Problem Statement
I'm currently working with a result about Kullback-Leibler divergence. Let $X$ be an discrete random variable taking values in $\mathcal{X} := \{0,1\}^p$, with $X = (X_1, X_2,...,X_p)$. Suppose that the distribution of X and its components $\{X_i\}_{i=1}^p$ are $g_X$ and $\{g_{X_i}\}_{i=1}^p$ respectively, then show that: $$ KL(g_X||\prod_{i=1}^pg_{X_i}) = \sum_{i=1}^nH(X_i)-H(X) $$
My Proof Attempt
$$ KL(g_X||\prod_{i=1}^pg_{X_i}) = \sum_{x\in\mathcal{X}} g_X(x)\log\left(\frac{g_X(x)}{\prod_{i=1}^pg_{X_i}(x)}\right) $$ $$ = -H(X)-\sum_{x\in\mathcal{X}}g_X(x)\log\left(\prod_{i=1}^pg_{X_i}(x)\right) $$ $$ = -H(X)-\sum_{x\in\mathcal{X}}g_X(x)\sum_{i=1}^p\log(g_{X_i}(x)) $$ $$ = -H(X)-\sum_{i=1}^pg_X(x)\log(g_{X_i}(x)) $$ Now here is the step I have a problem with... Can I say that $\sum_{i=1}^pg_X(x)\log(g_{X_i}(x)) = \sum_{i=1}^pH(X_i)$? This feels like it should require independence, but the notes I'm working through do not make such assumption. Maybe I've done something wrong before this step?
Further Comments
If the result holds, does anybody know a way to upper bound this? In context, the $X_i$ follow a Bernoulli distribution with parameter $p(R_i)$, where the $R_i$ are identically (but not independently) distributed $\mathbb{R}$-valued random variables. In my application the $X_i$ represent the existence of edges in a random geometric graph, and $R_i$ represent distances between randomly distributed points (to explain why they are not independent).