The KL is given by: $D_{\mathrm{KL}}(P\|Q) = \int_X p \, \log \frac{p}{q} \, d\mu.$
The PDF of a Lognormal distribution is given by: $P = \frac 1 x \cdot \frac 1 {\sigma\sqrt{2\pi\,}} \exp\left( -\frac{(\ln x-\mu)^2}{2\sigma^2} \right).$
My Question is if it is possible to obtain an analytical solution for the KL with the given PDFs. I've searched the google and found nothing on it, except this post
Where the solution to my question is given as:
$$ D(f_i\|f_j)= \frac1{2\sigma_j^2}\left[(\mu_i-\mu_j)^2+\sigma_i^2-\sigma_j^2\right] + \ln \frac{\sigma_j}{\sigma_i}, $$ But I think this is wrong. I calculated numerically the KL for two Lognormal distributions and it does not match with this solution.
With these parameter:
$p(x)=\text{Lognormal}\left(0,1\right),~~q(x)=\text{Lognormal}\left(5,4\right)$
I get numerically:
$D_{KL_{numerical}} = 1.256$
The expression on the other hand yields:
$D_{KL_{analytical}} = 2.776$
Maybe someone else could provide more insight here because I was unable to derive an analytical solution for the KL of two Lognormal distributions and it was asked at least one time already here.
If someone has some clever tricks for how to derive it, i will try my best and, if successful, post the solution here.
EDIT:
thanks to kjetil b halvorsen who pointed out that in this thread this paper was named to contain the same solution for my problem as it was stated in the post i was mentioning originally.
Still the problem that i can't validate this solution persists. Can someone maybe confirm the given solution or give me a hint how to derive it ?
