Last updated: 2020-04-15

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KL divergence direction

Assume \(p\) is true distribution and \(q\) is the approximated one. The forward KL is \(KL(p||q)\) and reverse KL is \(KL(q||p)\).

What’s the difference between them? The forward KL is a sum of \(log\frac{p}{q}\), weighted by \(p\). We want to choose \(q\) to minimize \(KL(p||q)\), so \(q\) will try to cover everywhere \(p>0\), otherwise \(KL(p||q)\) will be large. On the other hand, the reverse KL is weighted by \(q\), so whenever \(q>0\), \(q\) and \(p\) should be close.

Forward KL is called zero avoiding while reverse KL is called zero forcing. A classicial example is from GAN tutorial page 24 figure 14, by Ian.

Why variational inference uses reverse KL?

Because forward KL is intractable.