Monte Carlo integration

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class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/MonteCarloIntegrationCircle.svg/330px-MonteCarloIntegrationCircle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/MonteCarloIntegrationCircle.svg/440px-MonteCarloIntegrationCircle.svg.png 2x" data-file-width="480" data-file-height="466" /></a><figcaption>An illustration of Monte Carlo integration. In this example, the domain <i>D</i> is the inner circle and the domain E is the square. Because the square's area (4) can be easily calculated, the area of the circle (π*1.0<sup>2</sup>) can be estimated by the ratio (0.8) of the points inside the circle (40) to the total number of points (50), yielding an approximation for the circle's area of 4*0.8 = 3.2 ≈ π.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>Monte Carlo integration</b> is a technique for <a href="/wiki/Numerical_quadrature" class="mw-redirect" title="Numerical quadrature">numerical integration</a> using <a href="/wiki/Pseudorandomness" title="Pseudorandomness">random numbers</a>. It is a particular <a href="/wiki/Monte_Carlo_method" title="Monte Carlo method">Monte Carlo method</a> that numerically computes a <a href="/wiki/Definite_integral" class="mw-redirect" title="Definite integral">definite integral</a>. While other algorithms usually evaluate the integrand at a regular grid,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Monte Carlo randomly chooses points at which the integrand is evaluated.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> This method is particularly useful for higher-dimensional integrals.<sup id="cite_ref-newman1999ch2_3-0" class="reference"><a href="#cite_note-newman1999ch2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>There are different methods to perform a Monte Carlo integration, such as <a href="/wiki/Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniform sampling</a>, <a href="/wiki/Stratified_sampling" title="Stratified sampling">stratified sampling</a>, <a href="/wiki/Importance_sampling" title="Importance sampling">importance sampling</a>, <a href="/wiki/Particle_filter" title="Particle filter">sequential Monte Carlo</a> (also known as a particle filter), and <a href="/wiki/Mean-field_particle_methods" title="Mean-field particle methods">mean-field particle methods</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Monte_Carlo_integration&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In numerical integration, methods such as the <a href="/wiki/Trapezoidal_rule" title="Trapezoidal rule">trapezoidal rule</a> use a <a href="/wiki/Deterministic_algorithm" title="Deterministic algorithm">deterministic approach</a>. Monte Carlo integration, on the other hand, employs a <a href="/wiki/Stochastic" title="Stochastic">non-deterministic</a> 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. </p><p>The problem Monte Carlo integration addresses is the computation of a <a href="/wiki/Multiple_integral" title="Multiple integral">multidimensional definite integral</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=\int _{\Omega }f({\overline {\mathbf {x} }})\,d{\overline {\mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=\int _{\Omega }f({\overline {\mathbf {x} }})\,d{\overline {\mathbf {x} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e55a9f3415b94c0ccb94ac2d818014d424f92f07" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.111ex; height:5.676ex;" alt="{\displaystyle I=\int _{\Omega }f({\overline {\mathbf {x} }})\,d{\overline {\mathbf {x} }}}"></span> where Ω, a subset of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a87a024931038d1858dc22e8a194e5978c3412e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.353ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{m}}"></span>, has volume <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\int _{\Omega }d{\overline {\mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\int _{\Omega }d{\overline {\mathbf {x} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12a46c1bf758d1eb67ff118e49db8403246d7e0e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.726ex; height:5.676ex;" alt="{\displaystyle V=\int _{\Omega }d{\overline {\mathbf {x} }}}"></span> </p><p>The naive Monte Carlo approach is to sample points uniformly on Ω:<sup id="cite_ref-newman1999ch1_4-0" class="reference"><a href="#cite_note-newman1999ch1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> given <i>N</i> uniform samples, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathbf {x} }}_{1},\cdots ,{\overline {\mathbf {x} }}_{N}\in \Omega ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathbf {x} }}_{1},\cdots ,{\overline {\mathbf {x} }}_{N}\in \Omega ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4158d757a3dc0f3d40298471e875175a4ec77e2a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.141ex; height:2.676ex;" alt="{\displaystyle {\overline {\mathbf {x} }}_{1},\cdots ,{\overline {\mathbf {x} }}_{N}\in \Omega ,}"></span> </p><p><i>I</i> can be approximated by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\approx Q_{N}\equiv V{\frac {1}{N}}\sum _{i=1}^{N}f({\overline {\mathbf {x} }}_{i})=V\langle f\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>≈</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>≡</mo> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>V</mi> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I\approx Q_{N}\equiv V{\frac {1}{N}}\sum _{i=1}^{N}f({\overline {\mathbf {x} }}_{i})=V\langle f\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5c8b81372aab98e20c35cdf5d6a39ec6aec8105" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.749ex; height:7.343ex;" alt="{\displaystyle I\approx Q_{N}\equiv V{\frac {1}{N}}\sum _{i=1}^{N}f({\overline {\mathbf {x} }}_{i})=V\langle f\rangle .}"></span> </p><p>This is because the <a href="/wiki/Law_of_large_numbers" title="Law of large numbers">law of large numbers</a> ensures that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{N\to \infty }Q_{N}=I.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>=</mo> <mi>I</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{N\to \infty }Q_{N}=I.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24fefa747e85e25ee267a89d8886789cc0fda3f8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.579ex; height:3.843ex;" alt="{\displaystyle \lim _{N\to \infty }Q_{N}=I.}"></span> </p><p>Given the estimation of <i>I</i> from <i>Q<sub>N</sub></i>, the error bars of <i>Q<sub>N</sub></i> can be estimated by the <a href="/wiki/Sample_variance#Population_variance_and_sample_variance" class="mw-redirect" title="Sample variance">sample variance</a> using the <a href="/wiki/Bias_of_an_estimator#Sample_variance" title="Bias of an estimator">unbiased estimate of the variance</a>. