Does the following identity $$ I(X_1,X_2 ; Y_1, Y_2) = I(X_1; Y_1) + I(X_2; Y_2) $$ hold for mutual information for $(X_1, Y_1)$ and $(X_2, Y_2)$ independent?
Attempt: $p(x_1, x_2, y_1, y_2) = p(x_1, y_1) \cdot p(x_2, y_2)$ implies $p(x_1, x_2) = p(x_1) \cdot p(x_2)$ and $p(y_1, y_2) = p(y_1) \cdot p(y_2)$, where I use lower case $p$ to denote the probability of specific outcomes, e.g. $p(x)$ for $\Pr\{X=x\}$. Thus $$ \sum_{x_1, x_2, y_1, y_2} p(x_1, x_2, y_1, y_2) \cdot \log \frac{p(x_1, x_2, y_1, y_2)}{p(x_1, x_2) \cdot p(y_1, y_2)} = \\ \sum_{x_1, x_2, y_1, y_2} p(x_1, x_2, y_1, y_2) \cdot \log \frac{p(x_1, y_1)}{p(x_1) \cdot p(y_1)} \cdot \frac{p(x_2, y_2)}{p(x_2) \cdot p(y_2)} $$
