Kullback-leibler divergence

tags
Applied maths

Definition

The KL divergence is not symmetric. For \(P, Q\) defined on the same probability space \(\mathcal{X}\), KL of \(Q\) from \(P\) is \[ KL(P, Q) = \sum_{x \in \mathcal{X}} P(x) \log\left( \frac{P(x)}{Q(x)} \right) \]

It has two main interpretations:

  • It is the information gain from using the right probability distribution \(P\) instead of \(Q\) or the amount of information lost by approximating \(P\) with \(Q\).
  • The average difference in code length for a sequence following \(P\) and using a code optimized for \(Q\) to encode it.

Forward and reverse KL

Eric Jang’s blog has an interesting visual explanation of the difference between forward and backward KL.

{{< img src="/img/forwardvsreversedKL.jpg" title=“Difference between forward and backward KL” width=800 alt=“Difference between forward and backward KL” class=“center” >}}

So in summary, minimizing forward-KL “stretches” your variational distribution \(Q(Z)\) to cover over the entire \(P(Z)\) like a tarp, while minimizing reverse-KL “squeezes” the \(Q(Z)\) under \(P(Z)\).

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