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{{distinguish|Monte Carlo algorithm}}

{{Short description|Probabilistic problem-solving algorithm}}

'''Monte Carlo methods''', or '''Monte Carlo experiments''', are a broad class of [[computation]]al [[algorithm]]s that rely on repeated [[random sampling]] to obtain numerical results. The underlying concept is to use [[randomness]] to solve problems that might be [[deterministic system|deterministic]] in principle. They are often used in [[physics|physical]] and [[mathematics|mathematical]] problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes:<ref>{{cite journal|last1 = Kroese|first1 = D. P.|last2=Brereton|first2=T.|last3 = Taimre|first3 = T.|last4 = Botev|first4 = Z. I. |year =2014 |title=Why the Monte Carlo method is so important today |journal = WIREs Comput Stat|volume=6|issue = 6|pages = 386–392 |doi=10.1002/wics.1314|url = https://semanticscholar.org/paper/7a56b632de84d0b81f283750b11609a042890639}}</ref> [[optimization]], [[numerical integration]], and generating draws from a [[probability distribution]].

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution.

蒙特卡罗方法，或称蒙特卡罗方法方法，是一类广泛的计算算法，它依赖于重复的随机抽样来获得数值结果。其基本概念是利用随机性来解决原则上可能是确定性的问题。它们通常用于物理和数学问题，当很难或不可能使用其他方法时，它们最有用。蒙特卡罗方法主要用于3个问题类: 优化，数值积分，从概率分布生成绘图。

In physics-related problems, Monte Carlo methods are useful for simulating systems with many [[coupling (physics)|coupled]] [[degrees of freedom]], such as fluids, disordered materials, strongly coupled solids, and cellular structures (see [[cellular Potts model]], [[interacting particle systems]], [[McKean–Vlasov process]]es, [[kinetic theory of gases|kinetic models of gases]]). Other examples include modeling phenomena with significant [[uncertainty]] in inputs such as the calculation of [[risk]] in business and, in mathematics, evaluation of multidimensional [[Integral|definite integral]]s with complicated [[boundary conditions]]. In application to systems engineering problems (space, [[oil exploration]], aircraft design, etc.), Monte Carlo–based predictions of failure, [[cost overrun]]s and schedule overruns are routinely better than human intuition or alternative "soft" methods.<ref>{{cite journal|last1 = Hubbard|first1 = Douglas|last2=Samuelson|first2 = Douglas A. |date = October 2009 |title=Modeling Without Measurements |url=http://viewer.zmags.com/publication/357348e6#/357348e6/28|journal = OR/MS Today|pages = 28–33}}</ref>

In physics-related problems, Monte Carlo methods are useful for simulating systems with many coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures (see cellular Potts model, interacting particle systems, McKean–Vlasov processes, kinetic models of gases). Other examples include modeling phenomena with significant uncertainty in inputs such as the calculation of risk in business and, in mathematics, evaluation of multidimensional definite integrals with complicated boundary conditions. In application to systems engineering problems (space, oil exploration, aircraft design, etc.), Monte Carlo–based predictions of failure, cost overruns and schedule overruns are routinely better than human intuition or alternative "soft" methods.

在与物理学相关的问题中，蒙特卡罗方法可用于模拟具有多个耦合自由度的系统，如流体、无序材料、强耦合固体和细胞结构(参见细胞波茨模型、相互作用粒子系统、 McKean-Vlasov 过程、气体动力学模型)。其他例子包括在输入中存在明显不确定性的建模现象，例如商业中的风险计算，以及在数学中对具有复杂边界条件的多维定积分的评价。应用于系统工程问题(空间、石油勘探、飞机设计等)。)、蒙特卡洛——基于失败、成本超支和进度超支的预测通常比人类直觉或其他“软”方法更好。

In principle, Monte Carlo methods can be used to solve any problem having a probabilistic interpretation. By the [[law of large numbers]], integrals described by the [[expected value]] of some random variable can be approximated by taking the [[Sample mean and sample covariance|empirical mean]] (a.k.a. the sample mean) of independent samples of the variable. When the [[probability distribution]] of the variable is parametrized, mathematicians often use a [[Markov chain Monte Carlo]] (MCMC) sampler.<ref>{{Cite journal|title = Equation of State Calculations by Fast Computing Machines|journal = The Journal of Chemical Physics|date = 1953-06-01|issn = 0021-9606|pages = 1087–1092|volume = 21|issue = 6|doi = 10.1063/1.1699114|first = Nicholas|last = Metropolis|first2 = Arianna W.|last2 = Rosenbluth|first3 = Marshall N.|last3 = Rosenbluth|first4 = Augusta H.|last4 = Teller|first5 = Edward|last5 = Teller|bibcode=1953JChPh..21.1087M|url = https://semanticscholar.org/paper/f6a13f116e270dde9d67848495f801cdb8efa25d}}</ref><ref>{{Cite journal|title = Monte Carlo sampling methods using Markov chains and their applications|journal = Biometrika|date = 1970-04-01|issn = 0006-3444|pages = 97–109|volume = 57|issue = 1|doi = 10.1093/biomet/57.1.97|first = W. K.|last = Hastings|bibcode = 1970Bimka..57...97H|url = https://semanticscholar.org/paper/143d2e02ab91ae6259576ac50b664b8647af8988}}</ref><ref>{{Cite journal|title = The Multiple-Try Method and Local Optimization in Metropolis Sampling|journal = Journal of the American Statistical Association|date = 2000-03-01|issn = 0162-1459|pages = 121–134|volume = 95|issue = 449|doi = 10.1080/01621459.2000.10473908|first = Jun S.|last = Liu|first2 = Faming|last2 = Liang|first3 = Wing Hung|last3 = Wong|url = https://semanticscholar.org/paper/ff17c129a8d32bb7dc7206230da612e94bd24b9f}}</ref><ref>{{Cite journal|title = On the flexibility of the design of multiple try Metropolis schemes|journal = Computational Statistics|date = 2013-07-11|issn = 0943-4062|pages = 2797–2823|volume = 28|issue = 6|doi = 10.1007/s00180-013-0429-2|first = Luca|last = Martino|first2 = Jesse|last2 = Read|arxiv = 1201.0646}}</ref> The central idea is to design a judicious [[Markov chain]] model with a prescribed [[stationary probability distribution]]. That is, in the limit, the samples being generated by the MCMC method will be samples from the desired (target) distribution.<ref>{{cite journal | last1 = Spall | first1 = J. C. | year = 2003 | title = Estimation via Markov Chain Monte Carlo | doi = 10.1109/MCS.2003.1188770 | journal = IEEE Control Systems Magazine | volume = 23 | issue = 2| pages = 34–45 }}</ref><ref>{{Cite journal |doi = 10.1109/MCS.2018.2876959|title = Stationarity and Convergence of the Metropolis-Hastings Algorithm: Insights into Theoretical Aspects|journal = IEEE Control Systems Magazine |volume = 39|pages = 56–67|year = 2019|last1 = Hill|first1 = Stacy D.|last2 = Spall|first2 = James C.}}</ref> By the [[ergodic theorem]], the stationary distribution is approximated by the [[empirical measure]]s of the random states of the MCMC sampler.

In principle, Monte Carlo methods can be used to solve any problem having a probabilistic interpretation. By the law of large numbers, integrals described by the expected value of some random variable can be approximated by taking the empirical mean (a.k.a. the sample mean) of independent samples of the variable. When the probability distribution of the variable is parametrized, mathematicians often use a Markov chain Monte Carlo (MCMC) sampler. The central idea is to design a judicious Markov chain model with a prescribed stationary probability distribution. That is, in the limit, the samples being generated by the MCMC method will be samples from the desired (target) distribution. By the ergodic theorem, the stationary distribution is approximated by the empirical measures of the random states of the MCMC sampler.

原则上，蒙特卡罗方法可以用来解决任何具有概率解释的问题。利用大数定律，用随机变量的期望值所描述的积分可以用经验均值来近似。样本平均值)的独立样本的变量。当变量的概率分布被参数化时，数学家们经常使用马尔科夫蒙特卡洛取样器。其核心思想是设计一个具有规定平稳概率分布的明智的马尔可夫链模型。也就是说，在极限情况下，由 MCMC 方法生成的样本将是来自期望(目标)分布的样本。利用遍历定理，用 MCMC 采样器随机状态的经验测度近似平稳分布。

In other problems, the objective is generating draws from a sequence of probability distributions satisfying a nonlinear evolution equation. These flows of probability distributions can always be interpreted as the distributions of the random states of a [[Markov process]] whose transition probabilities depend on the distributions of the current random states (see [[McKean–Vlasov process]]es, [[particle filter|nonlinear filtering equation]]).<ref name="kol10">{{cite book|last = Kolokoltsov|first = Vassili|title = Nonlinear Markov processes|year = 2010|publisher = Cambridge Univ. Press|pages = 375}}</ref><ref name="dp13">{{cite book|last = Del Moral|first = Pierre|title = Mean field simulation for Monte Carlo integration|year = 2013|publisher = Chapman & Hall/CRC Press|quote = Monographs on Statistics & Applied Probability|url = http://www.crcpress.com/product/isbn/9781466504059|pages = 626}}</ref> In other instances we are given a flow of probability distributions with an increasing level of sampling complexity (path spaces models with an increasing time horizon, Boltzmann–Gibbs measures associated with decreasing temperature parameters, and many others). These models can also be seen as the evolution of the law of the random states of a nonlinear Markov chain.<ref name="dp13" /><ref>{{Cite journal|title = Sequential Monte Carlo samplers | last = Del Moral | first = P | last2 = Doucet | first2 = A | last3 = Jasra | first3 = A | year = 2006 |doi=10.1111/j.1467-9868.2006.00553.x|volume=68| issue = 3 |journal=Journal of the Royal Statistical Society, Series B|pages=411–436|arxiv = cond-mat/0212648}}</ref> A natural way to simulate these sophisticated nonlinear Markov processes is to sample multiple copies of the process, replacing in the evolution equation the unknown distributions of the random states by the sampled [[empirical measure]]s. In contrast with traditional Monte Carlo and MCMC methodologies these [[Mean field particle methods|mean field particle]] techniques rely on sequential interacting samples. The terminology ''mean field'' reflects the fact that each of the ''samples'' (a.k.a. particles, individuals, walkers, agents, creatures, or phenotypes) interacts with the empirical measures of the process. When the size of the system tends to infinity, these random empirical measures converge to the deterministic distribution of the random states of the nonlinear Markov chain, so that the statistical interaction between particles vanishes.

In other problems, the objective is generating draws from a sequence of probability distributions satisfying a nonlinear evolution equation. These flows of probability distributions can always be interpreted as the distributions of the random states of a Markov process whose transition probabilities depend on the distributions of the current random states (see McKean–Vlasov processes, nonlinear filtering equation). In other instances we are given a flow of probability distributions with an increasing level of sampling complexity (path spaces models with an increasing time horizon, Boltzmann–Gibbs measures associated with decreasing temperature parameters, and many others). These models can also be seen as the evolution of the law of the random states of a nonlinear Markov chain. A natural way to simulate these sophisticated nonlinear Markov processes is to sample multiple copies of the process, replacing in the evolution equation the unknown distributions of the random states by the sampled empirical measures. In contrast with traditional Monte Carlo and MCMC methodologies these mean field particle techniques rely on sequential interacting samples. The terminology mean field reflects the fact that each of the samples (a.k.a. particles, individuals, walkers, agents, creatures, or phenotypes) interacts with the empirical measures of the process. When the size of the system tends to infinity, these random empirical measures converge to the deterministic distribution of the random states of the nonlinear Markov chain, so that the statistical interaction between particles vanishes.

在其他问题中，目标是从满足非线性发展方程的概率分布序列生成图。这些概率分布流总是可以解释为马尔可夫过程的随机状态的分布，其转移概率依赖于当前随机状态的分布(见 McKean-Vlasov 过程，非线性滤波方程)。在其他情况下，我们给出了一个随着采样复杂度的增加而增加的概率分布流(增加时间范围的路径空间模型，与降低温度参数有关的 Boltzmann-Gibbs 测度，以及许多其他)。这些模型也可以看作是一个非线性马尔可夫链的随机状态规律的演化。模拟这些复杂的非线性马尔可夫过程的一个自然的方法是对这个过程的多个副本进行抽样，用抽样的经验测量代替发展方程中未知的随机状态分布。与传统的蒙特卡罗和 MCMC 方法相比，这些平均场粒子技术依赖于连续的相互作用样本。术语的意思是领域反映了这样一个事实，即每个样本(又称为。粒子，个体，步行者，代理人，生物，或表型)与过程的经验措施相互作用。当系统规模趋于无穷大时，这些随机经验测度收敛于非线性马尔可夫链随机状态的确定性分布，从而使粒子之间的统计相互作用消失。

== Overview ==

Monte Carlo methods vary, but tend to follow a particular pattern:

Monte Carlo methods vary, but tend to follow a particular pattern:

蒙特卡罗方法各不相同，但趋向于遵循一个特定的模式:

# Define a domain of possible inputs

Define a domain of possible inputs

定义可能输入的域

# Generate inputs randomly from a [[probability distribution]] over the domain

Generate inputs randomly from a probability distribution over the domain

从域上的概率分布随机生成输入

# Perform a [[Deterministic algorithm|deterministic]] computation on the inputs

Perform a deterministic computation on the inputs

对输入进行确定性计算

# Aggregate the results

Aggregate the results

汇总结果

[[File:Pi 30K.gif|thumb|right| Monte Carlo method applied to approximating the value of {{pi}}.]]

Monte Carlo method applied to approximating the value of .