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mrow> <mo>[</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\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].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebb48c8d189433cb952c036657cd2dbabd7376ed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:49.949ex; height:7.343ex;" alt="{\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].}"></span> which leads to <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Var} (Q_{N})={\frac {V^{2}}{N^{2}}}\sum _{i=1}^{N}\mathrm {Var} (f)=V^{2}{\frac {\mathrm {Var} (f)}{N}}=V^{2}{\frac {\mathrm {E} (\sigma _{N}^{2})}{N}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mi>N</mi> </mfrac> </mrow> <mo>=</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> <mi>N</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Var} (Q_{N})={\frac {V^{2}}{N^{2}}}\sum _{i=1}^{N}\mathrm {Var} (f)=V^{2}{\frac {\mathrm {Var} (f)}{N}}=V^{2}{\frac {\mathrm {E} (\sigma _{N}^{2})}{N}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e47f6480fbf3ab076e4783c6c8c88c6db2713766" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:55.077ex; height:7.343ex;" alt="{\displaystyle \mathrm {Var} (Q_{N})={\frac {V^{2}}{N^{2}}}\sum _{i=1}^{N}\mathrm {Var} (f)=V^{2}{\frac {\mathrm {Var} (f)}{N}}=V^{2}{\frac {\mathrm {E} (\sigma _{N}^{2})}{N}}.}"></span> </p><p>Since the sequence <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{\mathrm {E} (\sigma _{1}^{2}),\mathrm {E} (\sigma _{2}^{2}),\mathrm {E} (\sigma _{3}^{2}),\ldots \right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\mathrm {E} (\sigma _{1}^{2}),\mathrm {E} (\sigma _{2}^{2}),\mathrm {E} (\sigma _{3}^{2}),\ldots \right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2ff61242e001f3069825b3e9adf46cbb18708b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.866ex; height:3.176ex;" alt="{\displaystyle \left\{\mathrm {E} (\sigma _{1}^{2}),\mathrm {E} (\sigma _{2}^{2}),\mathrm {E} (\sigma _{3}^{2}),\ldots \right\}}"></span> is bounded due to being identically equal to <i>Var(f)</i>, as long as this is assumed finite, this variance decreases asymptotically to zero as 1/<i>N</i>. The estimation of the error of <i>Q<sub>N</sub></i> is thus <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta Q_{N}\approx {\sqrt {\mathrm {Var} (Q_{N})}}=V{\frac {\sqrt {\mathrm {Var} (f)}}{\sqrt {N}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>=</mo> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </msqrt> <msqrt> <mi>N</mi> </msqrt> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta Q_{N}\approx {\sqrt {\mathrm {Var} (Q_{N})}}=V{\frac {\sqrt {\mathrm {Var} (f)}}{\sqrt {N}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9807451f66d67ebad88fd1b0456ffbabe9e8afe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:34.754ex; height:7.176ex;" alt="{\displaystyle \delta Q_{N}\approx {\sqrt {\mathrm {Var} (Q_{N})}}=V{\frac {\sqrt {\mathrm {Var} (f)}}{\sqrt {N}}},}"></span> which decreases as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{\sqrt {N}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msqrt> <mi>N</mi> </msqrt> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\sqrt {N}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4b3147bfcee9c85e5579979c6ffacdb85954f6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.664ex; height:4.176ex;" alt="{\displaystyle {\tfrac {1}{\sqrt {N}}}}"></span>. This is <a href="/wiki/Standard_error_of_the_mean" class="mw-redirect" title="Standard error of the mean">standard error of the mean</a> multiplied with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>. This result does not depend on the number of dimensions of the integral, which is the promised advantage of Monte Carlo integration against most deterministic methods that depend exponentially on the dimension.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> It is important to notice that, unlike in deterministic methods, the estimate of the error is not a strict error bound; random sampling may not uncover all the important features of the integrand that can result in an underestimate of the error. </p><p>While the naive Monte Carlo works for simple examples, an improvement over deterministic algorithms can only be accomplished with algorithms that use problem-specific sampling distributions. With an appropriate sample distribution it is possible to exploit the fact that almost all higher-dimensional integrands are very localized and only small subspace notably contributes to the integral.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> A large part of the Monte Carlo literature is dedicated in developing strategies to improve the error estimates. In particular, stratified sampling—dividing the region in sub-domains—and importance sampling—sampling from non-uniform distributions—are two examples of such techniques. </p> <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Monte_Carlo_integration&action=edit&section=2" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Relative_error_of_a_Monte_Carlo_integration_to_calculate_pi.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/7/72/Relative_error_of_a_Monte_Carlo_integration_to_calculate_pi.svg/350px-Relative_error_of_a_Monte_Carlo_integration_to_calculate_pi.svg.png" decoding="async" width="350" height="270" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/7/72/Relative_error_of_a_Monte_Carlo_integration_to_calculate_pi.svg/525px-Relative_error_of_a_Monte_Carlo_integration_to_calculate_pi.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/7/72/Relative_error_of_a_Monte_Carlo_integration_to_calculate_pi.svg/700px-Relative_error_of_a_Monte_Carlo_integration_to_calculate_pi.svg.png 2x" data-file-width="792" data-file-height="612" /></a><figcaption>Relative error as a function of the number of samples, showing the scaling <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{\sqrt {N}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msqrt> <mi>N</mi> </msqrt> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\sqrt {N}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4b3147bfcee9c85e5579979c6ffacdb85954f6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.664ex; height:4.176ex;" alt="{\displaystyle {\tfrac {1}{\sqrt {N}}}}"></span></figcaption></figure> <p>A paradigmatic example of a Monte Carlo integration is the estimation of π. Consider the function <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\left(x,y\right)={\begin{cases}1&{\text{if }}x^{2}+y^{2}\leq 1\\0&{\text{else}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>else</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\left(x,y\right)={\begin{cases}1&{\text{if }}x^{2}+y^{2}\leq 1\\0&{\text{else}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82972206329e3db684ab691e3d65dac553f18201" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.497ex; height:6.176ex;" alt="{\displaystyle H\left(x,y\right)={\begin{cases}1&{\text{if }}x^{2}+y^{2}\leq 1\\0&{\text{else}}\end{cases}}}"></span> and the set Ω = [−1,1] × [−1,1] with <i>V</i> = 4. Notice that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\pi }=\int _{\Omega }H(x,y)dxdy=\pi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> <mi>d</mi> <mi>y</mi> <mo>=</mo> <mi>π</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\pi }=\int _{\Omega }H(x,y)dxdy=\pi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbf66b8c02cdf0e450d40998e37dcfea0464df4e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.78ex; height:5.676ex;" alt="{\displaystyle I_{\pi }=\int _{\Omega }H(x,y)dxdy=\pi .