Say we have two variables $X_{1}X\_1$ and $X_{2}X\_2$ that we are using as predictors for a target variable $YY$. We know the total information that $X_{1}X\_1$ and $X_{2}X\_2$ can give about $YY$ is $I(X_1,X_2\mathrm{\mid }Y)I(X\_1,\; X\_2|\; Y)$ but we would also like to know how $X_{1}X\_1$ and $X_{2}X\_2$ contribute to the total information.

Why? Say we're trying to rob a bank. Our espionage efforts have found that the bank has multiple vaults each with a number $YY$. Opening a vault requires two bank officials to insert two separate codes, $X_{1}X\_1$ and $X_{2}X\_2$. Our social engineer found out that the two codes

$X_{1}X\_1$ represents test questions, $X_{2}X\_2$ images corresponding to the questions and $YY$ represents answers.

[Image of heart with an arrow pointing to a part of the heart. Question: Which part of the heart is this?]

Williams and Beer came up with the idea of