蒙特卡罗方法应用于估计。

For example, consider a [[circular sector#Quadrant|quadrant (circular sector)]] inscribed in a [[unit square]]. Given that the ratio of their areas is {{sfrac|{{pi}}|4}}, the value of [[pi|{{pi}}]] can be approximated using a Monte Carlo method:{{sfn|Kalos|Whitlock|2008}}

For example, consider a quadrant (circular sector) inscribed in a unit square. Given that the ratio of their areas is |4}}, the value of pi| can be approximated using a Monte Carlo method:

例如，考虑一个象限(圆形扇区)内接在一个单位正方形上。如果它们的面积比是 | 4}} ，那么 pi | 的值可以用蒙特卡罗方法来近似:

# Draw a square, then [[inscribed figure|inscribe]] a quadrant within it

Draw a square, then inscribe a quadrant within it

画一个正方形，然后在其中划出一个象限

# [[uniform distribution (continuous)|Uniformly]] scatter a given number of points over the square

Uniformly scatter a given number of points over the square

在正方形上均匀散布给定数量的点

# Count the number of points inside the quadrant, i.e. having a distance from the origin of less than 1

Count the number of points inside the quadrant, i.e. having a distance from the origin of less than 1

计算象限内的点数，即。距离原点小于1的

# The ratio of the inside-count and the total-sample-count is an estimate of the ratio of the two areas, {{sfrac|{{pi}}|4}}. Multiply the result by 4 to estimate {{pi}}.

The ratio of the inside-count and the total-sample-count is an estimate of the ratio of the two areas, |4}}. Multiply the result by 4 to estimate .

内部计数和总样本计数的比值是两个区域比值的估计，| 4}。用4乘以结果来估计。

In this procedure the domain of inputs is the square that circumscribes the quadrant. We generate random inputs by scattering grains over the square then perform a computation on each input (test whether it falls within the quadrant). Aggregating the results yields our final result, the approximation of {{pi}}.

In this procedure the domain of inputs is the square that circumscribes the quadrant. We generate random inputs by scattering grains over the square then perform a computation on each input (test whether it falls within the quadrant). Aggregating the results yields our final result, the approximation of .

在这个过程中，输入域是限定象限的正方形。我们通过将颗粒散射到正方形上来产生随机输入，然后对每个输入执行计算(测试它是否在象限内)。聚合结果会产生我们的最终结果，近似为。

There are two important considerations:

There are two important considerations:

有两个重要的考虑因素:

# If the points are not uniformly distributed, then the approximation will be poor.

If the points are not uniformly distributed, then the approximation will be poor.

如果这些点不是均匀分布的，那么近似就会很差。

# There are many points. The approximation is generally poor if only a few points are randomly placed in the whole square. On average, the approximation improves as more points are placed.

There are many points. The approximation is generally poor if only a few points are randomly placed in the whole square. On average, the approximation improves as more points are placed.

这里有很多要点。如果整个正方形中只有几个点是随机放置的，那么这个近似值通常是很差的。平均而言，随着放置更多的点，近似值会有所改进。

Uses of Monte Carlo methods require large amounts of random numbers, and it was their use that spurred the development of [[pseudorandom number generator]]s{{Citation needed|reason=I can't find any reference in the linked page saying that PRNG development was spurred by Monte Carlo methods and there's no reference here. This assertion, between the commas, needs a quotable source backing up that PRNGs development happened faster than it would have otherwise done specifically because of the use of MC methods. This statement seems to me to be conjecture, in the absence of a citation.|date=November 2019}}, which were far quicker to use than the tables of random numbers that had been previously used for statistical sampling.

Uses of Monte Carlo methods require large amounts of random numbers, and it was their use that spurred the development of pseudorandom number generators, which were far quicker to use than the tables of random numbers that had been previously used for statistical sampling.

蒙特卡罗方法的使用需要大量的随机数，正是它们的使用促进了伪随机数生成器的发展，伪随机数生成器的使用要比以前用于统计抽样的随机数表快得多。

== History ==

Before the Monte Carlo method was developed, simulations tested a previously understood deterministic problem, and statistical sampling was used to estimate uncertainties in the simulations. Monte Carlo simulations invert this approach, solving deterministic problems using a [[probabilistic]] [[meta-algorithm|analog]] (see [[Simulated annealing]]).

Before the Monte Carlo method was developed, simulations tested a previously understood deterministic problem, and statistical sampling was used to estimate uncertainties in the simulations. Monte Carlo simulations invert this approach, solving deterministic problems using a probabilistic analog (see Simulated annealing).

在蒙特卡罗方法模型开发之前，模拟测试了一个先前已知的确定性问题，并且使用统计抽样来估计模拟中的不确定性。蒙特卡罗模拟颠覆了这种方法，使用概率模拟来解决确定性问题(见模拟退火)。





<！ ——在早期的草案中提到了威廉·戈塞对小样本的研究，但并不清楚它们之间是如何联系的。-->

An early variant of the Monte Carlo method can be seen in the [[Buffon's needle]] experiment, in which {{pi}} can be estimated by dropping needles on a floor made of parallel and equidistant strips. In the 1930s, [[Enrico Fermi]] first experimented with the Monte Carlo method while studying neutron diffusion, but did not publish anything on it.{{sfn|Metropolis|1987}}

An early variant of the Monte Carlo method can be seen in the Buffon's needle experiment, in which can be estimated by dropping needles on a floor made of parallel and equidistant strips. In the 1930s, Enrico Fermi first experimented with the Monte Carlo method while studying neutron diffusion, but did not publish anything on it.

蒙特卡罗方法的早期变体可以在布冯的针实验中看到，在这个实验中，可以通过将针放在由平行和等距条组成的地板上来估计。在20世纪30年代，Enrico Fermi 在研究中子扩散的时候第一次实验了蒙特卡罗方法，但是没有发表任何关于中子扩散的文章。

The modern version of the Markov Chain Monte Carlo method was invented in the late 1940s by [[Stanislaw Ulam]], while he was working on nuclear weapons projects at the [[Los Alamos National Laboratory]]. Immediately after Ulam's breakthrough, [[John von Neumann]] understood its importance and programmed the [[ENIAC]] computer to carry out Monte Carlo calculations. In 1946, nuclear weapons physicists at [[Los Alamos Scientific Laboratory]] were investigating "neutron diffusion in fissionable material".{{sfn|Metropolis|1987}} Despite having most of the necessary data, such as the average distance a neutron would travel in a substance before it collided with an atomic nucleus, and how much energy the neutron was likely to give off following a collision, the Los Alamos physicists were unable to solve the problem using conventional, deterministic mathematical methods. Ulam had the idea of using random experiments. He recounts his inspiration as follows:

The modern version of the Markov Chain Monte Carlo method was invented in the late 1940s by Stanislaw Ulam, while he was working on nuclear weapons projects at the Los Alamos National Laboratory. Immediately after Ulam's breakthrough, John von Neumann understood its importance and programmed the ENIAC computer to carry out Monte Carlo calculations. In 1946, nuclear weapons physicists at Los Alamos Scientific Laboratory were investigating "neutron diffusion in fissionable material". Despite having most of the necessary data, such as the average distance a neutron would travel in a substance before it collided with an atomic nucleus, and how much energy the neutron was likely to give off following a collision, the Los Alamos physicists were unable to solve the problem using conventional, deterministic mathematical methods. Ulam had the idea of using random experiments. He recounts his inspiration as follows:

20世纪40年代后期，Stanislaw Ulam 发明了现代版的马尔科夫蒙特卡洛方法，当时他正在洛斯阿拉莫斯国家实验室研究核武器项目。在乌兰取得突破性进展之后，约翰·冯·诺伊曼马上意识到了它的重要性，并为 ENIAC 计算机编写了程序，使其能够进行蒙特卡洛计算。1946年，洛斯阿拉莫斯国家实验室的核武器物理学家正在研究“裂变材料中的中子扩散”。尽管拥有大部分必要的数据，比如中子在与原子核碰撞之前在物质中的平均运动距离，以及碰撞后中子可能释放出多少能量，洛斯阿拉莫斯的物理学家们还是无法用传统的、确定性的数学方法解决这个问题。乌拉姆有了使用随机实验的想法。他的灵感如下:

{{quote|The first thoughts and attempts I made to practice [the Monte Carlo Method] were suggested by a question which occurred to me in 1946 as I was convalescing from an illness and playing solitaires. The question was what are the chances that a [[Canfield (solitaire)|Canfield solitaire]] laid out with 52 cards will come out successfully? After spending a lot of time trying to estimate them by pure combinatorial calculations, I wondered whether a more practical method than "abstract thinking" might not be to lay it out say one hundred times and simply observe and count the number of successful plays. This was already possible to envisage with the beginning of the new era of fast computers, and I immediately thought of problems of neutron diffusion and other questions of mathematical physics, and more generally how to change processes described by certain differential equations into an equivalent form interpretable as a succession of random operations. Later [in 1946], I described the idea to [[John von Neumann]], and we began to plan actual calculations.{{sfn|Eckhardt|1987}}}}

Being secret, the work of von Neumann and Ulam required a code name.{{sfn|Mazhdrakov|Benov|Valkanov|2018|p=250}} A colleague of von Neumann and Ulam, [[Nicholas Metropolis]], suggested using the name ''Monte Carlo'', which refers to the [[Monte Carlo Casino]] in [[Monaco]] where Ulam's uncle would borrow money from relatives to gamble.{{sfn|Metropolis|1987}} Using [[A Million Random Digits with 100,000 Normal Deviates|lists of "truly random" random numbers]] was extremely slow, but von Neumann developed a way to calculate [[pseudorandom number]]s, using the [[middle-square method]]. Though this method has been criticized as crude, von Neumann was aware of this: he justified it as being faster than any other method at his disposal, and also noted that when it went awry it did so obviously, unlike methods that could be subtly incorrect.<ref>{{cite book |last = Peragine |first = Michael |title = The Universal Mind: The Evolution of Machine Intelligence and Human Psychology |year = 2013 |publisher = Xiphias Press |url = https://books.google.com/?id=Dvb0DAAAQBAJ&pg=PT201&lpg=PT201&dq=he+justified+it+as+being+faster+than+any+other+method+at+his+disposal,+and+also+noted+that+when+it+went+awry+it+did+so+obviously,+unlike+methods+that+could+be+subtly+incorrect.#v=onepage&q=he%20justified%20it%20as%20being%20faster%20than%20any%20other%20method%20at%20his%20disposal%2C%20and%20also%20noted%20that%20when%20it%20went%20awry%20it%20did%20so%20obviously%2C%20unlike%20methods%20that%20could%20be%20subtly%20incorrect.&f=false |access-date = 2018-12-17 }}</ref>

Being secret, the work of von Neumann and Ulam required a code name. A colleague of von Neumann and Ulam, Nicholas Metropolis, suggested using the name Monte Carlo, which refers to the Monte Carlo Casino in Monaco where Ulam's uncle would borrow money from relatives to gamble. Using lists of "truly random" random numbers was extremely slow, but von Neumann developed a way to calculate pseudorandom numbers, using the middle-square method. Though this method has been criticized as crude, von Neumann was aware of this: he justified it as being faster than any other method at his disposal, and also noted that when it went awry it did so obviously, unlike methods that could be subtly incorrect.

冯 · 诺依曼和乌拉姆的工作是秘密进行的，需要一个代号。冯 · 诺依曼和乌拉姆的一位同事尼古拉斯·梅特罗波利斯建议使用蒙特卡洛这个名字，这个名字指的是摩纳哥的蒙特卡洛赌场，在那里乌拉姆的叔叔会从亲戚那里借钱去赌博。使用“真正随机”的随机数列表是非常慢的，但冯 · 诺依曼开发了一种计算伪随机数生成器的方法，使用平方取中法。尽管这种方法被批评为粗糙，冯 · 诺依曼意识到了这一点: 他证明这种方法比他所支配的任何其他方法都要快，并指出，当它出错时，它的效果是如此明显，不像方法可能会微妙地不正确。

Monte Carlo methods were central to the [[simulation]]s required for the [[Manhattan Project]], though severely limited by the computational tools at the time. In the 1950s they were used at [[Los Alamos National Laboratory|Los Alamos]] for early work relating to the development of the [[hydrogen bomb]], and became popularized in the fields of [[physics]], [[physical chemistry]], and [[operations research]]. The [[Rand Corporation]] and the [[U.S. Air Force]] were two of the major organizations responsible for funding and disseminating information on Monte Carlo methods during this time, and they began to find a wide application in many different fields.

Monte Carlo methods were central to the simulations required for the Manhattan Project, though severely limited by the computational tools at the time. In the 1950s they were used at Los Alamos for early work relating to the development of the hydrogen bomb, and became popularized in the fields of physics, physical chemistry, and operations research. The Rand Corporation and the U.S. Air Force were two of the major organizations responsible for funding and disseminating information on Monte Carlo methods during this time, and they began to find a wide application in many different fields.