}"></span> </p><p>Thus, a crude way of calculating the value of π with Monte Carlo integration is to pick <i>N</i> random numbers on Ω and compute <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{N}=4{\frac {1}{N}}\sum _{i=1}^{N}H(x_{i},y_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mi>H</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{N}=4{\frac {1}{N}}\sum _{i=1}^{N}H(x_{i},y_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cb0c69cb720a0eeeea3ad838f544e7a92c85bf7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.795ex; height:7.343ex;" alt="{\displaystyle Q_{N}=4{\frac {1}{N}}\sum _{i=1}^{N}H(x_{i},y_{i})}"></span> </p><p>In the figure on the right, the relative error <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {Q_{N}-\pi }{\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>−</mo> <mi>π</mi> </mrow> <mi>π</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {Q_{N}-\pi }{\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cab3fecbe3df3abfa06741fd07335dad4fe07dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.705ex; height:3.843ex;" alt="{\displaystyle {\tfrac {Q_{N}-\pi }{\pi }}}"></span> is measured as a function of <i>N</i>, confirming the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{\sqrt {N}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msqrt> <mi>N</mi> </msqrt> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\sqrt {N}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4b3147bfcee9c85e5579979c6ffacdb85954f6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.664ex; height:4.176ex;" alt="{\displaystyle {\tfrac {1}{\sqrt {N}}}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="C/C++_example"><span id="C.2FC.2B.2B_example"></span>C/C++ example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Monte_Carlo_integration&action=edit&section=3" title="Edit section: C/C++ example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-highlight mw-highlight-lang-c mw-content-ltr" dir="ltr"><pre><span></span><span class="cp">#include</span><span class="w"> </span><span class="cpf"><stdio.h></span> <span class="cp">#include</span><span class="w"> </span><span class="cpf"><stdlib.h></span> <span class="cp">#include</span><span class="w"> </span><span class="cpf"><time.h></span> <span class="kt">int</span><span class="w"> </span><span class="nf">main</span><span class="p">()</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="c1">// Initialize the number of counts to 0, setting the total number to be 100000 in the loop.</span> <span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">throws</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">99999</span><span class="p">,</span><span class="w"> </span><span class="n">insideCircle</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span> <span class="w"> </span><span class="kt">double</span><span class="w"> </span><span class="n">randX</span><span class="p">,</span><span class="w"> </span><span class="n">randY</span><span class="p">,</span><span class="w"> </span><span class="n">pi</span><span class="p">;</span> <span class="w"> </span><span class="n">srand</span><span class="p">(</span><span class="n">time</span><span class="p">(</span><span class="nb">NULL</span><span class="p">));</span> <span class="w"> </span><span class="c1">// Checks for each random pair of x and y if they are inside circle of radius 1.</span> <span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">throws</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="n">randX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rand</span><span class="p">()</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="p">)</span><span class="w"> </span><span class="n">RAND_MAX</span><span class="p">;</span> <span class="w"> </span><span class="n">randY</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rand</span><span class="p">()</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="p">)</span><span class="w"> </span><span class="n">RAND_MAX</span><span class="p">;</span> <span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">randX</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">randX</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">randY</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">randY</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span> <span class="w"> </span><span class="n">insideCircle</span><span class="o">++</span><span class="p">;</span> <span class="w"> </span><span class="p">}</span> <span class="w"> </span><span class="p">}</span> <span class="w"> </span><span class="c1">// Calculating pi and printing.</span> <span class="w"> </span><span class="n">pi</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">4.0</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">insideCircle</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">throws</span><span class="p">;</span> <span class="w"> </span><span class="n">printf</span><span class="p">(</span><span class="s">"%lf</span><span class="se">\n</span><span class="s">"</span><span class="p">,</span><span class="w"> </span><span class="n">pi</span><span class="p">);</span> <span class="p">}</span> </pre></div> <div class="mw-heading mw-heading3"><h3 id="Python_example">Python example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Monte_Carlo_integration&action=edit&section=4" title="Edit section: Python example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Made in <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a>. </p> <div class="mw-highlight mw-highlight-lang-numpy mw-content-ltr" dir="ltr"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span> <span class="n">rng</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">default_rng</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="n">throws</span> <span class="o">=</span> <span class="mi">2000</span> <span class="n">radius</span> <span class="o">=</span> <span class="mi">1</span> <span class="c1"># Choose random X and Y data centered around 0,0</span> <span class="n">x</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="kp">uniform</span><span class="p">(</span><span class="o">-</span><span class="n">radius</span><span class="p">,</span> <span class="n">radius</span><span class="p">,</span> <span class="n">throws</span><span class="p">)</span> <span class="n">y</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="kp">uniform</span><span class="p">(</span><span class="o">-</span><span class="n">radius</span><span class="p">,</span> <span class="n">radius</span><span class="p">,</span> <span class="n">throws</span><span class="p">)</span> <span class="c1"># Count the times (x, y) is inside the circle,</span> <span class="c1"># which happens when sqrt(x^2 + y^2) <= radius.