In bandit theory one often encounters KL divergences in upper and lower bounds, for example in (https://arxiv.org/pdf/1302.1611.pdf, theorem 5). There two models are considered:
$\nu = \mathcal{N}(0,1) \otimes \mathcal{N}(-\Delta, 1)$ and $\nu' = \mathcal{N}(-\Delta, 1) \otimes \mathcal{N}(0,1)$.
These are however two product distributions. How does one calculate a KL divergence between two product distributions? Following from the proof of thrm. 5 in the aforementioned paper, the outcome should be $\Delta^2$, which is the same as the KL divergence calculated between one Gaussian $\mathcal{N}(0,1)$ and one other Gaussian $\mathcal{N}(-\Delta, 1)$.
Say that I want to calculate the KL divergence between two product distributions, each containing infinitely many Gaussian distributions, of which a proportion p with distributions $\mathcal{N}(0,1)$ and a proportion 1-p with distributions $\mathcal{N}(-\Delta, 1)$, how would this work?
Thanks!
I found the answer. It is simple the sum of the two KL's: If $\nu = p_1 \otimes p_2$ and $\nu' = q_1 \otimes q_2$, then KL$(\nu, \nu') = $KL$(p_1, q_1) + $KL$(p_2, q_2)$. The answer to my next question would be infinity.