Attacking the Final Problem

In the previous lesson, we finally wrote a tiny handful of lines of code to implement importance sampling correctly; if we have a distribution p that we’re sampling from, and a function f that we’re running those samples through, we can compute the expected value of f even if there are “black swan” regions in p.

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Helper Distribution q

All we need is a helper distribution q that has the same support as p, but no black swans.

Great. How are we going to find that?

A variation of this problem has also occurred before: what should the initial and proposal distributions be when using Metropolis? If we’re using Metropolis to compute a posterior from a prior, then we can use the prior as the initial distribution. But it’s not at all clear in general how to choose a high-quality proposal distribution; there’s some art there.

There is also some art in choosing appropriate helper distributions when doing importance sampling. Let’s once again take a look at our “black swan” situation: