Consider $q(x)$ be a Multinomial distribution over $\{1, \ldots, k\}$ with parameters $\{\theta_1,\ldots, \theta_k\}$. And p(x) over $\{1,\ldots, k\}$ with distribution $p(x)=\frac{1}{k}$. Then what is the KL divergence?
I know that for a discrete distribution, $KL(p(x)||q(x))=\sum p(x)\log\frac{p(x)}{q(x)}$. But as for the KL divergence on multinomial distributions, is the $x$ becomes a vector $\mathbf{x}$? Then what is the log of the ratio? and will the answer of the KL divergence also be a vector? Thanks so much.