paper a day : importance sampling in rare-event simulations

Introduction to importance sampling in rare-event simulations
Mark Denny
European Journal of Physics, 22 (2001) : 403–411

This paper is about importance sampling (IS), which a method to improve the error behavior of Monte Carlo (MC) methods. In engineering systems, getting good simulation results for rare events (such as decoding error) on the order of 10-10 would require an obscene amount of computation if you just did things the naive way. For example, the quality of a numerical bound on the tail probability of a random variable gets worse and worse as you look farther and farther out. Importance sampling is a method of reweighting the distribution to either get a smaller error in the regime of interest and/or uniformize the estimation error. This paper gives some motivation, a simple IS algorithm, analysis, and some simulations. It’s pretty readable, and I went from knowing nothing about importance sampling to getting a decent idea of how to use it in practice, along with its potential problems and benefits.

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a quote that gives some comfort

From Van H. Vu’s “Concentration of Non-Lipschitz Functions and Applications,” Random Structures and Algorithms 20 : 262–316, 2002 :

Finding this hypothesis was actually the hardest part of our work, requiring some insight and a numerous number of attempts. As the reader will see after reading Section 5, a proper induction hypothesis not only allows us to derive a powerful result, but also reduces the proof to a rather routine verification of a few conditions.

I have spent a long time reading through difficult proofs with numerous lemmata, wondering why it had to be so complicated. Some things have to be proved by brute force. For others, just phrasing the problem in the right way can make the proof seem trivial. Some might say “well, I could have done that,” but the more accurate response is “I wish I had thought to do it that way.”