Suppose $p$ and $q$ are discrete distributions over $k$ outcomes, $q$ is given, for which $p$ does the following hold?
\begin{equation*} p_1 \log q_1 + \ldots + p_k \log q_k \ge q_1 \log q_1 + \ldots + q_k \log q_k\end{equation*}
Suppose $p$ and $q$ are discrete distributions over $k$ outcomes, $q$ is given, for which $p$ does the following hold?
\begin{equation*} p_1 \log q_1 + \ldots + p_k \log q_k \ge q_1 \log q_1 + \ldots + q_k \log q_k\end{equation*}
p such that:
$\prod_i q_i^{p_i-q_i} \ge 1$
I arrive at this point with the following calculations:
First of all I rewrite the equation:
$\sum_i p_i \log q_i \ge \sum_i q_i \log q_i$
And
$\sum_i (p_i \log q_i - q_i \log q_i) = \sum_i (\log q_i^{p_i-q_i}) = \log \prod_i q_i^{p_i-q_i} \ge 0$
The most simple deduction about p is that
$p_i \ge q_i , \forall i$
with
$q_i \ge 1$