3
$\begingroup$

Suppose through $\ell_1$ minimization I obtained two sparse probability distributions $P, Q$ which may contain many zero terms. Then I would like to compute the KL-Divergence of them $D(P || Q) = \sum_i {p_i\log(\frac{p_i}{q_i})}$. However, since the probability distribution is sparse, it might occur that $p_i \not= 0$ and $q_i = 0$. In that case, KL-Divergence is not well defined. One solution is to incorporate Dirichlet Prior. However, I am afraid by doing so the sparsity of the probability distributions is violated. Is there any other way to compute the KL-Divergence of the two probability distributions?

$\endgroup$
2
$\begingroup$

Have you looked into Jensen-Shannon Divergence? It avoids the situation you described by adding a midpoint. Formally we have, $$\mathrm{JSD}(P||Q) = \frac{1}{2} \mathrm{KL}(P||M) + \frac{1}{2} \mathrm{KL}(Q||M)$$ where $M = \frac{1}{2}(P + Q)$. Informally, JSD is nice when you want to compare distributions that seem to have the same shape, but dont completely overlap.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.