</span> <span class="n">inside_circle</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">count_nonzero</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="kp">hypot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span> <span class="o"><=</span> <span class="n">radius</span><span class="p">)</span> <span class="c1"># Calculate area and print; should be closer to Pi with increasing number of throws</span> <span class="n">area</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">radius</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">inside_circle</span> <span class="o">/</span> <span class="n">throws</span> <span class="nb">print</span><span class="p">(</span><span class="n">area</span><span class="p">)</span> </pre></div> <div class="mw-heading mw-heading3"><h3 id="Wolfram_Mathematica_example">Wolfram Mathematica example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Monte_Carlo_integration&action=edit&section=5" title="Edit section: Wolfram Mathematica example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The code below describes a process of integrating the function <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {1}{1+\sinh(2x)\log(x)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {1}{1+\sinh(2x)\log(x)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfa17460cd52c704154fb277ba32d418ab516758" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.357ex; height:6.009ex;" alt="{\displaystyle f(x)={\frac {1}{1+\sinh(2x)\log(x)^{2}}}}"></span> from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.8<x<3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.8</mn> <mo><</mo> <mi>x</mi> <mo><</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.8<x<3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40dfb8c47a399538849f5d4dc0d0e20735054bd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.661ex; height:2.176ex;" alt="{\displaystyle 0.8<x<3}"></span> using the Monte-Carlo method in <a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a>: </p> <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre><span></span><span class="n">func</span><span class="p">[</span><span class="nv">x_</span><span class="p">]</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">Sinh</span><span class="p">[</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">Log</span><span class="p">[</span><span class="n">x</span><span class="p">])</span><span class="o">^</span><span class="mi">2</span><span class="p">);</span> <span class="c">(*Sample from truncated normal distribution to speed up convergence*)</span> <span class="n">Distrib</span><span class="p">[</span><span class="nv">x_</span><span class="p">,</span><span class="w"> </span><span class="nv">average_</span><span class="p">,</span><span class="w"> </span><span class="nv">var_</span><span class="p">]</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="n">PDF</span><span class="p">[</span><span class="n">NormalDistribution</span><span class="p">[</span><span class="n">average</span><span class="p">,</span><span class="w"> </span><span class="n">var</span><span class="p">],</span><span class="w"> </span><span class="mf">1.1</span><span class="o">*</span><span class="n">x</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">0.1</span><span class="p">];</span> <span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">10</span><span class="p">;</span> <span class="n">RV</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">RandomVariate</span><span class="p">[</span><span class="n">TruncatedDistribution</span><span class="p">[{</span><span class="mf">0.8</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">},</span><span class="w"> </span><span class="n">NormalDistribution</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mf">0.399</span><span class="p">]],</span><span class="w"> </span><span class="n">n</span><span class="p">];</span> <span class="n">Int</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="o">/</span><span class="n">n</span><span class="w"> </span><span class="n">Total</span><span class="p">[</span><span class="n">func</span><span class="p">[</span><span class="n">RV</span><span class="p">]</span><span class="o">/</span><span class="n">Distrib</span><span class="p">[</span><span class="n">RV</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mf">0.399</span><span class="p">]]</span><span class="o">*</span><span class="n">Integrate</span><span class="p">[</span><span class="n">Distrib</span><span class="p">[</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mf">0.399</span><span class="p">],</span><span class="w"> </span><span class="p">{</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="mf">0.8</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">}]</span> <span class="n">NIntegrate</span><span class="p">[</span><span class="n">func</span><span class="p">[</span><span class="n">x</span><span class="p">],</span><span class="w"> </span><span class="p">{</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="mf">0.8</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">}]</span><span class="w"> </span><span class="c">(*Compare with real answer*)</span> </pre></div> <div class="mw-heading mw-heading2"><h2 id="Recursive_stratified_sampling">Recursive stratified sampling</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Monte_Carlo_integration&action=edit&section=6" title="Edit section: Recursive stratified sampling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Stratified_sampling" title="Stratified sampling">Stratified sampling</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Strata.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/f/f3/Strata.png/220px-Strata.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f3/Strata.png/330px-Strata.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f3/Strata.png/440px-Strata.png 2x" data-file-width="570" data-file-height="570" /></a><figcaption>An illustration of Recursive Stratified Sampling. In this example, the function: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)={\begin{cases}1&x^{2}+y^{2}<1\\0&x^{2}+y^{2}\geq 1\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≥</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)={\begin{cases}1&x^{2}+y^{2}<1\\0&x^{2}+y^{2}\geq 1\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb5457fb9d0a166b735db8e7584d282fb1f53275" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.385ex; height:6.176ex;" alt="{\displaystyle f(x,y)={\begin{cases}1&x^{2}+y^{2}<1\\0&x^{2}+y^{2}\geq 1\end{cases}}}"></span> <br /> from the above illustration was integrated within a unit square using the suggested algorithm. The sampled points were recorded and plotted. Clearly stratified sampling algorithm concentrates the points in the regions where the variation of the function is largest.