蒙特卡罗方法是曼哈顿计划所需要的模拟的中心，尽管在当时受到计算工具的严重限制。20世纪50年代，它们在洛斯阿拉莫斯用于研制氢弹的早期工作，并在物理学、物理化学和运筹学领域得到普及。兰德公司和美国空军是当时负责资助和传播蒙特卡洛方法信息的两个主要组织，他们开始在许多不同的领域找到广泛的应用。

The theory of more sophisticated mean field type particle Monte Carlo methods had certainly started by the mid-1960s, with the work of [[Henry McKean|Henry P. McKean Jr.]] on Markov interpretations of a class of nonlinear parabolic partial differential equations arising in fluid mechanics.<ref name="mck67">{{cite journal |last = McKean |first = Henry, P. |title = Propagation of chaos for a class of non-linear parabolic equations |journal = Lecture Series in Differential Equations, Catholic Univ. |year = 1967 |volume = 7 |pages = 41–57 }}</ref><ref>{{cite journal |last1 = McKean |first1 = Henry, P. |title = A class of Markov processes associated with nonlinear parabolic equations |journal = Proc. Natl. Acad. Sci. USA |year = 1966 |volume = 56 |issue = 6 |pages = 1907–1911 |doi = 10.1073/pnas.56.6.1907 |pmid = 16591437 |pmc = 220210 |bibcode = 1966PNAS...56.1907M }}</ref> We also quote an earlier pioneering article by [[Ted Harris (mathematician)|Theodore E. Harris]] and Herman Kahn, published in 1951, using mean field [[genetic algorithm|genetic]]-type Monte Carlo methods for estimating particle transmission energies.<ref>{{cite journal |last1 = Herman |first1 = Kahn |last2 = Theodore |first2 = Harris E. |title = Estimation of particle transmission by random sampling |journal = Natl. Bur. Stand. Appl. Math. Ser. |year = 1951 |volume = 12 |pages = 27–30 |url = https://dornsifecms.usc.edu/assets/sites/520/docs/kahnharris.pdf }}</ref> Mean field genetic type Monte Carlo methodologies are also used as heuristic natural search algorithms (a.k.a. [[metaheuristic]]) in evolutionary computing. The origins of these mean field computational techniques can be traced to 1950 and 1954 with the work of [[Alan Turing]] on genetic type mutation-selection learning machines<ref>{{cite journal |last = Turing |first = Alan M. |title = Computing machinery and intelligence|journal = Mind|volume = LIX |issue = 238 |pages = 433–460 |doi = 10.1093/mind/LIX.236.433 |year = 1950 }}</ref> and the articles by [[Nils Aall Barricelli]] at the [[Institute for Advanced Study]] in [[Princeton, New Jersey]].<ref>{{cite journal |last = Barricelli |first = Nils Aall |year = 1954 |author-link = Nils Aall Barricelli |title = Esempi numerici di processi di evoluzione |journal = Methodos |pages = 45–68 }}</ref><ref>{{cite journal |last = Barricelli |first = Nils Aall |year = 1957 |author-link = Nils Aall Barricelli |title = Symbiogenetic evolution processes realized by artificial methods |journal = Methodos |pages = 143–182 }}</ref>

The theory of more sophisticated mean field type particle Monte Carlo methods had certainly started by the mid-1960s, with the work of Henry P. McKean Jr. on Markov interpretations of a class of nonlinear parabolic partial differential equations arising in fluid mechanics. We also quote an earlier pioneering article by Theodore E. Harris and Herman Kahn, published in 1951, using mean field genetic-type Monte Carlo methods for estimating particle transmission energies. Mean field genetic type Monte Carlo methodologies are also used as heuristic natural search algorithms (a.k.a. metaheuristic) in evolutionary computing. The origins of these mean field computational techniques can be traced to 1950 and 1954 with the work of Alan Turing on genetic type mutation-selection learning machines and the articles by Nils Aall Barricelli at the Institute for Advanced Study in Princeton, New Jersey.

更为复杂的平均场型粒子蒙特卡罗方法的理论自20世纪60年代中期开始，由 Henry p. McKean jr. 关于流体力学中一类非线性抛物型偏微分方程的 Markov 解释的工作开始。我们还引用了西奥多 · e · 哈里斯(Theodore e. Harris)和赫尔曼 · 卡恩(Herman Kahn)在1951年发表的一篇开创性文章，该文使用平均场遗传型蒙特卡罗方法估算粒子传输能量。平均场遗传型蒙特卡罗方法也被用作启发式自然搜索算法。进化计算中的元启发式算法。这些平均场计算技术的起源可以追溯到1950年和1954年，其中包括阿兰 · 图灵关于基因类型突变选择学习机的工作，以及新泽西州普林斯顿高级研究所的尼尔斯 · 阿尔 · 巴里切利的文章。

[[Quantum Monte Carlo]], and more specifically [[Diffusion Monte Carlo|diffusion Monte Carlo methods]] can also be interpreted as a mean field particle Monte Carlo approximation of [[Richard Feynman|Feynman]]–[[Mark Kac|Kac]] path integrals.<ref name="dp04">{{cite book |last = Del Moral |first = Pierre|title = Feynman–Kac formulae. Genealogical and interacting particle approximations |year = 2004 |publisher = Springer |quote = Series: Probability and Applications |url = https://www.springer.com/mathematics/probability/book/978-0-387-20268-6 |page = 575 |isbn = 9780387202686|series = Probability and Its Applications}}</ref><ref name="dmm002">{{cite book

Quantum Monte Carlo, and more specifically diffusion Monte Carlo methods can also be interpreted as a mean field particle Monte Carlo approximation of Feynman–Kac path integrals.<ref name="dmm002">{{cite book

量子蒙特卡罗法，特别是扩散蒙特卡罗方法也可以解释为平均场粒子蒙特卡罗近似的费曼-卡克路径积分。 002"{ cite book

| last1 = Del Moral | first1 = P.

| last1 = Del Moral | first1 = P.

1 p.

| last2 = Miclo | first2 = L.

| last2 = Miclo | first2 = L.

最后2米高 | 最初2米高。

| contribution = Branching and interacting particle systems approximations of Feynman–Kac formulae with applications to non-linear filtering

| contribution = Branching and interacting particle systems approximations of Feynman–Kac formulae with applications to non-linear filtering

| 贡献分支和相互作用粒子系统近似的 Feynman-Kac 公式及其在非线性滤波中的应用

| contribution-url = http://archive.numdam.org/item/SPS_2000__34__1_0

| contribution-url = http://archive.numdam.org/item/SPS_2000__34__1_0

| 贡献- http://archive.numdam.org/item/sps_2000__34__1_0

| doi = 10.1007/BFb0103798

| doi = 10.1007/BFb0103798

10.1007 / BFb0103798

| mr = 1768060

| mr = 1768060

1768060先生

| pages = 1–145

| pages = 1–145

第1-145页

| publisher = Springer |location = Berlin

| publisher = Springer |location = Berlin

| 出版商 Springer | 位置: 柏林

| series = Lecture Notes in Mathematics

| series = Lecture Notes in Mathematics

| 数学系列讲义

| title = Séminaire de Probabilités, XXXIV

| title = Séminaire de Probabilités, XXXIV

| title = Séminaire de Probabilités, XXXIV

| volume = 1729

| volume = 1729

第1729卷

| year = 2000

| year = 2000

2000年

|isbn = 978-3-540-67314-9

|isbn = 978-3-540-67314-9

[国际标准图书馆编号978-3-540-67314-9]

| url = http://www.numdam.org/item/SPS_2000__34__1_0/

| url = http://www.numdam.org/item/SPS_2000__34__1_0/

Http://www.numdam.org/item/sps_2000__34__1_0/

}}</ref><ref name="dmm00m">{{cite journal|last1 = Del Moral|first1 = Pierre|last2 = Miclo|first2 = Laurent|title = A Moran particle system approximation of Feynman–Kac formulae.|journal = Stochastic Processes and Their Applications |year = 2000|volume = 86|issue = 2|pages = 193–216|doi = 10.1016/S0304-4149(99)00094-0|doi-access = free}}</ref><ref name="dm-esaim03">{{cite journal|last1 = Del Moral|first1 = Pierre|title = Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman–Kac semigroups|journal = ESAIM Probability & Statistics|date = 2003|volume = 7|pages = 171–208|url = http://journals.cambridge.org/download.php?file=%2FPSS%2FPSS7%2FS1292810003000016a.pdf&code=a0dbaa7ffca871126dc05fe2f918880a|doi = 10.1051/ps:2003001|doi-access = free}}</ref><ref name="caffarel1">{{cite journal|last1 = Assaraf|first1 = Roland|last2 = Caffarel|first2 = Michel|last3 = Khelif|first3 = Anatole|title = Diffusion Monte Carlo Methods with a fixed number of walkers|journal = Phys. Rev. E|url = http://qmcchem.ups-tlse.fr/files/caffarel/31.pdf|date = 2000|volume = 61|issue = 4|pages = 4566–4575|doi = 10.1103/physreve.61.4566|pmid = 11088257|bibcode = 2000PhRvE..61.4566A|url-status = dead|archiveurl = https://web.archive.org/web/20141107015724/http://qmcchem.ups-tlse.fr/files/caffarel/31.pdf|archivedate = 2014-11-07 }}</ref><ref name="caffarel2">{{cite journal|last1 = Caffarel|first1 = Michel|last2 = Ceperley|first2 = David |last3 = Kalos|first3 = Malvin|title = Comment on Feynman–Kac Path-Integral Calculation of the Ground-State Energies of Atoms|journal = Phys. Rev. Lett.|date = 1993|volume = 71|issue = 13|doi = 10.1103/physrevlett.71.2159|bibcode = 1993PhRvL..71.2159C|pages=2159|pmid=10054598}}</ref><ref name="h84">{{cite journal |last = Hetherington|first = Jack, H.|title = Observations on the statistical iteration of matrices|journal = Phys. Rev. A |date = 1984|volume = 30|issue = 2713|doi = 10.1103/PhysRevA.30.2713|pages = 2713–2719|bibcode = 1984PhRvA..30.2713H}}</ref> The origins of Quantum Monte Carlo methods are often attributed to Enrico Fermi and [[Robert D. Richtmyer|Robert Richtmyer]] who developed in 1948 a mean field particle interpretation of neutron-chain reactions,<ref>{{cite journal|last1 = Fermi|first1 = Enrique|last2 = Richtmyer|first2 = Robert, D.|title = Note on census-taking in Monte Carlo calculations|journal = LAM|date = 1948|volume = 805|issue = A|url = http://scienze-como.uninsubria.it/bressanini/montecarlo-history/fermi-1948.pdf|quote = Declassified report Los Alamos Archive}}</ref> but the first heuristic-like and genetic type particle algorithm (a.k.a. Resampled or Reconfiguration Monte Carlo methods) for estimating ground state energies of quantum systems (in reduced matrix models) is due to Jack H. Hetherington in 1984<ref name="h84" /> In molecular chemistry, the use of genetic heuristic-like particle methodologies (a.k.a. pruning and enrichment strategies) can be traced back to 1955 with the seminal work of [[Marshall Rosenbluth|Marshall N. Rosenbluth]] and [[Arianna W. Rosenbluth]].<ref name=":0">{{cite journal |last1 = Rosenbluth|first1 = Marshall, N.|last2 = Rosenbluth|first2 = Arianna, W.|title = Monte-Carlo calculations of the average extension of macromolecular chains|journal = J. Chem. Phys.|date = 1955|volume = 23|issue = 2|pages = 356–359|bibcode = 1955JChPh..23..356R|doi = 10.1063/1.1741967 |url = https://semanticscholar.org/paper/1570c85ba9aca1cb413ada31e215e0917c3ccba7}}</ref>

}}</ref> The origins of Quantum Monte Carlo methods are often attributed to Enrico Fermi and Robert Richtmyer who developed in 1948 a mean field particle interpretation of neutron-chain reactions, but the first heuristic-like and genetic type particle algorithm (a.k.a. Resampled or Reconfiguration Monte Carlo methods) for estimating ground state energies of quantum systems (in reduced matrix models) is due to Jack H. Hetherington in 1984

量子蒙特卡罗法粒子方法的起源通常归功于 Enrico Fermi 和 Robert Richtmyer，他们在1948年发明了中子链反应的平均场粒子解释，但是第一个启发式的遗传类型粒子算法。用于估计量子系统基态能量的重构蒙特卡罗方法(简化矩阵模型)是1984年杰克 · h · 赫瑟林顿提出的

The use of [[Sequential Monte Carlo method|Sequential Monte Carlo]] in advanced [[signal processing]] and [[Bayesian inference]] is more recent. It was in 1993, that Gordon et al., published in their seminal work<ref>{{Cite journal|title = Novel approach to nonlinear/non-Gaussian Bayesian state estimation |journal = IEE Proceedings F - Radar and Signal Processing|date = April 1993|issn = 0956-375X|pages = 107–113|volume = 140|issue = 2|first = N.J.|last = Gordon|first2 = D.J.|last2 = Salmond|first3 = A.F.M.|last3 = Smith|doi=10.1049/ip-f-2.1993.0015|url = https://semanticscholar.org/paper/65484334a5cd4cabf6e5f7a17f606f07e2acf625}}</ref> the first application of a Monte Carlo [[Resampling (statistics)|resampling]] algorithm in Bayesian statistical inference. The authors named their algorithm 'the bootstrap filter', and demonstrated that compared to other filtering methods, their bootstrap algorithm does not require any assumption about that state-space or the noise of the system. We also quote another pioneering article in this field of Genshiro Kitagawa on a related "Monte Carlo filter",<ref>{{cite journal

The use of Sequential Monte Carlo in advanced signal processing and Bayesian inference is more recent. It was in 1993, that Gordon et al., published in their seminal work the first application of a Monte Carlo resampling algorithm in Bayesian statistical inference. The authors named their algorithm 'the bootstrap filter', and demonstrated that compared to other filtering methods, their bootstrap algorithm does not require any assumption about that state-space or the noise of the system. We also quote another pioneering article in this field of Genshiro Kitagawa on a related "Monte Carlo filter",<ref>{{cite journal

