Say you're trying to maximize a likelihood \(p_{\theta}(x)\), but you only have an unnormalized version \(\hat{p_{\theta}}\) for which \(p_{\theta}(x) = \frac{\hat{p_\theta}(x)}{N_\theta}\). How do you pick \(\theta\)? Well, you can rely on the magic of self normalized importance sampling.

Take a Monte Carlo estimate of the expectation, and you're good to go. Specifically, you can maximize

A special case is when \(q(x)\) is uniform, where this simplifies to \(\log \hat{p_{\theta}}(x) - b\log E_{q(x)}\hat{p_{\theta}}(x)\) for constant \(b\). This is just the negative sampling rule from Mikolov's famous skip-gram paper!

Maximizing Implicit Likelihoods

Okay, that's cool, but what if we don't even have an unnormalized \(\hat{p_{\theta}}(x)\) and \(q(x)\)? What if we just had a simulator \(q(x)\) that can spit out samples, but doesn't know anything about densities?

We'd like to minimize the KL divergence of between whatever density our sampler \(q\) is capturing and the true data distribution \(p\).

Well, that expression looks familiar. Once again, we need to maximize a likelihood ratio! Only this time, we can't evaluate \(q(x)\). Instead, we can notice that \(\log p(x)/q(x)\) is just the log odds of a sample \(x\) coming from \(p\) rather than \(q\). And estimating the log odds of some event occurring is easy: just build a discriminative binary classifier! Specifically, let \(u(x)=p(x)\) if \(y=1\) and \(u(x)=q(x)\) if \(y=0\). Then

So now we have two objectives to minimize: the one for the classifier \(u(y=1 \vert x)\) and the one for the implicit model \(q(x)\). That's just a GAN!