KL-divergence: P||Q vs. Q||P

Question

Assume, that we have several data generating measures $P_{1}, \dots, P_{k}$ and $Q$, all defined on the same probability space. Next, assume, we have the same amount of independently sampled data from $P_{1}, \dots, P_{k}$ and some data from $Q$ and we aim to find which distribution $P_{1}, \dots, P_{k}$ is the closest to $Q$ is a sense of KL-divergence.

KL-divergence, $D_{KL}(P_{i}||Q) = \int_{-\infty}^{\infty}p(x)\log\left(\frac{p(x)}{q(x)}\right)dx \neq D_{KL}(Q||P_{i})$, is not symmetric.

Therefore, if we compare $Q$ to all $P_{i}$, which one $D_{KL}(P_{i}||Q)$ or $D_{KL}(Q||P_{i})$, for $i = 1, \dots, k$ is correct to consider as the criterion?

From what I know, in AIK criterion one goes for $D_{KL}(Q||P_{i})$ case.

UPDATE:

My confusion is partly from the following fact that KL is a premetric, it generates a topology on the space of probability distributions. Let us consider the sequence of measures $U_{1}, \dots, U_{n}$. Then if $$ \lim_{i\to\infty}D_{KL}(U_{i}||Q) = 0 $$ then $$ U_{n} \xrightarrow{d} Q. $$

Answer 1

In $$\DeclareMathOperator{\E}{\mathbb{E}} D_{KL}(P || Q) = \int_{-\infty}^{\infty}p(x)\log\left(\frac{p(x)}{q(x)}\right)\;dx = \E_{P}\log\left(\frac{p(X)}{q(X)}\right) $$ we see this is the expectation of the loglikelihood ratio when $P$ is the truth, see Intuition on the Kullback-Leibler (KL) Divergence.

If, in hypothesis test language, $P$ is the null while $Q$ is the alternative: So $D_{KL}(P || Q)$ is divergence of $Q$ from (null) truth, while $D_{KL}(Q || P)$ is divergence when the alternative hypothesis is taken as truth. Then your question:

which distribution $P_1,\dotsc,P_k$ is the closest to $Q$ is a sense of KL-divergence?

If this means you want a model which is difficult to distinguish from $Q$ when/if $Q$ is the truth, you needs $D_{KL}(Q || P)$. Remember, the first argument is the truth (This is a way of saying that we calculate the divergence calculating an expectation assuming that the distribution generating $X$ is the distribution given in the first argument. That is, the truth about what is generating $X$.)