在高级信号处理和贝叶斯推断中使用 Sequential Monte Carlo 是最近才出现的。这是在1993年，Gordon 等人，在他们的开创性工作中发表的第一个应用蒙特卡罗重采样算法在贝叶斯推论统计学。作者将他们的算法命名为“自举过滤器” ，并证明了与其他过滤方法相比，他们的自举过滤算法不需要任何关于系统状态空间或噪声的假设。我们还引用了北川玄史郎在这一领域的另一篇开创性的文章，关于一个相关的“蒙特卡罗过滤器” ，参见{ cite journal

|last = Kitagawa|first = G.|year = 1996|title = Monte carlo filter and smoother for non-Gaussian nonlinear state space models|volume = 5|issue = 1|journal = Journal of Computational and Graphical Statistics|pages = 1–25|doi = 10.2307/1390750|jstor = 1390750}}

|last = Kitagawa|first = G.|year = 1996|title = Monte carlo filter and smoother for non-Gaussian nonlinear state space models|volume = 5|issue = 1|journal = Journal of Computational and Graphical Statistics|pages = 1–25|doi = 10.2307/1390750|jstor = 1390750}}

非高斯非线性状态空间模型的蒙特卡罗滤波器和光滑器 | 卷5 | 第1期 | 计算和图形统计杂志 | 第1-25页 | doi 10.2307 / 1390750 | jstor 1390750}

</ref> and the ones by Pierre Del Moral<ref name="dm9622">{{cite journal|last1 = Del Moral|first1 = Pierre|title = Non Linear Filtering: Interacting Particle Solution.|journal = Markov Processes and Related Fields|date = 1996|volume = 2|issue = 4|pages = 555–580|url = http://web.maths.unsw.edu.au/~peterdel-moral/mprfs.pdf}}</ref> and Himilcon Carvalho, Pierre Del Moral, André Monin and Gérard Salut<ref>{{cite journal|last1 = Carvalho|first1 = Himilcon|last2 = Del Moral|first2 = Pierre|last3 = Monin|first3 = André|last4 = Salut|first4 = Gérard|title = Optimal Non-linear Filtering in GPS/INS Integration.|journal = IEEE Transactions on Aerospace and Electronic Systems|date = July 1997|volume = 33|issue = 3|pages = 835|url = http://homepages.laas.fr/monin/Version_anglaise/Publications_files/GPS.pdf|bibcode = 1997ITAES..33..835C|doi = 10.1109/7.599254}}</ref> on particle filters published in the mid-1990s. Particle filters were also developed in signal processing in 1989–1992 by P. Del Moral, J. C. Noyer, G. Rigal, and G. Salut in the LAAS-CNRS in a series of restricted and classified research reports with STCAN (Service Technique des Constructions et Armes Navales), the IT company DIGILOG, and the [https://www.laas.fr/public/en LAAS-CNRS] (the Laboratory for Analysis and Architecture of Systems) on radar/sonar and GPS signal processing problems.<ref>P. Del Moral, G. Rigal, and G. Salut. "Estimation and nonlinear optimal control: An unified framework for particle solutions". LAAS-CNRS, Toulouse, Research Report no. 91137, DRET-DIGILOG- LAAS/CNRS contract, April (1991).</ref><ref>P. Del Moral, G. Rigal, and G. Salut. "Nonlinear and non Gaussian particle filters applied to inertial platform repositioning." LAAS-CNRS, Toulouse, Research Report no. 92207, STCAN/DIGILOG-LAAS/CNRS Convention STCAN no. A.91.77.013, (94p.) September (1991).</ref><ref>P. Del Moral, G. Rigal, and G. Salut. "Estimation and nonlinear optimal control: Particle resolution in filtering and estimation: Experimental results". Convention DRET no. 89.34.553.00.470.75.01, Research report no.2 (54p.), January (1992).</ref><ref>P. Del Moral, G. Rigal, and G. Salut. "Estimation and nonlinear optimal control: Particle resolution in filtering and estimation: Theoretical results".

</ref> and the ones by Pierre Del Moral and Himilcon Carvalho, Pierre Del Moral, André Monin and Gérard Salut on particle filters published in the mid-1990s. Particle filters were also developed in signal processing in 1989–1992 by P. Del Moral, J. C. Noyer, G. Rigal, and G. Salut in the LAAS-CNRS in a series of restricted and classified research reports with STCAN (Service Technique des Constructions et Armes Navales), the IT company DIGILOG, and the [https://www.laas.fr/public/en LAAS-CNRS] (the Laboratory for Analysis and Architecture of Systems) on radar/sonar and GPS signal processing problems.<ref>P. Del Moral, G. Rigal, and G. Salut. "Estimation and nonlinear optimal control: Particle resolution in filtering and estimation: Theoretical results".

1990年代中期，Pierre Del Moral 和 Himilcon Carvalho，Pierre Del Moral，andr Monin 和 g rard Salut 在粒子过滤器上发表的文章。1989-1992年，p. Del Moral、 j. c. Noyer、 g. Rigal 和 g. Salut 也在信号处理领域开发了粒子滤波器，这些粒子滤波器是由 LAAS-CNRS 的 p. Del Moral、 j. c. Noyer、 g. Rigal 和 g. Salut 与 STCAN (Service Technique des construction et Armes Navales)、 IT 公司 DIGILOG 和 https://www.laas.fr/public/en LAAS-CNRS (the Laboratory for Analysis and Architecture of Systems)就雷达 / 声纳和 GPS 信号处理问题共同撰写的一系列受限制和机密的研究报告。参考文献 p。德尔 · 莫勒尔，g · 里加尔，g · 萨鲁特。“估计和非线性最优控制: 滤波和估计中的粒子分辨率: 理论结果”。

Convention DRET no. 89.34.553.00.470.75.01, Research report no.3 (123p.), October (1992).</ref><ref>P. Del Moral, J.-Ch. Noyer, G. Rigal, and G. Salut. "Particle filters in radar signal processing: detection, estimation and air targets recognition". LAAS-CNRS, Toulouse, Research report no. 92495, December (1992).</ref><ref>P. Del Moral, G. Rigal, and G. Salut. "Estimation and nonlinear optimal control: Particle resolution in filtering and estimation". Studies on: Filtering, optimal control, and maximum likelihood estimation. Convention DRET no. 89.34.553.00.470.75.01. Research report no.4 (210p.), January (1993).</ref> These Sequential Monte Carlo methodologies can be interpreted as an acceptance-rejection sampler equipped with an interacting recycling mechanism.

Convention DRET no. 89.34.553.00.470.75.01, Research report no.3 (123p.), October (1992).</ref> These Sequential Monte Carlo methodologies can be interpreted as an acceptance-rejection sampler equipped with an interacting recycling mechanism.

第号公约。89.34.553.00.470.75.01，第三号研究报告书(123p。) ，10月(1992年)。 / ref 这些序贯蒙特卡罗方法可以被解释为一个配备有相互作用回收机制的拒收取样器。

From 1950 to 1996, all the publications on Sequential Monte Carlo methodologies, including the pruning and resample Monte Carlo methods introduced in computational physics and molecular chemistry, present natural and heuristic-like algorithms applied to different situations without a single proof of their consistency, nor a discussion on the bias of the estimates and on genealogical and ancestral tree based algorithms. The mathematical foundations and the first rigorous analysis of these particle algorithms are due to Pierre Del Moral<ref name="dm9622"/><ref name=":22">{{cite journal|last1 = Del Moral|first1 = Pierre|title = Measure Valued Processes and Interacting Particle Systems. Application to Non Linear Filtering Problems|journal = Annals of Applied Probability|date = 1998|edition = Publications du Laboratoire de Statistique et Probabilités, 96-15 (1996)|volume = 8|issue = 2|pages = 438–495|url = http://projecteuclid.org/download/pdf_1/euclid.aoap/1028903535|doi = 10.1214/aoap/1028903535|citeseerx = 10.1.1.55.5257}}</ref> in 1996. Branching type particle methodologies with varying population sizes were also developed in the end of the 1990s by Dan Crisan, Jessica Gaines and Terry Lyons,<ref name=":42">{{cite journal|last1 = Crisan|first1 = Dan|last2 = Gaines|first2 = Jessica|last3 = Lyons|first3 = Terry|title = Convergence of a branching particle method to the solution of the Zakai|journal = SIAM Journal on Applied Mathematics|date = 1998|volume = 58|issue = 5|pages = 1568–1590|doi = 10.1137/s0036139996307371|url = https://semanticscholar.org/paper/99e8759a243cd0568b0f32cbace2ad0525b16bb6}}</ref><ref>{{cite journal|last1 = Crisan|first1 = Dan|last2 = Lyons|first2 = Terry|title = Nonlinear filtering and measure-valued processes|journal = Probability Theory and Related Fields|date = 1997|volume = 109|issue = 2|pages = 217–244|doi = 10.1007/s004400050131}}</ref><ref>{{cite journal|last1 = Crisan|first1 = Dan|last2 = Lyons|first2 = Terry|title = A particle approximation of the solution of the Kushner–Stratonovitch equation|journal = Probability Theory and Related Fields|date = 1999|volume = 115|issue = 4|pages = 549–578|doi = 10.1007/s004400050249}}</ref> and by Dan Crisan, Pierre Del Moral and Terry Lyons.<ref name=":52">{{cite journal|last1 = Crisan|first1 = Dan|last2 = Del Moral|first2 = Pierre|last3 = Lyons|first3 = Terry|title = Discrete filtering using branching and interacting particle systems|journal = Markov Processes and Related Fields|date = 1999|volume = 5|issue = 3|pages = 293–318|url = http://web.maths.unsw.edu.au/~peterdel-moral/crisan98discrete.pdf}}</ref> Further developments in this field were developed in 2000 by P. Del Moral, A. Guionnet and L. Miclo.<ref name="dmm002"/><ref name="dg99">{{cite journal|last1 = Del Moral|first1 = Pierre|last2 = Guionnet|first2 = Alice|title = On the stability of Measure Valued Processes with Applications to filtering|journal = C. R. Acad. Sci. Paris|date = 1999|volume = 39|issue = 1|pages = 429–434}}</ref><ref name="dg01">{{cite journal|last1 = Del Moral|first1 = Pierre|last2 = Guionnet|first2 = Alice|title = On the stability of interacting processes with applications to filtering and genetic algorithms|journal = Annales de l'Institut Henri Poincaré|date = 2001|volume = 37|issue = 2|pages = 155–194|url = http://web.maths.unsw.edu.au/~peterdel-moral/ihp.ps|doi = 10.1016/s0246-0203(00)01064-5|bibcode=2001AnIHP..37..155D}}</ref>

From 1950 to 1996, all the publications on Sequential Monte Carlo methodologies, including the pruning and resample Monte Carlo methods introduced in computational physics and molecular chemistry, present natural and heuristic-like algorithms applied to different situations without a single proof of their consistency, nor a discussion on the bias of the estimates and on genealogical and ancestral tree based algorithms. The mathematical foundations and the first rigorous analysis of these particle algorithms are due to Pierre Del Moral in 1996. Branching type particle methodologies with varying population sizes were also developed in the end of the 1990s by Dan Crisan, Jessica Gaines and Terry Lyons, and by Dan Crisan, Pierre Del Moral and Terry Lyons. Further developments in this field were developed in 2000 by P. Del Moral, A. Guionnet and L. Miclo.

从1950年到1996年，所有关于序贯蒙特卡罗方法的出版物，包括计算物理学和分子化学中引入的修剪和重采样的蒙特卡罗方法，目前的自然和启发式算法适用于不同的情况，没有一个单一的证明其一致性，也没有讨论估计的偏差和基于系谱和祖先树的算法。这些粒子算法的数学基础和第一次严格的分析是由于1996年的皮埃尔德尔道德。在20世纪90年代末，Dan Crisan，Jessica Gaines 和 Terry Lyons，以及 Dan Crisan，Pierre Del Moral 和 Terry Lyons 也发展了不同种群大小的分支型粒子方法学。2000年，p. Del Moral、 a. Guionnet 和 l. Miclo 在这一领域有了进一步的发展。

==Definitions==

There is no consensus on how ''Monte Carlo'' should be defined. For example, Ripley<ref name=Ripley>{{harvnb|Ripley|1987}}</ref> defines most probabilistic modeling as ''[[stochastic simulation]]'', with ''Monte Carlo'' being reserved for [[Monte Carlo integration]] and Monte Carlo statistical tests. [[Shlomo Sawilowsky|Sawilowsky]]<ref name=Sawilowsky>{{harvnb|Sawilowsky|2003}}</ref> distinguishes between a [[simulation]], a Monte Carlo method, and a Monte Carlo simulation: a simulation is a fictitious representation of reality, a Monte Carlo method is a technique that can be used to solve a mathematical or statistical problem, and a Monte Carlo simulation uses repeated sampling to obtain the statistical properties of some phenomenon (or behavior). Examples:

There is no consensus on how Monte Carlo should be defined. For example, Ripley defines most probabilistic modeling as stochastic simulation, with Monte Carlo being reserved for Monte Carlo integration and Monte Carlo statistical tests. Sawilowsky distinguishes between a simulation, a Monte Carlo method, and a Monte Carlo simulation: a simulation is a fictitious representation of reality, a Monte Carlo method is a technique that can be used to solve a mathematical or statistical problem, and a Monte Carlo simulation uses repeated sampling to obtain the statistical properties of some phenomenon (or behavior). Examples:

对于蒙特卡洛应该如何定义还没有达成共识。例如，Ripley 将大多数概率模型定义为随机模拟，蒙特卡罗模拟被保留给蒙地卡罗积分和蒙特卡罗统计检验。Sawilowsky 区分了模拟、蒙特卡罗方法和蒙特卡洛模拟: 模拟是对现实的虚拟表示，蒙特卡罗方法是一种可用于解决数学或统计问题的技术，蒙特卡洛模拟使用重复抽样来获得某种现象(或行为)的统计特性。例子:

*Simulation: Drawing '''one''' pseudo-random uniform variable from the interval [0,1] can be used to simulate the tossing of a coin: If the value is less than or equal to 0.50 designate the outcome as heads, but if the value is greater than 0.50 designate the outcome as tails. This is a simulation, but not a Monte Carlo simulation.

