# 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
$Y$
:
$I(X,Y) = H(X) - H(X|Y)$
The Venn Diagram below illustrates this concept: The Conditional Entropy
$H(X|Y)$
measures, in bits, the Expected Log-Loss associated with variable
$X$
once we have information on variable
$Y$
:
$H(X|Y) = - \sum\limits_{y \in Y} {p(y)\sum\limits_{x \in X} {p(x|y){{\log }_2}} } \left( {p(x|y)} \right)$
Hence, the Conditional Entropy is a key element in defining the Mutual Information between
$X$
and
$Y$
.
Note that
$I(X,Y) = H(X) - H(X|Y)$
is equivalent to:
$I(X,Y) = \sum\limits_{x \in X} {\sum\limits_{y \in Y} {p(x,y){{\log }_2}} } {{p(x,y)} \over {p(x)p(y)}}$
and furthermore equivalent to:
$I(X,Y) = \sum\limits_{y \in Y} {p(y)\sum\limits_{x \in X} {p(x|y){{\log }_2}} } {{p(x|y)} \over {p(x)}}$
This allows computing the Mutual Information between any two variables.

### Usage

For a given network, BayesiaLab can report the Mutual Information in several contexts:
• Main Menu > 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. • Main Menu > Analysis > Report > Relationship Analysis: • The Mutual Information can also be shown by selecting Main Menu > Analysis > Visual > Overall > Arc > Mutual Information and then clicking the Show Arc Comments icon or selecting Main 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.  