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Yes, mutual information is perfectly symmetric. The formula $I(X;Y) = I(Y;X)$ holds true.

Conditions: Applies to any pair of random variables $X$ and $Y$ for which mutual information is defined.

Yes, mutual information is perfectly symmetric. The formula $I(X;Y) = I(Y;X)$ holds true.

Conditions: Applies to any pair of random variables $X$ and $Y$ for which mutual information is defined.

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.