\[ \newcommand{\Exg}{\operatorname{\mathbb{E}}} \newcommand{\Ex}{\mathbb{E}} \newcommand{\Ind}{\mathbb{I}} \newcommand{\Var}{\operatorname{Var}} \newcommand{\Cov}{\operatorname{Cov}} \newcommand{\Corr}{\operatorname{Corr}} \newcommand{\ee}{\mathrm{e}} \]

8  Importance sampling I

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Summary:

• Importance sampling estimates \(\Exg \phi(X)\) by sampling from a different distribution \(Y\).

• The importance sampling estimator is \({\displaystyle \widehat{\theta}_n^{\mathrm{IS}} = \frac{1}{n} \sum_{i=1}^n \frac{f(Y_i)}{g(Y_i)}\,\phi(Y_i)}\).

• The importance sampling estimator is unbiased with mean-square error \[ \operatorname{MSE}\big(\widehat{\theta}_n^{\mathrm{IS}}\big) = \frac{1}{n} \operatorname{Var}\left( \frac{f(Y)}{g(Y)}\,\phi(Y) \right) . \]

Solutions are now available for Problem Sheet 1.

Read more: Voss, An Introduction to Statistical Computing, Subsection 3.3.1.