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Long story short, importance sampling is a method for approximating integrals using samples from a probability distribution. It is widely used in statistics, since many quantities of interest can be expressed as integrals. The idea behind importance sampling is to first express the integral in terms of some probability measure, then change the measure to something easier to work with. The token example of one such quantity is the expectation of a random quantity $f(X)$ with respect to some proba...