Importance sampling

In statistics, importance sampling is a general technique for estimating properties of a particular distribution, while only having samples generated from a different distribution than the distribution of interest. It is related to umbrella sampling in computational physics. Depending on the application, the term may refer to the process of sampling from this alternative distribution, the process of inference, or both.

Let ${\displaystyle X:\Omega \to \mathbb {R} }$ be a random variable in some probability space ${\displaystyle (\Omega ,{\mathcal {F}},P)}$. We wish to estimate the expected value of X under P, denoted E[X;P]. If we have random samples ${\displaystyle x_{1},\ldots ,x_{n}}$, generated according to P, then an empirical estimate of E[X;P] is

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