any event. Formally, \( {\cup{ \pi}} = {\cup{ \rho}}\) implies \(\pi = \rho\) for any partitions \(\pi\) and \(\rho\) in \(\operatorname{dom}{ \theta}\).
  2. \(\operatorname{dom}{ \theta} = \bigcup_{i=0}^\infty \operatorname{dom}_{ i}{ \theta}\)

The domain of a scope of relevance is all relevant questions. Each real value assigned to a question is a degree of relevance. The second condition in the definition means the partitions (questions) are dividing the probability space \(\Omega\) in a nested hierarchical manner. Or in other words, either the event of all outcomes (\(\Omega\)) or the event of an answer to a question is the event under which another question can be asked.