I have seen formulas that either sum or subtract entropy values (eg Information Gain). However, I have not seen a "multiplication/division of entropy values" anywhere. I would like to know if there is any formula/theorem that multiplies/divides entropy values. Additionally, I would like to know if there is any mathematical reasoning/meaning behind such operation. Thanks.
I couldn't think of any situations where I would multiply entropy values, and skimming the beginning of Elements of Information Theory didn't reveal any either. This isn't terribly surprising, as I cannot imagine too many situations where a quantity with units "bits squared" would be useful. (This is actually a pretty common situation--we often add or subtract temperatures, but rarely multiply them by other temperatures).
On the other hand, it is not unheard of to divide entropy values. If H(X) / H(Y) = 2, you could plausibly claim that X carries twice the information of Y. A similar calculation is used when computing the C4.5's information gain ratio, which it uses for building/pruning decision trees.