forward and reverse KL divergence for variational inference I have a question regarding the forward or reverse KL divergence used in variational inference. In accordance with the following lecture notes, reverse KL can cause q under-estimate the support of p while forward KL can cause the overestimate. I am not clear about the underlying logic.


So you want an approximation $q$ of a distribution $p$, obtained by minimizing the forward or backward KL-divergence. Denote $S_p$ the support of the distribution $p$ and $S_q$ the support of $q$.
If $q$ minimizes a KL-divergence (backward or forward), then necessarily this KL must be finite (since infinity can't be the minimum of any function).

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*Suppose that $KL(q||a)$ is finite. Then, if $x\in S_p$, i.e., $p(x) > 0$, then $q(x) > 0$ (otherwise $KL(q||p)$ would be infinite) such that $x\in S_q$. So $S_p\subset S_q$ : the support of $q$ is equal or larger than the support of $p$.


*Now suppose that $KL(p||q)$ is finite. Then if $x\in S_q$,i.e. $q(x) > 0$, then $p(x) > 0$ (otherwise $KL(p||q)$ is inifinite), such that $x\in S_p$. So this time $S_q \subset S_p$: the support of $q$ is smaller or equal than the support of $p$.
If you want to impose that $q$ correctly estimates the support of $p$, you can minimize $D_KL(p||q) + D_KL(q||p)$.
