Mutual Information

The Mutual Information $I(X, Y)$ measures the amount of information gained on variable $X$ (the reduction in the Expected Log-Loss) by observing variable $Y$:

$\displaystyle I(X,Y) = H(X) - H(X \mid Y)$

The Venn Diagram below illustrates this concept:

The Conditional Entropy $H(X \mid Y)$ measures, in bits, the Expected Log-Loss associated with variable $X$ once we have information on variable $Y$:

$\displaystyle H(X \mid Y) = - \sum\limits_{y \in Y} {p(y)\sum\limits_{x \in X} {p(x \mid y){{\log }_2}} } \left( {p(x \mid y)} \right)$

Hence, the Conditional Entropy is a key element in defining the Mutual Information between $X$ and $Y$.

Note that

$\displaystyle I(X,Y) = H(X) - H(X \mid Y)$

is equivalent to:

$\displaystyle I(X,Y) = \sum\limits_{x \in X} {\sum\limits_{y \in Y} {p(x,y){{\log }_2}} } {{p(x,y)} \over {p(x)p(y)}}$

and furthermore equivalent to:

$\displaystyle I(X,Y) = \sum\limits_{y \in Y} {p(y)\sum\limits_{x \in X} {p(x \mid y){{\log }_2}} } {{p(x \mid y)} \over {p(x)}}$

This allows computing the Mutual Information between any two variables.

Usage

For a given network, BayesiaLab can report the Mutual Information in several contexts:

• Menus > Analysis > Report > Target > Relationship with Target Node.
• Note that this table shows the Mutual Information of each node, e.g., XRay, Dyspnea, etc., only with regard to the Target Node, Cancer.

• Menus > Analysis > Report > Relationship Analysis:

• The Mutual Information can also be shown by selecting Menus > Analysis > Visual > Overall > Arc > Mutual Information and then clicking the Show Arc Comments icon or selecting Menu > View > Show Arc Comments.

• Note that the corresponding options under Preferences > Analysis > Visual Analysis > Arc's Mutual Information Analysis have to be selected first:

• In Preferences, Child refers to the Relative Mutual Information from the Parent onto the Child node, i.e., in the direction of the arc.
• Conversely, Parent refers to the Relative Mutual Information from the Child onto the Parent node, i.e., in the opposite direction of the arc.