*Monte Carlo method: Pouring out a box of coins on a table, and then computing the ratio of coins that land heads versus tails is a Monte Carlo method of determining the behavior of repeated coin tosses, but it is not a simulation.

*Monte Carlo simulation: Drawing '''a large number''' of pseudo-random uniform variables from the interval [0,1] at one time, or once at many different times, and assigning values less than or equal to 0.50 as heads and greater than 0.50 as tails, is a ''Monte Carlo simulation'' of the behavior of repeatedly tossing a coin.

Kalos and Whitlock<ref name="Kalos">{{harvnb|Kalos|Whitlock|2008}}</ref> point out that such distinctions are not always easy to maintain. For example, the emission of radiation from atoms is a natural stochastic process. It can be simulated directly, or its average behavior can be described by stochastic equations that can themselves be solved using Monte Carlo methods. "Indeed, the same computer code can be viewed simultaneously as a 'natural simulation' or as a solution of the equations by natural sampling."

Kalos and Whitlock point out that such distinctions are not always easy to maintain. For example, the emission of radiation from atoms is a natural stochastic process. It can be simulated directly, or its average behavior can be described by stochastic equations that can themselves be solved using Monte Carlo methods. "Indeed, the same computer code can be viewed simultaneously as a 'natural simulation' or as a solution of the equations by natural sampling."

卡洛斯和惠特洛克指出，这种区别并不总是容易维持。例如，来自原子的辐射是一种自然的随机过程。它可以直接模拟，也可以用随机方程描述其平均行为，这些随机方程本身可以用蒙特卡罗方法求解。“实际上，同样的计算机代码可以同时被看作是‘自然模拟’ ，或者是通过自然抽样解方程。”

===Monte Carlo and random numbers===

The main idea behind this method is that the results are computed based on repeated random sampling and statistical analysis. The Monte Carlo simulation is, in fact, random experimentations, in the case that, the results of these experiments are not well known.

The main idea behind this method is that the results are computed based on repeated random sampling and statistical analysis. The Monte Carlo simulation is, in fact, random experimentations, in the case that, the results of these experiments are not well known.

该方法的主要思想是在重复随机抽样和统计分析的基础上计算结果。蒙特卡罗模拟实际上是随机实验，在这种情况下，这些实验的结果并不为人所知。

Monte Carlo simulations are typically characterized by many unknown parameters, many of which are difficult to obtain experimentally.<ref name="usaus">{{cite journal|last1 = Shojaeefard|first1 = MH| last2 = Khalkhali|first2 = A| last3 =Yarmohammadisatri|first3 = Sadegh|title = An efficient sensitivity analysis method for modified geometry of Macpherson suspension based on Pearson Correlation Coefficient|journal = Vehicle System Dynamics|volume = 55|issue = 6|pages = 827–852|doi = 10.1080/00423114.2017.1283046|year = 2017|bibcode = 2017VSD....55..827S}}</ref> Monte Carlo simulation methods do not always require [[Random number generation#"True" random numbers vs. pseudo-random numbers|truly random number]]s to be useful (although, for some applications such as [[primality testing]], unpredictability is vital).<ref>{{harvnb|Davenport|1992}}</ref> Many of the most useful techniques use deterministic, [[pseudorandom number generator|pseudorandom]] sequences, making it easy to test and re-run simulations. The only quality usually necessary to make good [[simulation]]s is for the pseudo-random sequence to appear "random enough" in a certain sense.

Monte Carlo simulations are typically characterized by many unknown parameters, many of which are difficult to obtain experimentally. Monte Carlo simulation methods do not always require truly random numbers to be useful (although, for some applications such as primality testing, unpredictability is vital). Many of the most useful techniques use deterministic, pseudorandom sequences, making it easy to test and re-run simulations. The only quality usually necessary to make good simulations is for the pseudo-random sequence to appear "random enough" in a certain sense.

蒙特卡罗模拟通常需要拥有属性许多未知参数，其中许多参数很难通过实验获得。蒙特卡罗模拟方法并不总是要求真正的随机数是有用的(尽管，对于素数测试等一些应用，不可预测性是至关重要的)。许多最有用的技术使用确定性，伪随机序列，使它很容易测试和重新运行模拟。伪随机序列在某种意义上表现为“足够随机” ，这通常是进行良好模拟所必需的唯一品质。

What this means depends on the application, but typically they should pass a series of statistical tests. Testing that the numbers are [[Uniform distribution (continuous)|uniformly distributed]] or follow another desired distribution when a large enough number of elements of the sequence are considered is one of the simplest and most common ones. Weak correlations between successive samples are also often desirable/necessary.

What this means depends on the application, but typically they should pass a series of statistical tests. Testing that the numbers are uniformly distributed or follow another desired distribution when a large enough number of elements of the sequence are considered is one of the simplest and most common ones. Weak correlations between successive samples are also often desirable/necessary.

这意味着什么取决于应用程序，但通常应该通过一系列统计测试。当考虑序列中足够多的元素时，检验这些数是否均匀分布或遵循另一个期望的分布是最简单和最常见的方法之一。连续样本之间的弱相关性通常也是可取的 / 必要的。

Sawilowsky lists the characteristics of a high-quality Monte Carlo simulation:<ref name=Sawilowsky/>

Sawilowsky lists the characteristics of a high-quality Monte Carlo simulation:

萨维罗斯基列出了高质量蒙特卡罗模拟的特点:

*the (pseudo-random) number generator has certain characteristics (e.g. a long "period" before the sequence repeats)

*the (pseudo-random) number generator produces values that pass tests for randomness

*there are enough samples to ensure accurate results

*the proper sampling technique is used

*the algorithm used is valid for what is being modeled

*it simulates the phenomenon in question.

[[Pseudo-random number sampling]] algorithms are used to transform uniformly distributed pseudo-random numbers into numbers that are distributed according to a given [[probability distribution]].

Pseudo-random number sampling algorithms are used to transform uniformly distributed pseudo-random numbers into numbers that are distributed according to a given probability distribution.

伪随机数抽样算法用于将均匀分布的伪随机数转换为按给定概率分布分布的数。

[[Low-discrepancy sequences]] are often used instead of random sampling from a space as they ensure even coverage and normally have a faster order of convergence than Monte Carlo simulations using random or pseudorandom sequences. Methods based on their use are called [[quasi-Monte Carlo method]]s.

Low-discrepancy sequences are often used instead of random sampling from a space as they ensure even coverage and normally have a faster order of convergence than Monte Carlo simulations using random or pseudorandom sequences. Methods based on their use are called quasi-Monte Carlo methods.

与使用随机或伪随机序列的蒙特卡罗模拟相比，低差异序列常常被用来代替空间中的随机采样，因为它们确保了均匀的覆盖，并且通常具有更快的收敛阶。基于它们使用的方法称为拟蒙特卡罗方法。

In an effort to assess the impact of random number quality on Monte Carlo simulation outcomes, astrophysical researchers tested cryptographically-secure pseudorandom numbers generated via Intel's [[RDRAND]] instruction set, as compared to those derived from algorithms, like the [[Mersenne Twister]], in Monte Carlo simulations of radio flares from [[brown dwarfs]]. RDRAND is the closest pseudorandom number generator to a true random number generator. No statistically significant difference was found between models generated with typical pseudorandom number generators and RDRAND for trials consisting of the generation of 107 random numbers.<ref>{{cite journal|last1=Route|first1=Matthew|title=Radio-flaring Ultracool Dwarf Population Synthesis|journal=The Astrophysical Journal|date=August 10, 2017|volume=845|issue=1|page=66|doi=10.3847/1538-4357/aa7ede|arxiv=1707.02212|bibcode=2017ApJ...845...66R}}</ref>

In an effort to assess the impact of random number quality on Monte Carlo simulation outcomes, astrophysical researchers tested cryptographically-secure pseudorandom numbers generated via Intel's RDRAND instruction set, as compared to those derived from algorithms, like the Mersenne Twister, in Monte Carlo simulations of radio flares from brown dwarfs. RDRAND is the closest pseudorandom number generator to a true random number generator. No statistically significant difference was found between models generated with typical pseudorandom number generators and RDRAND for trials consisting of the generation of 107 random numbers.

为了评估随机数质量对蒙特卡罗模拟结果的影响，天体物理学研究人员测试了通过 Intel 的 RDRAND 指令集生成的密码安全的伪随机数生成器，与来自算法的比较，比如在蒙特卡罗模拟褐矮星的无线电耀斑中的梅森旋转算法。是最接近真正随机数生成器的伪随机数生成器。使用典型伪随机数生成器生成的模型与产生10个 sup 7 / sup 随机数的 RDRAND 模型之间无统计学意义上的差异。

=== Monte Carlo simulation versus "what if" scenarios ===

There are ways of using probabilities that are definitely not Monte Carlo simulations – for example, deterministic modeling using single-point estimates. Each uncertain variable within a model is assigned a "best guess" estimate. Scenarios (such as best, worst, or most likely case) for each input variable are chosen and the results recorded.<ref>{{harvnb|Vose|2000|page=13}}</ref>

There are ways of using probabilities that are definitely not Monte Carlo simulations – for example, deterministic modeling using single-point estimates. Each uncertain variable within a model is assigned a "best guess" estimate. Scenarios (such as best, worst, or most likely case) for each input variable are chosen and the results recorded.

使用概率的方法肯定不是蒙特卡洛模拟——例如，使用单点估计的确定性建模。模型中的每个不确定变量都被赋予一个“最佳猜测”估计。为每个输入变量选择场景(如最佳、最差或最可能的情况)并记录结果。

By contrast, Monte Carlo simulations sample from a [[probability distribution]] for each variable to produce hundreds or thousands of possible outcomes. The results are analyzed to get probabilities of different outcomes occurring.<ref>{{harvnb|Vose|2000|page=16}}</ref> For example, a comparison of a spreadsheet cost construction model run using traditional "what if" scenarios, and then running the comparison again with Monte Carlo simulation and [[triangular distribution|triangular probability distribution]]s shows that the Monte Carlo analysis has a narrower range than the "what if" analysis.{{Examples|date=May 2012}} This is because the "what if" analysis gives equal weight to all scenarios (see [[Corporate finance#Quantifying uncertainty|quantifying uncertainty in corporate finance]]), while the Monte Carlo method hardly samples in the very low probability regions. The samples in such regions are called "rare events".

By contrast, Monte Carlo simulations sample from a probability distribution for each variable to produce hundreds or thousands of possible outcomes. The results are analyzed to get probabilities of different outcomes occurring. For example, a comparison of a spreadsheet cost construction model run using traditional "what if" scenarios, and then running the comparison again with Monte Carlo simulation and triangular probability distributions shows that the Monte Carlo analysis has a narrower range than the "what if" analysis. This is because the "what if" analysis gives equal weight to all scenarios (see quantifying uncertainty in corporate finance), while the Monte Carlo method hardly samples in the very low probability regions. The samples in such regions are called "rare events".

相比之下，蒙特卡罗模拟从概率分布中抽取每个变量的样本，产生数百或数千个可能的结果。对结果进行分析，得到不同结果发生的概率。例如，对使用传统”如果”情景运行的电子表格成本构造模型进行比较，然后再与蒙特卡罗模拟和三角概率分布进行比较，结果表明蒙特卡罗分析的范围比”如果”分析的范围窄。这是因为“如果”分析对所有情景给予了同等的权重(见量化公司融资的不确定性) ，而蒙特卡罗方法银行几乎不在非常低的概率区域进行抽样调查。这些地区的样品被称为“稀有事件”。

==Applications==

Monte Carlo methods are especially useful for simulating phenomena with significant [[uncertainty]] in inputs and systems with many [[coupling (physics)|coupled]] degrees of freedom. Areas of application include:

Monte Carlo methods are especially useful for simulating phenomena with significant uncertainty in inputs and systems with many coupled degrees of freedom. Areas of application include:

蒙特卡罗方法尤其适用于模拟输入和多自由度耦合系统中具有明显不确定性的现象。申请范围包括:

