My background is computer science. I am fairly new to monte carlo sampling methods and, although I understand the math, I have hard time coming up with intuitive examples for importance sampling. More precisely, could someone provide examples of:

  1. an original distribution one cannot sample from but one can estimate
  2. an importance distribution which can be sampled from and adequate for this original distribution.

Suppose you want to simulate the mean of a standard normal distribution that is truncated to the unit interval $[0,1]$.

An inefficient way would be to take draws from $N(0,1)$, but only keep draws in [0,1]. Then you calculate the mean using only the data you kept.

A more efficient way would be to draw from $U(0,1)$, and calculate the importance weights, which you can use in calculating a weighted mean.


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