</figcaption></figure> <p><b>Recursive stratified sampling</b> is a generalization of one-dimensional <a href="/wiki/Adaptive_quadrature" title="Adaptive quadrature">adaptive quadratures</a> to multi-dimensional integrals. On each recursion step the integral and the error are estimated using a plain Monte Carlo algorithm. If the error estimate is larger than the required accuracy the integration volume is divided into sub-volumes and the procedure is recursively applied to sub-volumes. </p><p>The ordinary 'dividing by two' strategy does not work for multi-dimensions as the number of sub-volumes grows far too quickly to keep track. Instead one estimates along which dimension a subdivision should bring the most dividends and only subdivides the volume along this dimension. </p><p>The stratified sampling algorithm concentrates the sampling points in the regions where the variance of the function is largest thus reducing the grand variance and making the sampling more effective, as shown on the illustration. </p><p>The popular MISER routine implements a similar algorithm. </p> <div class="mw-heading mw-heading3"><h3 id="MISER_Monte_Carlo">MISER Monte Carlo</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Monte_Carlo_integration&action=edit&section=7" title="Edit section: MISER Monte Carlo"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The MISER algorithm is based on recursive <a href="/wiki/Stratified_sampling" title="Stratified sampling">stratified sampling</a>. This technique aims to reduce the overall integration error by concentrating integration points in the regions of highest variance.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>The idea of stratified sampling begins with the observation that for two <a href="/wiki/Disjoint_sets" title="Disjoint sets">disjoint</a> regions <i>a</i> and <i>b</i> with Monte Carlo estimates of the integral <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{a}(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{a}(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4760713fbae5dc298a0a1138ae9d16b8b11206e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.905ex; height:2.843ex;" alt="{\displaystyle E_{a}(f)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{b}(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{b}(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f00103fd9330c1decc4e97dc45d0f2bdd53dd26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.741ex; height:2.843ex;" alt="{\displaystyle E_{b}(f)}"></span> and variances <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{a}^{2}(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{a}^{2}(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c0c3a6a55508af5e14272a98a559e8825a9503a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.517ex; height:3.009ex;" alt="{\displaystyle \sigma _{a}^{2}(f)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{b}^{2}(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{b}^{2}(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fe0f8d8671aaeba1d8698ab9b0c6da9d9aa387a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.473ex; height:3.176ex;" alt="{\displaystyle \sigma _{b}^{2}(f)}"></span>, the variance Var(<i>f</i>) of the combined estimate <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(f)={\tfrac {1}{2}}\left(E_{a}(f)+E_{b}(f)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(f)={\tfrac {1}{2}}\left(E_{a}(f)+E_{b}(f)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b76837409e61891de1396eaf80c50c3a7b869b84" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.303ex; height:3.509ex;" alt="{\displaystyle E(f)={\tfrac {1}{2}}\left(E_{a}(f)+E_{b}(f)\right)}"></span> is given by, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Var} (f)={\frac {\sigma _{a}^{2}(f)}{4N_{a}}}+{\frac {\sigma _{b}^{2}(f)}{4N_{b}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>4</mn> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>4</mn> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Var} (f)={\frac {\sigma _{a}^{2}(f)}{4N_{a}}}+{\frac {\sigma _{b}^{2}(f)}{4N_{b}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a65e09b4a2add91312d19f2ac36909c1e44d6a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.506ex; height:6.509ex;" alt="{\displaystyle \mathrm {Var} (f)={\frac {\sigma _{a}^{2}(f)}{4N_{a}}}+{\frac {\sigma _{b}^{2}(f)}{4N_{b}}}}"></span> </p><p>It can be shown that this variance is minimized by distributing the points such that, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {N_{a}}{N_{a}+N_{b}}}={\frac {\sigma _{a}}{\sigma _{a}+\sigma _{b}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {N_{a}}{N_{a}+N_{b}}}={\frac {\sigma _{a}}{\sigma _{a}+\sigma _{b}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e90e5a0e8b88f17d5eced29a10bc4a9b2c4e0ca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.918ex; height:5.676ex;" alt="{\displaystyle {\frac {N_{a}}{N_{a}+N_{b}}}={\frac {\sigma _{a}}{\sigma _{a}+\sigma _{b}}}}"></span> </p><p>Hence the smallest error estimate is obtained by allocating sample points in proportion to the standard deviation of the function in each sub-region. </p><p>The MISER algorithm proceeds by bisecting the integration region along one coordinate axis to give two sub-regions at each step. The direction is chosen by examining all <i>d</i> possible bisections and selecting the one which will minimize the combined variance of the two sub-regions. The variance in the sub-regions is estimated by sampling with a fraction of the total number of points available to the current step. The same procedure is then repeated recursively for each of the two half-spaces from the best bisection. The remaining sample points are allocated to the sub-regions using the formula for <i>N<sub>a</sub></i> and <i>N<sub>b</sub></i>. This recursive allocation of integration points continues down to a user-specified depth where each sub-region is integrated using a plain Monte Carlo estimate. These individual values and their error estimates are then combined upwards to give an overall result and an estimate of its error. </p> <div class="mw-heading mw-heading2"><h2 id="Importance_sampling">Importance sampling</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Monte_Carlo_integration&action=edit&section=8" title="Edit section: Importance sampling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Importance_sampling" title="Importance sampling">Importance sampling</a></div> <p>There are a variety of importance sampling algorithms, such as </p> <div class="mw-heading mw-heading3"><h3 id="Importance_sampling_algorithm">Importance sampling algorithm</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Monte_Carlo_integration&action=edit&section=9" title="Edit section: Importance sampling algorithm"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Importance sampling provides a very important tool to perform Monte-Carlo integration.<sup id="cite_ref-newman1999ch2_3-1" class="reference"><a href="#cite_note-newman1999ch2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-kr11_8-0" class="reference"><a href="#cite_note-kr11-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The main result of importance sampling to this method is that the uniform sampling of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathbf {x} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70488c2676d937484ba86d6c828b68c0b2c4a32d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.526ex; height:2.343ex;" alt="{\displaystyle {\overline {\mathbf {x} }}}"></span> is a particular case of a more generic choice, on which the samples are drawn from any distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p({\overline {\mathbf {x} }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p({\overline {\mathbf {x} }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/739745cddbb437316ba11466d0fdb84f6cc1421b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.594ex; height:2.843ex;" alt="{\displaystyle p({\overline {\mathbf {x} }})}"></span>. The idea is that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p({\overline {\mathbf {x} }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p({\overline {\mathbf {x} }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/739745cddbb437316ba11466d0fdb84f6cc1421b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.594ex; height:2.843ex;" alt="{\displaystyle p({\overline {\mathbf {x} }})}"></span> can be chosen to decrease the variance of the measurement <i>Q<sub>N</sub></i>. </p><p>Consider the following example where one would like to numerically integrate a gaussian function, centered at 0, with σ = 1, from −1000 to 1000. Naturally, if the samples are drawn uniformly on the interval [−1000, 1000], only a very small part of them would be significant to the integral. This can be improved by choosing a different distribution from where the samples are chosen, for instance by sampling according to a gaussian distribution centered at 0, with σ = 1. Of course the "right" choice strongly depends on the integrand. </p><p>Formally, given a set of samples chosen from a distribution <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p({\overline {\mathbf {x} }}):\qquad {\overline {\mathbf {x} }}_{1},\cdots ,{\overline {\mathbf {x} }}_{N}\in V,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">)</mo> <mo>:</mo> <mspace width="2em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>∈</mo> <mi>V</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p({\overline {\mathbf {x} }}):\qquad {\overline {\mathbf {x} }}_{1},\cdots ,{\overline {\mathbf {x} }}_{N}\in V,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0dcd57fa9156c7a83a08a0c2bfbd8bab9f211fe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:27.427ex; height:2.843ex;" alt="{\displaystyle p({\overline {\mathbf {x} }}):\qquad {\overline {\mathbf {x} }}_{1},\cdots ,{\overline {\mathbf {x} }}_{N}\in V,}"></span> the estimator for <i>I</i> is given by<sup id="cite_ref-newman1999ch2_3-2" class="reference"><a href="#cite_note-newman1999ch2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{N}\equiv {\frac {1}{N}}\sum _{i=1}^{N}{\frac {f({\overline {\mathbf {x} }}_{i})}{p({\overline {\mathbf {x} }}_{i})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{N}\equiv {\frac {1}{N}}\sum _{i=1}^{N}{\frac {f({\overline {\mathbf {x} }}_{i})}{p({\overline {\mathbf {x} }}_{i})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bf3d1eca0f8c6a897398f625fdc2bf31b0422e2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.907ex; height:7.343ex;" alt="{\displaystyle Q_{N}\equiv {\frac {1}{N}}\sum _{i=1}^{N}{\frac {f({\overline {\mathbf {x} }}_{i})}{p({\overline {\mathbf {x} }}_{i})}}}"></span> </p><p>Intuitively, this says that if we pick a particular sample twice as much as other samples, we weight it half as much as the other samples. This estimator is naturally valid for uniform sampling, the case where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p({\overline {\mathbf {x} }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p({\overline {\mathbf {x} }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/739745cddbb437316ba11466d0fdb84f6cc1421b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.594ex; height:2.843ex;" alt="{\displaystyle p({\overline {\mathbf {x} }})}"></span> is constant. </p><p>The <a href="/wiki/Metropolis%E2%80%93Hastings_algorithm" title="Metropolis–Hastings algorithm">Metropolis–Hastings algorithm</a> is one of the most used algorithms to generate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathbf {x} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70488c2676d937484ba86d6c828b68c0b2c4a32d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.526ex; height:2.343ex;" alt="{\displaystyle {\overline {\mathbf {x} }}}"></span> from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p({\overline {\mathbf {x} }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p({\overline {\mathbf {x} }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/739745cddbb437316ba11466d0fdb84f6cc1421b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.594ex; height:2.843ex;" alt="{\displaystyle p({\overline {\mathbf {x} }})}"></span>,<sup id="cite_ref-newman1999ch2_3-3" class="reference"><a href="#cite_note-newman1999ch2-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> thus providing an efficient way of computing integrals. </p> <div class="mw-heading mw-heading3"><h3 id="VEGAS_Monte_Carlo">VEGAS Monte Carlo</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Monte_Carlo_integration&action=edit&section=10" title="Edit section: VEGAS Monte Carlo"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/VEGAS_algorithm" title="VEGAS algorithm">VEGAS algorithm</a></div> <p>The VEGAS algorithm approximates the exact distribution by making a number of passes over the integration region which creates the histogram of the function <i>f</i>. Each histogram is used to define a sampling distribution for the next pass. Asymptotically this procedure converges to the desired distribution.<sup id="cite_ref-Lepage,_1978_9-0" class="reference"><a href="#cite_note-Lepage,_1978-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> In order to avoid the number of histogram bins growing like <i>K<sup>d</sup></i>, the probability distribution is approximated by a separable function: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x_{1},x_{2},\ldots )=g_{1}(x_{1})g_{2}(x_{2})\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x_{1},x_{2},\ldots )=g_{1}(x_{1})g_{2}(x_{2})\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80efe7b24ff76634082c08b46af4759804e2ebe9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.406ex; height:2.843ex;" alt="{\displaystyle g(x_{1},x_{2},\ldots )=g_{1}(x_{1})g_{2}(x_{2})\ldots }"></span> so that the number of bins required is only <i>Kd</i>. This is equivalent to locating the peaks of the function from the projections of the integrand onto the coordinate axes. The efficiency of VEGAS depends on the validity of this assumption. It is most efficient when the peaks of the integrand are well-localized. If an integrand can be rewritten in a form which is approximately separable this will increase the efficiency of integration with VEGAS. VEGAS incorporates a number of additional features, and combines both stratified sampling and importance sampling.<sup id="cite_ref-Lepage,_1978_9-1" class="reference"><a href="#cite_note-Lepage,_1978-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Monte_Carlo_integration&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Quasi-Monte_Carlo_method" title="Quasi-Monte Carlo method">Quasi-Monte Carlo method</a></li> <li><a href="/wiki/Auxiliary_field_Monte_Carlo" class="mw-redirect" title="Auxiliary field Monte Carlo">Auxiliary field Monte Carlo</a></li> <li><a href="/wiki/Monte_Carlo_method_in_statistical_physics" class="mw-redirect" title="Monte Carlo method in statistical physics">Monte Carlo method in statistical physics</a></li> <li><a href="/wiki/Monte_Carlo_method" title="Monte Carlo method">Monte Carlo method</a></li> <li><a href="/wiki/Variance_reduction" title="Variance reduction">Variance reduction</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Monte_Carlo_integration&action=edit&section=12" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFPressTeukolskyVetterlingFlannery2007">Press et al. 2007</a>, Chap. 4</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFPressTeukolskyVetterlingFlannery2007">Press et al. 2007</a>, Chap. 7</span> </li> <li id="cite_note-newman1999ch2-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-newman1999ch2_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-newman1999ch2_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-newman1999ch2_3-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-newman1999ch2_3-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFNewmanBarkema1999">Newman & Barkema 1999</a>, Chap. 2</span> </li> <li id="cite_note-newman1999ch1-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-newman1999ch1_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFNewmanBarkema1999">Newman & Barkema 1999</a>, Chap. 1</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="#CITEREFPressTeukolskyVetterlingFlannery2007">Press et al. 2007</a></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFMacKay2003">MacKay 2003</a>, pp. 284–292</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFPressFarrar1990">Press & Farrar 1990</a>, pp. 190–195</span> </li> <li id="cite_note-kr11-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-kr11_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKroeseTaimreBotev2011">Kroese, Taimre & Botev 2011</a></span> </li> <li id="cite_note-Lepage,_1978-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-Lepage,_1978_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Lepage,_1978_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFLepage1978">Lepage 1978</a></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Monte_Carlo_integration&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFCaflisch1998" class="citation journal cs1"><a href="/wiki/Russel_E._Caflisch" title="Russel E. 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"Monte Carlo and quasi-Monte Carlo methods". <i><a href="/wiki/Acta_Numerica" title="Acta Numerica">Acta Numerica</a></i>. <b>7</b>: <span class="nowrap">1–</span>49. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998AcNum...7....1C">1998AcNum...7....1C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0962492900002804">10.1017/S0962492900002804</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:5708790">5708790</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Numerica&rft.atitle=Monte+Carlo+and+quasi-Monte+Carlo+methods&rft.volume=7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E49&rft.date=1998&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A5708790%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS0962492900002804&rft_id=info%3Abibcode%2F1998AcNum...7....1C&rft.aulast=Caflisch&rft.aufirst=R.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMonte+Carlo+integration" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeinzierl2000" class="citation arxiv cs1">Weinzierl, S. (2000). "Introduction to Monte Carlo methods". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-ph/0006269">hep-ph/0006269</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Introduction+to+Monte+Carlo+methods&rft.date=2000&rft_id=info%3Aarxiv%2Fhep-ph%2F0006269&rft.aulast=Weinzierl&rft.aufirst=S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMonte+Carlo+integration" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPressFarrar1990" class="citation journal cs1">Press, W. H.; Farrar, G. R. (1990). <a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.4822899">"Recursive Stratified Sampling for Multidimensional Monte Carlo Integration"</a>. <i>Computers in Physics</i>. <b>4</b> (2): 190. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1990ComPh...4..190P">1990ComPh...4..190P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.4822899">10.1063/1.4822899</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computers+in+Physics&rft.atitle=Recursive+Stratified+Sampling+for+Multidimensional+Monte+Carlo+Integration&rft.volume=4&rft.issue=2&rft.pages=190&rft.date=1990&rft_id=info%3Adoi%2F10.1063%2F1.4822899&rft_id=info%3Abibcode%2F1990ComPh...4..190P&rft.aulast=Press&rft.aufirst=W.+H.&rft.au=Farrar%2C+G.+R.&rft_id=https%3A%2F%2Fdoi.org%2F10.1063%252F1.4822899&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMonte+Carlo+integration" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLepage1978" class="citation journal cs1">Lepage, G. 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"A New Algorithm for Adaptive Multidimensional Integration". <i><a href="/wiki/Journal_of_Computational_Physics" title="Journal of Computational Physics">Journal of Computational Physics</a></i>. <b>27</b> (2): <span class="nowrap">192–</span>203. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1978JCoPh..27..192L">1978JCoPh..27..192L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0021-9991%2878%2990004-9">10.1016/0021-9991(78)90004-9</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Computational+Physics&rft.atitle=A+New+Algorithm+for+Adaptive+Multidimensional+Integration&rft.volume=27&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E192-%3C%2Fspan%3E203&rft.date=1978&rft_id=info%3Adoi%2F10.1016%2F0021-9991%2878%2990004-9&rft_id=info%3Abibcode%2F1978JCoPh..27..192L&rft.aulast=Lepage&rft.aufirst=G.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMonte+Carlo+integration" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLepage1980" class="citation journal cs1">Lepage, G. P. (1980). "VEGAS: An Adaptive Multi-dimensional Integration Program". <i>Cornell Preprint CLNS 80-447</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Cornell+Preprint+CLNS+80-447&rft.atitle=VEGAS%3A+An+Adaptive+Multi-dimensional+Integration+Program&rft.date=1980&rft.aulast=Lepage&rft.aufirst=G.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMonte+Carlo+integration" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHammersleyHandscomb1964" class="citation book cs1">Hammersley, J. M.; Handscomb, D. C. (1964). <i>Monte Carlo Methods</i>. Methuen. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-416-52340-9" title="Special:BookSources/978-0-416-52340-9"><bdi>978-0-416-52340-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Monte+Carlo+Methods&rft.pub=Methuen&rft.date=1964&rft.isbn=978-0-416-52340-9&rft.aulast=Hammersley&rft.aufirst=J.+M.&rft.au=Handscomb%2C+D.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMonte+Carlo+integration" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKroeseTaimreBotev2011" class="citation book cs1"><a href="/wiki/Dirk_Kroese" title="Dirk Kroese">Kroese, D. P.</a>; Taimre, T.; Botev, Z. I. (2011). <i>Handbook of Monte Carlo Methods</i>. John Wiley & Sons.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Monte+Carlo+Methods&rft.pub=John+Wiley+%26+Sons&rft.date=2011&rft.aulast=Kroese&rft.aufirst=D.+P.&rft.au=Taimre%2C+T.&rft.au=Botev%2C+Z.+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMonte+Carlo+integration" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPressTeukolskyVetterlingFlannery2007" class="citation book cs1">Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). <i>Numerical Recipes: The Art of Scientific Computing</i> (3rd ed.). New York: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-88068-8" title="Special:BookSources/978-0-521-88068-8"><bdi>978-0-521-88068-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Recipes%3A+The+Art+of+Scientific+Computing&rft.place=New+York&rft.edition=3rd&rft.pub=Cambridge+University+Press&rft.date=2007&rft.isbn=978-0-521-88068-8&rft.aulast=Press&rft.aufirst=WH&rft.au=Teukolsky%2C+SA&rft.au=Vetterling%2C+WT&rft.au=Flannery%2C+BP&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMonte+Carlo+integration" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMacKay2003" class="citation book cs1"><a href="/wiki/David_MacKay_(scientist)" class="mw-redirect" title="David MacKay (scientist)">MacKay, David</a> (2003). <a rel="nofollow" class="external text" href="http://www.inference.org.uk/itprnn/book.pdf">"chapter 4.4 Typicality & chapter 29.1"</a> <span class="cs1-format">(PDF)</span>. <a rel="nofollow" class="external text" href="http://www.inference.phy.cam.ac.uk/mackay/itila/book.html"><i>Information Theory, Inference and Learning Algorithms</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-64298-9" title="Special:BookSources/978-0-521-64298-9"><bdi>978-0-521-64298-9</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2012999">2012999</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=chapter+4.4+Typicality+%26+chapter+29.1&rft.btitle=Information+Theory%2C+Inference+and+Learning+Algorithms&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=978-0-521-64298-9&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2012999%23id-name%3DMR&rft.aulast=MacKay&rft.aufirst=David&rft_id=http%3A%2F%2Fwww.inference.org.uk%2Fitprnn%2Fbook.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMonte+Carlo+integration" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNewmanBarkema1999" class="citation book cs1">Newman, MEJ; Barkema, GT (1999). <i>Monte Carlo Methods in Statistical Physics</i>. Clarendon Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Monte+Carlo+Methods+in+Statistical+Physics&rft.pub=Clarendon+Press&rft.date=1999&rft.aulast=Newman&rft.aufirst=MEJ&rft.au=Barkema%2C+GT&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMonte+Carlo+integration" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobertCasella2004" class="citation book cs1">Robert, CP; Casella, G (2004). <i>Monte Carlo Statistical Methods</i> (2nd ed.). Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4419-1939-7" title="Special:BookSources/978-1-4419-1939-7"><bdi>978-1-4419-1939-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Monte+Carlo+Statistical+Methods&rft.edition=2nd&rft.pub=Springer&rft.date=2004&rft.isbn=978-1-4419-1939-7&rft.aulast=Robert&rft.aufirst=CP&rft.au=Casella%2C+G&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMonte+Carlo+integration" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Monte_Carlo_integration&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20140202110318/http://www.cafemath.fr/mathblog/article.php?page=MonteCarlo.php">Café math : Monte Carlo Integration</a> : A blog article describing Monte Carlo integration (principle, hypothesis, confidence interval)</li> <li><a rel="nofollow" class="external text" href="http://www.boost.org/doc/libs/release/libs/math/doc/html/math_toolkit/naive_monte_carlo.html">Boost.Math : Naive Monte Carlo integration: Documentation for the C++ naive Monte-Carlo routines</a></li> <li><a rel="nofollow" class="external text" href="https://sites.google.com/view/chremos-group/applets/monte-carlo:">Monte Carlo applet applied in statistical physics problems</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Monte_Carlo_integration&oldid=1275083251">https://en.wikipedia.org/w/index.php?title=Monte_Carlo_integration&oldid=1275083251</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Monte_Carlo_methods" title="Category:Monte Carlo methods">Monte Carlo methods</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Articles_with_example_code" title="Category:Articles with example code">Articles with example code</a></li><li><a href="/wiki/Category:Articles_with_example_C_code" title="Category:Articles with example C code">Articles with example C code</a></li><li><a href="/wiki/Category:Articles_with_example_Python_(programming_language)_code" title="Category:Articles with example Python (programming language) code">Articles with example Python (programming language) code</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 10 February 2025, at 23:36<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Monte Carlo integration - Wikipedia