16.1 The Monte Carlo method

To estimate \(\theta = \mathbb E[g(X)]\), draw independent copies \(X_1, \ldots , X_n\) and form \(\hat \theta _n = \frac 1n\sum g(X_i)\). By the law of large numbers \(\hat \theta _n \to \theta \), and by the central limit theorem \(\hat \theta _n\) is approximately normal with standard deviation \(\sigma /\sqrt n\), where \(\sigma ^2 = \operatorname {Var}g(X)\) is estimated by the sample variance \(s^2\). So \[ \hat \theta _n \pm 1.96\,\frac s{\sqrt n} \] is an approximate \(95\%\) confidence interval. The error decreases like \(1/\sqrt n\), whatever the dimension of the problem: a hundred times more work buys one more correct digit. Integrals are expectations too: \(\int _0^1e^{-v^2}\,dv = \mathbb E[e^{-U^2}]\) for \(U\) uniform, estimated from \(100\,000\) uniforms as \(0.74684 \pm 0.00125\) against the exact \(0.74682\) (Software 16.5.1).

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