Attacking the Final Problem
Explore advanced techniques for addressing challenging problems in continuous distributions through improved importance sampling and helper distribution adjustments. Understand how to stretch and shift distributions to enhance expected value estimates in C# programming, enabling better handling of uncertain or complex probability regions.
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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.
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 ...