===Physical sciences===

{{Computational physics}}

{{See also|Monte Carlo method in statistical physics}}

Monte Carlo methods are very important in [[computational physics]], [[physical chemistry]], and related applied fields, and have diverse applications from complicated [[quantum chromodynamics]] calculations to designing [[heat shield]]s and [[aerodynamics|aerodynamic]] forms as well as in modeling radiation transport for radiation dosimetry calculations.<ref>{{cite journal | doi = 10.1088/0031-9155/59/4/R151 | pmid=24486639 | volume=59 | issue=4 | title=GPU-based high-performance computing for radiation therapy | journal=Physics in Medicine and Biology | pages=R151–R182|bibcode = 2014PMB....59R.151J | year=2014 | last1=Jia | first1=Xun | last2=Ziegenhein | first2=Peter | last3=Jiang | first3=Steve B | pmc=4003902 }}</ref><ref>{{cite journal | doi = 10.1088/0031-9155/59/6/R183 | volume=59 | issue=6 | title=Advances in kilovoltage x-ray beam dosimetry | journal=Physics in Medicine and Biology | pages=R183–R231|bibcode = 2014PMB....59R.183H | pmid=24584183 | date=Mar 2014| last1=Hill | first1=R | last2=Healy | first2=B | last3=Holloway | first3=L | last4=Kuncic | first4=Z | last5=Thwaites | first5=D | last6=Baldock | first6=C | url=https://semanticscholar.org/paper/fb231c3d9ade811d793b85623fd32c6ea126d5ff }}</ref><ref>{{cite journal | doi = 10.1088/0031-9155/51/13/R17 | pmid=16790908 | volume=51 | issue=13 | title=Fifty years of Monte Carlo simulations for medical physics | journal=Physics in Medicine and Biology | pages=R287–R301|bibcode = 2006PMB....51R.287R | year=2006 | last1=Rogers | first1=D W O | url=https://semanticscholar.org/paper/b6d08efc5f0818a01dc60637a4a6f8115482483e }}</ref> In [[statistical physics]] [[Monte Carlo molecular modeling]] is an alternative to computational [[molecular dynamics]], and Monte Carlo methods are used to compute [[statistical field theory|statistical field theories]] of simple particle and polymer systems.<ref name=":0" /><ref>{{harvnb|Baeurle|2009}}</ref> [[Quantum Monte Carlo]] methods solve the [[many-body problem]] for quantum systems.<ref name="kol10" /><ref name="dp13" /><ref name="dp04" /> In [[Radiation material science|radiation materials science]], the [[binary collision approximation]] for simulating [[ion implantation]] is usually based on a Monte Carlo approach to select the next colliding atom.<ref>{{Cite journal|last=Möller|first=W.|last2=Eckstein|first2=W.|date=1984-03-01|title=Tridyn — A TRIM simulation code including dynamic composition changes|journal=Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms|volume=2|issue=1|pages=814–818|doi=10.1016/0168-583X(84)90321-5|bibcode=1984NIMPB...2..814M}}</ref> In experimental [[particle physics]], Monte Carlo methods are used for designing [[particle detector|detectors]], understanding their behavior and comparing experimental data to theory. In [[astrophysics]], they are used in such diverse manners as to model both [[galaxy]] evolution<ref>{{harvnb|MacGillivray|Dodd|1982}}</ref> and microwave radiation transmission through a rough planetary surface.<ref>{{harvnb|Golden|1979}}</ref> Monte Carlo methods are also used in the [[Ensemble forecasting|ensemble models]] that form the basis of modern [[Numerical weather prediction|weather forecasting]].

Monte Carlo methods are very important in computational physics, physical chemistry, and related applied fields, and have diverse applications from complicated quantum chromodynamics calculations to designing heat shields and aerodynamic forms as well as in modeling radiation transport for radiation dosimetry calculations. In statistical physics Monte Carlo molecular modeling is an alternative to computational molecular dynamics, and Monte Carlo methods are used to compute statistical field theories of simple particle and polymer systems. Quantum Monte Carlo methods solve the many-body problem for quantum systems. In experimental particle physics, Monte Carlo methods are used for designing detectors, understanding their behavior and comparing experimental data to theory. In astrophysics, they are used in such diverse manners as to model both galaxy evolution and microwave radiation transmission through a rough planetary surface. Monte Carlo methods are also used in the ensemble models that form the basis of modern weather forecasting.

蒙特卡罗方法在计算物理学、物理化学和相关应用领域中非常重要，并且有各种各样的应用，从复杂的量子色动力学计算到设计热屏和空气动力学形式，以及辐射剂量计算的辐射传输模型。在统计物理学中，蒙特卡罗分子模拟是计算分子动力学的一种替代方法，而蒙特卡罗方法被用来计算简单粒子和聚合物体系的统计场理论。量子蒙特卡罗法方法解决了量子系统的多体问题。在实验粒子物理学中，蒙特卡罗方法被用来设计探测器，了解它们的行为，并将实验数据与理论进行比较。在天体物理学中，它们以各种不同的方式被用来模拟星系演化和微波辐射通过粗糙行星表面的传输。蒙特卡罗方法也用于构成现代天气预报基础的集合模型中。

===Engineering===

Monte Carlo methods are widely used in engineering for [[sensitivity analysis]] and quantitative [[probabilistic]] analysis in [[Process design (chemical engineering)|process design]]. The need arises from the interactive, co-linear and non-linear behavior of typical process simulations. For example,

Monte Carlo methods are widely used in engineering for sensitivity analysis and quantitative probabilistic analysis in process design. The need arises from the interactive, co-linear and non-linear behavior of typical process simulations. For example,

蒙特卡罗方法在工程中被广泛应用于敏感度分析设计和工艺设计中的定量概率分析。这种需求来源于典型过程模拟的交互性、共线性和非线性行为。比如说,

* In [[microelectronics|microelectronics engineering]], Monte Carlo methods are applied to analyze correlated and uncorrelated variations in [[Analog signal|analog]] and [[Digital data|digital]] [[integrated circuits]].

* In [[geostatistics]] and [[geometallurgy]], Monte Carlo methods underpin the design of [[mineral processing]] [[process flow diagram|flowsheets]] and contribute to quantitative [[Quantitative risk analysis|risk analysis]].<ref name="mbv01">{{Cite book | last =Mazhdrakov | first =Metodi | last2 =Benov | first2 =Dobriyan |last3=Valkanov|first3=Nikolai | year =2018 | title =The Monte Carlo Method. Engineering Applications | publisher =ACMO Academic Press | volume = | pages = 250| isbn =978-619-90684-3-4 | doi = |url=https://books.google.com/?id=t0BqDwAAQBAJ&dq=the+monte+carlo+method+engineering+applications+mazhdrakov}}</ref>

* In [[wind energy]] yield analysis, the predicted energy output of a wind farm during its lifetime is calculated giving different levels of uncertainty ([[Percentile|P90]], P50, etc.)

* impacts of pollution are simulated<ref name="IntPanis1">{{harvnb|Int Panis|De Nocker|De Vlieger|Torfs|2001}}</ref> and diesel compared with petrol.<ref name="IntPanis2">{{harvnb|Int Panis|Rabl|De Nocker|Torfs|2002}}</ref>

* In [[fluid dynamics]], in particular [[gas dynamics|rarefied gas dynamics]], where the Boltzmann equation is solved for finite [[Knudsen number]] fluid flows using the [[direct simulation Monte Carlo]] <ref>G. A. Bird, Molecular Gas Dynamics, Clarendon, Oxford (1976)</ref> method in combination with highly efficient computational algorithms.<ref>{{cite journal | last1 = Dietrich | first1 = S. | last2 = Boyd | first2 = I. | year = 1996 | title = A Scalar optimized parallel implementation of the DSMC technique | url = | journal = Journal of Computational Physics | volume = 126 | issue = 2| pages = 328–42 | doi=10.1006/jcph.1996.0141|bibcode = 1996JCoPh.126..328D }}</ref>

* In [[autonomous robotics]], [[Monte Carlo localization]] can determine the position of a robot. It is often applied to stochastic filters such as the [[Kalman filter]] or [[particle filter]] that forms the heart of the [[Simultaneous localization and mapping|SLAM]] (simultaneous localization and mapping) algorithm.

* In [[telecommunications]], when planning a wireless network, design must be proved to work for a wide variety of scenarios that depend mainly on the number of users, their locations and the services they want to use. Monte Carlo methods are typically used to generate these users and their states. The network performance is then evaluated and, if results are not satisfactory, the network design goes through an optimization process.

* In [[reliability engineering]], Monte Carlo simulation is used to compute system-level response given the component-level response. For example, for a transportation network subject to an earthquake event, Monte Carlo simulation can be used to assess the ''k''-terminal reliability of the network given the failure probability of its components, e.g. bridges, roadways, etc.<ref>{{cite journal|last=Nabian|first=Mohammad Amin|last2=Meidani|first2=Hadi|date=2017-08-28|title=Deep Learning for Accelerated Reliability Analysis of Infrastructure Networks|journal=Computer-Aided Civil and Infrastructure Engineering|volume=33|issue=6|pages=443–458|arxiv=1708.08551|doi=10.1111/mice.12359|bibcode=2017arXiv170808551N}}</ref><ref>{{Cite journal|last=Nabian|first=Mohammad Amin|last2=Meidani|first2=Hadi|date=2018|title=Accelerating Stochastic Assessment of Post-Earthquake Transportation Network Connectivity via Machine-Learning-Based Surrogates|url=https://trid.trb.org/view/1496617|journal=Transportation Research Board 97th Annual Meeting|volume=|pages=|via=}}</ref><ref>{{Cite journal|last=Nabian|first=Mohammad Amin|last2=Meidani|first2=Hadi|date=2017|title=Uncertainty Quantification and PCA-Based Model Reduction for Parallel Monte Carlo Analysis of Infrastructure System Reliability|url=https://trid.trb.org/view/1439614|journal=Transportation Research Board 96th Annual Meeting|volume=|pages=|via=}}</ref>

* In [[signal processing]] and [[Bayesian inference]], [[particle filter]]s and [[Sequential Monte Carlo method|sequential Monte Carlo techniques]] are a class of [[mean field particle methods]] for sampling and computing the posterior distribution of a signal process given some noisy and partial observations using interacting [[empirical measure]]<nowiki/>s.

===Climate change and radiative forcing===

The [[IPCC|Intergovernmental Panel on Climate Change]] relies on Monte Carlo methods in [[probability density function]] analysis of [[radiative forcing]].

The Intergovernmental Panel on Climate Change relies on Monte Carlo methods in probability density function analysis of radiative forcing.

政府间气候变化专门委员会基金会依赖于蒙特卡罗方法对概率密度函数辐射效应进行分析。

{{Quote|text=Probability density function (PDF) of ERF due to total GHG, aerosol forcing and total anthropogenic forcing. The GHG consists of WMGHG, ozone and stratospheric water vapour. The PDFs are generated based on uncertainties provided in Table 8.6. The combination of the individual RF agents to derive total forcing over the Industrial Era are done by Monte Carlo simulations and based on the method in Boucher and Haywood (2001). PDF of the ERF from surface albedo changes and combined contrails and contrail-induced cirrus are included in the total anthropogenic forcing, but not shown as a separate PDF. We currently do not have ERF estimates for some forcing mechanisms: ozone, land use, solar, etc.<ref>{{cite book|title=Climate Change 2013 The Physical Science Basis|date=2013|publisher=Cambridge University Press|isbn=978-1-107-66182-0|page=697|url=http://www.climatechange2013.org/images/report/WG1AR5_ALL_FINAL.pdf|accessdate=2 March 2016}}</ref>}}

===Computational biology===

Monte Carlo methods are used in various fields of [[computational biology]], for example for [[Bayesian inference in phylogeny]], or for studying biological systems such as genomes, proteins,<ref>{{harvnb|Ojeda|et al.|2009}},</ref> or membranes.<ref>{{harvnb|Milik|Skolnick|1993}}</ref>

Monte Carlo methods are used in various fields of computational biology, for example for Bayesian inference in phylogeny, or for studying biological systems such as genomes, proteins, or membranes.

蒙特卡罗方法被用于计算生物学的各个领域，例如在系统发育学中的贝叶斯推断，或者用于基因组、蛋白质或膜等生物系统的研究。

The systems can be studied in the coarse-grained or ''ab initio'' frameworks depending on the desired accuracy.

The systems can be studied in the coarse-grained or ab initio frameworks depending on the desired accuracy.

该系统可以在粗粒度或从头开始框架中研究，这取决于所需的准确性。

Computer simulations allow us to monitor the local environment of a particular [[biomolecule|molecule]] to see if some [[chemical reaction]] is happening for instance. In cases where it is not feasible to conduct a physical experiment, [[thought experiment]]s can be conducted (for instance: breaking bonds, introducing impurities at specific sites, changing the local/global structure, or introducing external fields).

Computer simulations allow us to monitor the local environment of a particular molecule to see if some chemical reaction is happening for instance. In cases where it is not feasible to conduct a physical experiment, thought experiments can be conducted (for instance: breaking bonds, introducing impurities at specific sites, changing the local/global structure, or introducing external fields).

计算机模拟使我们能够监测特定分子的局部环境，看看是否正在发生某种化学反应。在无法进行物理实验的情况下，可以进行思维实验(例如: 打破键，在特定位置引入杂质，改变局部 / 全球结构，或引入外部场)。

===Computer graphics===

[[Path tracing]], occasionally referred to as Monte Carlo ray tracing, renders a 3D scene by randomly tracing samples of possible light paths. Repeated sampling of any given pixel will eventually cause the average of the samples to converge on the correct solution of the [[rendering equation]], making it one of the most physically accurate 3D graphics rendering methods in existence.

Path tracing, occasionally referred to as Monte Carlo ray tracing, renders a 3D scene by randomly tracing samples of possible light paths. Repeated sampling of any given pixel will eventually cause the average of the samples to converge on the correct solution of the rendering equation, making it one of the most physically accurate 3D graphics rendering methods in existence.

