coming from a computer science background and lacking some mathematical knowledge I got stuck trying to find bounds for the following difference:
$u - v = \sum_{i=1}^n k_i*logp_i - \sum_{i=1}^n k_i*logq_i$, where $p_i, q_i$ are probability values such that $\sum_{i=1}^n p_i = \sum_{i=1}^n q_i = 1$, $k$ is a vector of samples with $k_i$ being an indicator of how often label $i$ (which occurs with probability $p_i$ or $q_i$) was drawn. Additionally we know the number of draws $r = \sum_{i=1}^n k_i$.
After doing some initial research it seems highly related to concepts like statistical difference but I can't quite find a way to apply it to my particular problem. Ideally I'd like to be able to show that there exists an a,b such that $ a\lt u - v \lt b$ (or at least one of them).
I'm thankful for any pointer or ideas!