In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher-dimensional integrals. There are different methods to perform a Monte Carlo integration, such as uniform sampling, stratified sampling, importance sampling, sequential Monte Carlo (also known as a particle filter), and mean-field particle methods.
Overview In numerical integration, methods such as the trapezoidal rule use a deterministic approach. Monte Carlo integration, on the other hand, employs a non-deterministic approach: each realization provides a different outcome. In Monte Carlo, the final outcome is an approximation of the correct value with respective error bars, and the correct value is likely to be within those error bars. The problem Monte Carlo integration addresses is the computation of a multidimensional definite integral
I = ∫ Ω f ( x ¯ ) d x ¯ {\displaystyle I=\int _{\Omega }f({\overline {\mathbf {x} }})\,d{\overline {\mathbf {x} }}}
where Ω, a subset of R m {\displaystyle \mathbb {R} ^{m}} , has volume
V = ∫ Ω d x ¯ {\displaystyle V=\int _{\Omega }d{\overline {\mathbf {x} }}}
The naive Monte Carlo approach is to sample points uniformly on Ω: given N uniform samples,
x ¯ 1 , ⋯ , x ¯ N ∈ Ω , {\displaystyle {\overline {\mathbf {x} }}_{1},\cdots ,{\overline {\mathbf {x} }}_{N}\in \Omega ,}
I can be approximated by
I ≈ Q N ≡ V 1 N ∑ i = 1 N f ( x ¯ i ) = V ⟨ f ⟩ . {\displaystyle I\approx Q_{N}\equiv V{\frac {1}{N}}\sum _{i=1}^{N}f({\overline {\mathbf {x} }}_{i})=V\langle f\rangle .}
This is because the law of large numbers ensures that
lim N → ∞ Q N = I . {\displaystyle \lim _{N\to \infty }Q_{N}=I.}
Given the estimation of I from QN, the error bars of QN can be estimated by the sample variance using the unbiased estimate of the variance.
V a r ( f ) = E ( σ N 2 ) ≡ 1 N − 1 ∑ i = 1 N E [ ( f ( x ¯ i ) − ⟨ f ⟩ ) 2 ] . {\displaystyle \mathrm {Var} (f)=\mathrm {E} (\sigma _{N}^{2})\equiv {\frac {1}{N-1}}\sum _{i=1}^{N}\mathrm {E} \left[\left(f({\overline {\mathbf {x} }}_{i})-\langle f\rangle \right)^{2}\right].}
which leads to
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