路径追踪，偶尔被称为蒙特卡罗光线追踪，通过随机追踪可能的光路样本来呈现一个三维场景。对任何给定像素的重复采样最终将导致样本的平均值收敛到渲染方程的正确解，使其成为现存物理上最精确的3 d 图形渲染方法之一。

===Applied statistics===

The standards for Monte Carlo experiments in statistics were set by Sawilowsky.<ref>{{cite journal | last1 = Cassey | last2 = Smith | year = 2014 | title = Simulating confidence for the Ellison-Glaeser Index | url = | journal = Journal of Urban Economics | volume = 81 | issue = | page = 93 | doi = 10.1016/j.jue.2014.02.005}}</ref><ref>Grissom & Kim (2005)p. 131</ref><ref>Graysen Cline (2020), , Nonparametric statistical methods using R, UK: ED-Tech Press, {{ISBN|978-1-83947-325-8}}, p. 47</ref> In applied statistics, Monte Carlo methods may be used for at least four purposes:

The standards for Monte Carlo experiments in statistics were set by Sawilowsky. In applied statistics, Monte Carlo methods may be used for at least four purposes:

蒙特卡罗方法的统计标准是由 Sawilowsky 制定的。在应用统计学中，蒙特卡罗方法至少可用于四种目的:

#To compare competing statistics for small samples under realistic data conditions. Although [[type I error]] and power properties of statistics can be calculated for data drawn from classical theoretical distributions (''e.g.'', [[normal curve]], [[Cauchy distribution]]) for [[asymptotic]] conditions (''i. e'', infinite sample size and infinitesimally small treatment effect), real data often do not have such distributions.<ref>{{harvnb|Sawilowsky|Fahoome|2003}}</ref>

To compare competing statistics for small samples under realistic data conditions. Although type I error and power properties of statistics can be calculated for data drawn from classical theoretical distributions (e.g., normal curve, Cauchy distribution) for asymptotic conditions (i. e, infinite sample size and infinitesimally small treatment effect), real data often do not have such distributions.

比较在现实数据条件下小样本的竞争统计。虽然 i 型误差和统计的幂次特性可以计算从经典的理论分布(例如，正态曲线，柯西分布)的数据的渐近条件(即，无限的样本大小和无限小的处理效果) ，实际数据往往没有这样的分布。

#To provide implementations of [[Statistical hypothesis testing|hypothesis tests]] that are more efficient than exact tests such as [[permutation tests]] (which are often impossible to compute) while being more accurate than critical values for [[asymptotic distribution]]s.

To provide implementations of hypothesis tests that are more efficient than exact tests such as permutation tests (which are often impossible to compute) while being more accurate than critical values for asymptotic distributions.

提供比排列检验(通常无法计算)等精确检验更有效的假设检验的实现，同时比渐近分布的临界值更精确。

#To provide a random sample from the posterior distribution in [[Bayesian inference]]. This sample then approximates and summarizes all the essential features of the posterior.

To provide a random sample from the posterior distribution in Bayesian inference. This sample then approximates and summarizes all the essential features of the posterior.

提供一份来自后验概率贝叶斯推断的随机样本。这个样本然后估计和总结所有的基本特征后。

#To provide efficient random estimates of the Hessian matrix of the negative log-likelihood function that may be averaged to form an estimate of the [[Fisher information]] matrix.<ref>{{Cite journal |doi = 10.1198/106186005X78800|title = Monte Carlo Computation of the Fisher Information Matrix in Nonstandard Settings|journal = Journal of Computational and Graphical Statistics|volume = 14|issue = 4|pages = 889–909|year = 2005|last1 = Spall|first1 = James C.|citeseerx = 10.1.1.142.738}}</ref><ref>{{Cite journal |doi = 10.1016/j.csda.2009.09.018|title = Efficient Monte Carlo computation of Fisher information matrix using prior information|journal = Computational Statistics & Data Analysis|volume = 54|issue = 2|pages = 272–289|year = 2010|last1 = Das|first1 = Sonjoy|last2 = Spall|first2 = James C.|last3 = Ghanem|first3 = Roger}}</ref>

To provide efficient random estimates of the Hessian matrix of the negative log-likelihood function that may be averaged to form an estimate of the Fisher information matrix.

提供负对数似然函数的 Hessian 矩阵的有效的随机估计，这些估计可以平均得到费雪资讯矩阵的估计。

Monte Carlo methods are also a compromise between approximate randomization and permutation tests. An approximate [[randomization test]] is based on a specified subset of all permutations (which entails potentially enormous housekeeping of which permutations have been considered). The Monte Carlo approach is based on a specified number of randomly drawn permutations (exchanging a minor loss in precision if a permutation is drawn twice—or more frequently—for the efficiency of not having to track which permutations have already been selected).

Monte Carlo methods are also a compromise between approximate randomization and permutation tests. An approximate randomization test is based on a specified subset of all permutations (which entails potentially enormous housekeeping of which permutations have been considered). The Monte Carlo approach is based on a specified number of randomly drawn permutations (exchanging a minor loss in precision if a permutation is drawn twice—or more frequently—for the efficiency of not having to track which permutations have already been selected).

蒙特卡罗方法也是近似随机化和置换检验的折衷。近似随机化测试是基于所有排列的特定子集(这需要潜在的庞大的内务管理，其中排列已被考虑)。蒙特卡罗方法是基于一定数量的随机排列(如果排列被抽取两次或更频繁，精度会有轻微的损失，因为不必追踪哪些排列已经被选择)。

{{anchor|Monte Carlo tree search}}

===Artificial intelligence for games===

{{Main|Monte Carlo tree search}}

Monte Carlo methods have been developed into a technique called [[Monte-Carlo tree search]] that is useful for searching for the best move in a game. Possible moves are organized in a [[search tree]] and many random simulations are used to estimate the long-term potential of each move. A black box simulator represents the opponent's moves.<ref>{{cite web|url=http://sander.landofsand.com/publications/Monte-Carlo_Tree_Search_-_A_New_Framework_for_Game_AI.pdf|title=Monte-Carlo Tree Search: A New Framework for Game AI|author1=Guillaume Chaslot|author2=Sander Bakkes|author3=Istvan Szita|author4=Pieter Spronck|website=Sander.landofsand.com|accessdate=28 October 2017}}</ref>

Monte Carlo methods have been developed into a technique called Monte-Carlo tree search that is useful for searching for the best move in a game. Possible moves are organized in a search tree and many random simulations are used to estimate the long-term potential of each move. A black box simulator represents the opponent's moves.

蒙特卡罗方法已经发展成为一种叫做蒙特卡洛树搜索的技术，可以用来搜索游戏中的最佳移动。可能的移动被组织在一个搜索树和许多随机模拟被用来估计每个移动的长期潜力。一个黑盒模拟器代表对手的动作。

The Monte Carlo tree search (MCTS) method has four steps:<ref>{{cite web|url=http://mcts.ai/about/index.html|title=Monte Carlo Tree Search - About|access-date=2013-05-15|archive-url=https://web.archive.org/web/20151129023043/http://mcts.ai/about/index.html|archive-date=2015-11-29|url-status=dead}}</ref>

The Monte Carlo tree search (MCTS) method has four steps:

蒙特卡罗树搜索(Monte Carlo tree search，mct)方法有四个步骤:

#Starting at root node of the tree, select optimal child nodes until a leaf node is reached.

Starting at root node of the tree, select optimal child nodes until a leaf node is reached.

从树的根节点开始，选择最佳的子节点，直到达到叶节点。

#Expand the leaf node and choose one of its children.

Expand the leaf node and choose one of its children.

展开叶节点并选择其中一个子节点。

#Play a simulated game starting with that node.

Play a simulated game starting with that node.

以该节点开始玩一个模拟游戏。

#Use the results of that simulated game to update the node and its ancestors.

Use the results of that simulated game to update the node and its ancestors.

使用模拟游戏的结果来更新节点及其祖先。

The net effect, over the course of many simulated games, is that the value of a node representing a move will go up or down, hopefully corresponding to whether or not that node represents a good move.

The net effect, over the course of many simulated games, is that the value of a node representing a move will go up or down, hopefully corresponding to whether or not that node represents a good move.

在许多模拟游戏的过程中，净效应是代表移动的一个节点的值将上升或下降，希望与该节点是否代表一个好的移动相对应。

Monte Carlo Tree Search has been used successfully to play games such as [[Go (game)|Go]],<ref>{{cite book|title=Parallel Monte-Carlo Tree Search| doi=10.1007/978-3-540-87608-3_6|volume=5131|pages=60–71|series=Lecture Notes in Computer Science|year=2008|last1=Chaslot|first1=Guillaume M. J. -B|last2=Winands|first2=Mark H. M|last3=Van Den Herik|first3=H. Jaap|isbn=978-3-540-87607-6|citeseerx = 10.1.1.159.4373}}</ref> [[Tantrix]],<ref>{{cite report|url=https://www.tantrix.com/Tantrix/TRobot/MCTS%20Final%20Report.pdf|title=Monte-Carlo Tree Search in the game of Tantrix: Cosc490 Final Report|last=Bruns|first=Pete}}</ref> [[Battleship (game)|Battleship]],<ref>{{cite web|url=http://www0.cs.ucl.ac.uk/staff/D.Silver/web/Publications_files/pomcp.pdf|title=Monte-Carlo Planning in Large POMDPs|author1=David Silver|author2=Joel Veness|website=0.cs.ucl.ac.uk|accessdate=28 October 2017}}</ref> [[Havannah]],<ref>{{cite book|chapter=Improving Monte–Carlo Tree Search in Havannah| doi=10.1007/978-3-642-17928-0_10|volume=6515|pages=105–115|bibcode=2011LNCS.6515..105L|series=Lecture Notes in Computer Science|year=2011|last1=Lorentz|first1=Richard J|title=Computers and Games|isbn=978-3-642-17927-3}}</ref> and [[Arimaa]].<ref>{{cite web|url=http://www.arimaa.com/arimaa/papers/ThomasJakl/bc-thesis.pdf|author=Tomas Jakl|title=Arimaa challenge – comparison study of MCTS versus alpha-beta methods|website=Arimaa.com|accessdate=28 October 2017}}</ref>

Monte Carlo Tree Search has been used successfully to play games such as Go, Tantrix, Battleship, Havannah, and Arimaa.

蒙特卡洛树搜索已成功地用于游戏，如围棋，Tantrix，战舰，哈瓦那和 Arimaa。

{{See also|Computer Go}}

===Design and visuals===

Monte Carlo methods are also efficient in solving coupled integral differential equations of radiation fields and energy transport, and thus these methods have been used in [[global illumination]] computations that produce photo-realistic images of virtual 3D models, with applications in [[video game]]s, [[architecture]], [[design]], computer generated [[film]]s, and cinematic special effects.<ref>{{harvnb|Szirmay–Kalos|2008}}</ref>

Monte Carlo methods are also efficient in solving coupled integral differential equations of radiation fields and energy transport, and thus these methods have been used in global illumination computations that produce photo-realistic images of virtual 3D models, with applications in video games, architecture, design, computer generated films, and cinematic special effects.

蒙特卡罗方法在解决辐射场和能量传输的耦合积分微分方程方面也很有效，因此这些方法已经被用于全局光源计算，产生虚拟3 d 模型的照片般逼真的图像，应用于视频游戏、建筑、设计、计算机生成的电影和电影特效。

===Search and rescue===

The [[US Coast Guard]] utilizes Monte Carlo methods within its computer modeling software [[SAROPS]] in order to calculate the probable locations of vessels during [[search and rescue]] operations. Each simulation can generate as many as ten thousand data points that are randomly distributed based upon provided variables.<ref>{{cite web|url=http://insights.dice.com/2014/01/03/how-the-coast-guard-uses-analytics-to-search-for-those-lost-at-sea|title=How the Coast Guard Uses Analytics to Search for Those Lost at Sea|work=Dice Insights|date=2014-01-03}}</ref> Search patterns are then generated based upon extrapolations of these data in order to optimize the probability of containment (POC) and the probability of detection (POD), which together will equal an overall probability of success (POS). Ultimately this serves as a practical application of [[probability distribution]] in order to provide the swiftest and most expedient method of rescue, saving both lives and resources.<ref>{{cite web|url=http://www.ifremer.fr/web-com/sar2011/Presentations/SARWS2011_STONE_L.pdf|title=Search Modeling and Optimization in USCG's Search and Rescue Optimal Planning System (SAROPS)|author1=Lawrence D. Stone|author2=Thomas M. Kratzke|author3=John R. Frost|website=Ifremer.fr|accessdate=28 October 2017}}</ref>

The US Coast Guard utilizes Monte Carlo methods within its computer modeling software SAROPS in order to calculate the probable locations of vessels during search and rescue operations. Each simulation can generate as many as ten thousand data points that are randomly distributed based upon provided variables. Search patterns are then generated based upon extrapolations of these data in order to optimize the probability of containment (POC) and the probability of detection (POD), which together will equal an overall probability of success (POS). Ultimately this serves as a practical application of probability distribution in order to provide the swiftest and most expedient method of rescue, saving both lives and resources.

美国海岸警卫队在其计算机建模软件 SAROPS 中使用蒙特卡罗方法来计算搜救行动中可能的船只位置。每个模拟可以生成多达一万个数据点，这些数据点是根据提供的变量随机分布的。然后根据这些数据的推断生成搜索模式，以优化包容概率(POC)和检测概率(POD) ，这两者合起来等于总体成功概率(POS)。最终，这作为概率分布的一个实际应用，以提供最迅速和最便捷的救援方法，拯救生命和资源。

===Finance and business===

{{See also|Monte Carlo methods in finance| Quasi-Monte Carlo methods in finance| Monte Carlo methods for option pricing| Stochastic modelling (insurance) | Stochastic asset model}}

Monte Carlo simulation is commonly used to evaluate the risk and uncertainty that would affect the outcome of different decision options. Monte Carlo simulation allows the business risk analyst to incorporate the total effects of uncertainty in variables like sales volume, commodity and labour prices, interest and exchange rates, as well as the effect of distinct risk events like the cancellation of a contract or the change of a tax law.

Monte Carlo simulation is commonly used to evaluate the risk and uncertainty that would affect the outcome of different decision options. Monte Carlo simulation allows the business risk analyst to incorporate the total effects of uncertainty in variables like sales volume, commodity and labour prices, interest and exchange rates, as well as the effect of distinct risk events like the cancellation of a contract or the change of a tax law.

