What is Kullback–Leibler divergence, quasi-maximum likelihood?

The Kullback–Leibler divergence of a density $f$ from the true density $p$ is $D_{\mathrm{KL}}(p\,\|\,f) = \E_p[\ln(p(X)/f(X))] \ge 0$, zero only if $f = p$. Quasi-maximum likelihood maximises a likelihood that is not believed to be the true one.