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Conceptually, this formula identifies mutual information as the Kullback-Leibler (KL) divergence between the actual joint distribution $p(x,y)$ and the independent reference distribution $p(x)p(y)$. It measures how far the actual joint distribution is from pure independence.

Conditions: The displayed derivation is for discrete distributions using probability mass functions.; The ratio form requires $p(x,y)>0$ and $p(x)p(y)>0$ on the summed support.

Conceptually, this formula identifies mutual information as the Kullback-Leibler (KL) divergence between the actual joint distribution $p(x,y)$ and the independent reference distribution $p(x)p(y)$. It measures how far the actual joint distribution is from pure independence.

Conditions: The displayed derivation is for discrete distributions using probability mass functions.; The ratio form requires $p(x,y)>0$ and $p(x)p(y)>0$ on the summed support.

When $X$ and $Y$ are independent, knowing one tells you nothing about the other. Mathematically, the joint distribution factors into the product of the marginals, $p(x,y) = p(x) \cdot p(y)$.

Conditions: Applies to the case where $X$ and $Y$ are independent.; The displayed notation uses probability mass functions $p(x)$, $p(y)$, and $p(x,y)$.

When $X$ and $Y$ are independent, knowing one tells you nothing about the other. Mathematically, the joint distribution factors into the product of the marginals, $p(x,y) = p(x) \cdot p(y)$.

Conditions: Applies to the case where $X$ and $Y$ are independent.; The displayed notation uses probability mass functions $p(x)$, $p(y)$, and $p(x,y)$.

Mutual information $I(X;Y)$ is related to marginal entropies $H(X)$ and $H(Y)$, and joint entropy $H(X,Y)$ by the identity $I(X;Y) = H(X) + H(Y) - H(X,Y)$. This is derived from the Venn diagram where the union of the two entropy circles is the joint entropy, and their intersection is the mutual information.

Conditions: $X$ and $Y$ are random variables with finite entropies.; The diagram treats $H(X,Y)$ as the union of the two entropy circles and $I(X;Y)$ as their intersection.

Mutual information $I(X;Y)$ is related to marginal entropies $H(X)$ and $H(Y)$, and joint entropy $H(X,Y)$ by the identity $I(X;Y) = H(X) + H(Y) - H(X,Y)$. This is derived from the Venn diagram where the union of the two entropy circles is the joint entropy, and their intersection is the mutual information.

Conditions: $X$ and $Y$ are random variables with finite entropies.; The diagram treats $H(X,Y)$ as the union of the two entropy circles and $I(X;Y)$ as their intersection.