蒙特卡罗模拟通常用于评估影响不同决策方案结果的风险和不确定性。蒙特卡洛模拟允许商业风险分析师在销售量、商品和劳动力价格、利率和汇率等变量中考虑不确定性的总体影响，以及不同风险事件的影响，如合同的取消或税法的变化。

[[Monte Carlo methods in finance]] are often used to [[Corporate finance#Quantifying uncertainty|evaluate investments in projects]] at a business unit or corporate level, or to evaluate [[derivative (finance)|financial derivatives]]. They can be used to model [[project management|project schedules]], where simulations aggregate estimates for worst-case, best-case, and most likely durations for each task to determine outcomes for the overall project.[https://risk.octigo.pl/] Monte Carlo methods are also used in option pricing, default risk analysis.<ref>{{Cite book|title = An Introduction to Particle Methods with Financial Applications|publisher = Springer Berlin Heidelberg|journal = Numerical Methods in Finance|date = 2012|isbn = 978-3-642-25745-2|pages = 3–49|series = Springer Proceedings in Mathematics|volume = 12|first = René|last = Carmona|first2 = Pierre|last2 = Del Moral|first3 = Peng|last3 = Hu|first4 = Nadia|last4 = Oudjane|editor-first = René A.|editor-last = Carmona|editor2-first = Pierre Del|editor2-last = Moral|editor3-first = Peng|editor3-last = Hu|editor4-first = Nadia|display-editors = 3 |editor4-last = Oudjane|doi=10.1007/978-3-642-25746-9_1|citeseerx = 10.1.1.359.7957}}</ref><ref>{{Cite book |volume = 12|doi=10.1007/978-3-642-25746-9|series = Springer Proceedings in Mathematics|year = 2012|isbn = 978-3-642-25745-2|url = https://basepub.dauphine.fr/handle/123456789/11498|title=Numerical Methods in Finance|last1=Carmona|first1=René|last2=Del Moral|first2=Pierre|last3=Hu|first3=Peng|last4=Oudjane|first4=Nadia}}</ref><ref name="kr11">{{cite book|last1 = Kroese|first1 = D. P.|last2 = Taimre|first2 = T.|last3 = Botev|first3 = Z. I. |title = Handbook of Monte Carlo Methods|year = 2011|publisher = John Wiley & Sons}}</ref> Additionally, they can be used to estimate the financial impact of medical interventions.<ref>{{Cite journal |doi = 10.1371/journal.pone.0189718|pmid = 29284026|pmc = 5746244|title = A Monte Carlo simulation approach for estimating the health and economic impact of interventions provided at a student-run clinic|journal = PLOS One|volume = 12|issue = 12|pages = e0189718|year = 2017|last1 = Arenas|first1 = Daniel J.|last2 = Lett|first2 = Lanair A.|last3 = Klusaritz|first3 = Heather|last4 = Teitelman|first4 = Anne M.|bibcode = 2017PLoSO..1289718A}}</ref>

Monte Carlo methods in finance are often used to evaluate investments in projects at a business unit or corporate level, or to evaluate financial derivatives. They can be used to model project schedules, where simulations aggregate estimates for worst-case, best-case, and most likely durations for each task to determine outcomes for the overall project.[https://risk.octigo.pl/] Monte Carlo methods are also used in option pricing, default risk analysis. Additionally, they can be used to estimate the financial impact of medical interventions.

金融领域的蒙特卡罗方法通常用于评估一个业务单位或公司层面的项目投资，或评估金融衍生产品。它们可以用于对项目进度表进行建模，模拟对每个任务的最坏情况、最好情况和最可能的持续时间进行聚合估计，以确定整个项目的结果。蒙特卡罗方法也用于期权定价，违约风险分析 https://risk.octigo.pl/ 。此外，它们可以用来估计医疗干预的财务影响。

===Law===

A Monte Carlo approach was used for evaluating the potential value of a proposed program to help female petitioners in Wisconsin be successful in their applications for [[Harassment Restraining Order|harassment]] and [[Domestic Abuse Restraining Order|domestic abuse restraining orders]]. It was proposed to help women succeed in their petitions by providing them with greater advocacy thereby potentially reducing the risk of [[rape]] and [[physical assault]]. However, there were many variables in play that could not be estimated perfectly, including the effectiveness of restraining orders, the success rate of petitioners both with and without advocacy, and many others. The study ran trials that varied these variables to come up with an overall estimate of the success level of the proposed program as a whole.<ref>{{cite web|url=http://legalaidresearch.org/wp-content/uploads/Research-Increasing-Access-to-REstraining-Order-for-Low-Income-Victims-of-DV-A-Cost-Benefit-Analysis-of-the-Proposed-Domestic-Abuse-Grant-Program.pdf| title=Increasing Access to Restraining Orders for Low Income Victims of Domestic Violence: A Cost-Benefit Analysis of the Proposed Domestic Abuse Grant Program |publisher=[[State Bar of Wisconsin]] |date=December 2006 |accessdate=2016-12-12|last1=Elwart|first1=Liz|last2=Emerson|first2=Nina|last3=Enders|first3=Christina|last4=Fumia|first4=Dani|last5=Murphy|first5=Kevin}}</ref>

A Monte Carlo approach was used for evaluating the potential value of a proposed program to help female petitioners in Wisconsin be successful in their applications for harassment and domestic abuse restraining orders. It was proposed to help women succeed in their petitions by providing them with greater advocacy thereby potentially reducing the risk of rape and physical assault. However, there were many variables in play that could not be estimated perfectly, including the effectiveness of restraining orders, the success rate of petitioners both with and without advocacy, and many others. The study ran trials that varied these variables to come up with an overall estimate of the success level of the proposed program as a whole.

蒙特卡洛方法被用来评估一个拟议的方案的潜在价值，以帮助威斯康星州的女性请愿者成功地申请骚扰和家庭虐待限制令。提议帮助妇女成功地提出请愿，向她们提供更多的宣传，从而有可能减少强奸和人身攻击的风险。然而，还有很多变量无法完全估计，包括限制令的有效性，上访者的成功率，无论有没有主张，以及许多其他因素。这项研究通过改变这些变量进行了试验，得出了对整个计划成功程度的总体评估。

==Use in mathematics==

In general, the Monte Carlo methods are used in mathematics to solve various problems by generating suitable random numbers (see also [[Random number generation]]) and observing that fraction of the numbers that obeys some property or properties. The method is useful for obtaining numerical solutions to problems too complicated to solve analytically. The most common application of the Monte Carlo method is Monte Carlo integration.

In general, the Monte Carlo methods are used in mathematics to solve various problems by generating suitable random numbers (see also Random number generation) and observing that fraction of the numbers that obeys some property or properties. The method is useful for obtaining numerical solutions to problems too complicated to solve analytically. The most common application of the Monte Carlo method is Monte Carlo integration.

一般来说，蒙特卡罗方法在数学中通过产生合适的随机数(也见随机数产生)和观察符合某些性质的数字分数来解决各种问题。这种方法对于求解解析求解过于复杂的问题的数值解是有用的。蒙特卡罗方法最常用的应用是蒙地卡罗积分。

=== Integration ===

{{Main|Monte Carlo integration}}

[[File:Monte-carlo2.gif|thumb|Monte-Carlo integration works by comparing random points with the value of the function]]

Monte-Carlo integration works by comparing random points with the value of the function

蒙特卡罗积分是通过比较随机点和函数值来工作的

[[File:Monte-Carlo method (errors).png|thumb|Errors reduce by a factor of <math>\scriptstyle 1/\sqrt{N}</math>]]

Errors reduce by a factor of <math>\scriptstyle 1/\sqrt{N}</math>

错误减少一个数学 scriptstyle 1 / sqrt { n } / math 因子

Deterministic [[numerical integration]] algorithms work well in a small number of dimensions, but encounter two problems when the functions have many variables. First, the number of function evaluations needed increases rapidly with the number of dimensions. For example, if 10 evaluations provide adequate accuracy in one dimension, then [[googol|10100]] points are needed for 100 dimensions—far too many to be computed. This is called the [[curse of dimensionality]]. Second, the boundary of a multidimensional region may be very complicated, so it may not be feasible to reduce the problem to an [[iterated integral]].<ref name=Press>{{harvnb|Press|Teukolsky|Vetterling|Flannery|1996}}</ref> 100 [[dimension]]s is by no means unusual, since in many physical problems, a "dimension" is equivalent to a [[degrees of freedom (physics and chemistry)|degree of freedom]].

Deterministic numerical integration algorithms work well in a small number of dimensions, but encounter two problems when the functions have many variables. First, the number of function evaluations needed increases rapidly with the number of dimensions. For example, if 10 evaluations provide adequate accuracy in one dimension, then 10100 points are needed for 100 dimensions—far too many to be computed. This is called the curse of dimensionality. Second, the boundary of a multidimensional region may be very complicated, so it may not be feasible to reduce the problem to an iterated integral. 100 dimensions is by no means unusual, since in many physical problems, a "dimension" is equivalent to a degree of freedom.

确定性数值积分算法在少数维上运行良好，但在函数具有多个变量时会遇到两个问题。首先，随着维数的增加，需要进行的功能评估的数量迅速增加。例如，如果10个评估在一个维度上提供了足够的精度，那么100个维度需要10个支持100 / 支持点，这个数字太多了，无法计算。这就是所谓的维数灾难。其次，多维区域的边界可能非常复杂，因此将问题简化为迭代积分可能是不可行的。100维绝对不是不寻常的，因为在许多物理问题中，一个“维度”等同于一个自由度。

Monte Carlo methods provide a way out of this exponential increase in computation time. As long as the function in question is reasonably [[well-behaved]], it can be estimated by randomly selecting points in 100-dimensional space, and taking some kind of average of the function values at these points. By the [[central limit theorem]], this method displays <math>\scriptstyle 1/\sqrt{N}</math> convergence—i.e., quadrupling the number of sampled points halves the error, regardless of the number of dimensions.<ref name=Press/>

Monte Carlo methods provide a way out of this exponential increase in computation time. As long as the function in question is reasonably well-behaved, it can be estimated by randomly selecting points in 100-dimensional space, and taking some kind of average of the function values at these points. By the central limit theorem, this method displays <math>\scriptstyle 1/\sqrt{N}</math> convergence—i.e., quadrupling the number of sampled points halves the error, regardless of the number of dimensions.

蒙特卡罗方法提供了一种方法来摆脱这种指数增长的计算时间。只要所涉及的函数具有合理的性质，就可以在100维空间中随机选取一些点，并在这些点上取某种函数值的平均值来估计。通过中心极限定理，这个方法显示了 math scriptstyle 1 / sqrt { n } / math convergence---- 也就是说，不管维数多少，将采样点的数量翻两番，误差减半。

A refinement of this method, known as [[importance sampling]] in statistics, involves sampling the points randomly, but more frequently where the integrand is large. To do this precisely one would have to already know the integral, but one can approximate the integral by an integral of a similar function or use adaptive routines such as [[stratified sampling]], [[Monte Carlo integration#Recursive stratified sampling|recursive stratified sampling]], adaptive umbrella sampling<ref>{{cite journal|last=MEZEI|first=M|title=Adaptive umbrella sampling: Self-consistent determination of the non-Boltzmann bias|journal=Journal of Computational Physics|date=31 December 1986|volume=68|issue=1|pages=237–248|doi=10.1016/0021-9991(87)90054-4|bibcode = 1987JCoPh..68..237M}}</ref><ref>{{cite journal|last=Bartels|first=Christian|last2=Karplus|first2=Martin|title=Probability Distributions for Complex Systems: Adaptive Umbrella Sampling of the Potential Energy|journal=The Journal of Physical Chemistry B|date=31 December 1997|volume=102|issue=5|pages=865–880|doi=10.1021/jp972280j}}</ref> or the [[VEGAS algorithm]].

A refinement of this method, known as importance sampling in statistics, involves sampling the points randomly, but more frequently where the integrand is large. To do this precisely one would have to already know the integral, but one can approximate the integral by an integral of a similar function or use adaptive routines such as stratified sampling, recursive stratified sampling, adaptive umbrella sampling or the VEGAS algorithm.

这种方法的一个改进，称为统计学中的重要抽样，包括随机抽样点，但更多的是在被积函数很大的情况下。要做到这一点，你必须已经知道这个积分，但是你可以用类似函数的积分来近似这个积分，或者使用自适应例程，例如分层抽样，递归分层抽样，自适应伞抽样或者 VEGAS 算法。

A similar approach, the [[quasi-Monte Carlo method]], uses [[low-discrepancy sequence]]s. These sequences "fill" the area better and sample the most important points more frequently, so quasi-Monte Carlo methods can often converge on the integral more quickly.

A similar approach, the quasi-Monte Carlo method, uses low-discrepancy sequences. These sequences "fill" the area better and sample the most important points more frequently, so quasi-Monte Carlo methods can often converge on the integral more quickly.

一个类似的方法，拟蒙特卡罗方法，使用低差异序列。这些序列能更好地“填充”区域，更频繁地采样最重要的点，因此拟蒙特卡罗方法往往能更快地收敛于积分。

Another class of methods for sampling points in a volume is to simulate random walks over it ([[Markov chain Monte Carlo]]). Such methods include the [[Metropolis–Hastings algorithm]], [[Gibbs sampling]], [[Wang and Landau algorithm]], and interacting type MCMC methodologies such as the [[Particle filter|sequential Monte Carlo]] samplers.<ref>{{Cite journal|title = Sequential Monte Carlo samplers - Del Moral - Doucet - Jasra- 2006 - Journal of the Royal Statistical Society: Series B (Statistical Methodology) - Wiley Online Library|journal = Journal of the Royal Statistical Society, Series B|doi=10.1111/j.1467-9868.2006.00553.x|volume=68|issue = 3|pages=411–436|year = 2006|last1 = