# Proof help for multivariate mutual information as a sum of entropies

I'm following this paper on ICA and I got to equation (1) describing the multivariate mutual information contrast function as a sum of entropies.

$$J(Y) = \int p(y_1,...y_D)log(\frac{p(y_1,...y_D)}{p(y_1)p(y_2)...p(y_D))})du$$

where $$du = dy_1dy_2...dy_D$$

$$= KL(p(y_1,...y_D)\parallel\displaystyle\prod_{i=1}^{D}p(y_i))$$

$$= \displaystyle\sum_{i=1}^{D}H(Y_i) - H(Y_1,...Y_D)$$

This seems to be in line with this post referencing a metric called "total correlation". I'm curious as to where the first summation term comes from.

From the first line, I can see how $$J(Y)$$ could be written as

$$J(Y) = -\int p(y_1,...y_D)log(p(y_1)p(y_2)...p(y_D))du + \int p(y_1,...y_D)log(p(y_1,...y_D))du$$

so the second term appears to be joint entropy

$$-H(Y_1,...Y_D) = \int p(y_1,...y_D)log(p(y_1,...y_D))du$$

The first term is what I'd like clarification on.

Does

$$\displaystyle\sum_{i=1}^{D}H(Y_i) = -\int p(y_1,...y_D)log(p(y_1)p(y_2)...p(y_D))du$$

because

$$-\displaystyle\int p(y_1,...y_D)log(p(y_1)p(y_2)...p(y_D))du = \displaystyle\int_{y_1 \in Y_1}...\displaystyle\int_{y_D \in Y_D} p(y_1,...y_D)[log(p(y_1)) + ... + log(p(y_D))]dy_1...dy_D$$

and for each term every other $$y$$ is marginalized out? i.e. :

$$\displaystyle\int_{y_1 \in Y_1}...\displaystyle\int_{y_D \in Y_D} p(y_1,...y_D)[log(p(y_n))]dy_1...dy_D = \displaystyle\int_{y_n \in Y_n} p(y_n)log(p(y_n))dy_n = H(Y_n)$$

$$where, n \in 1...D$$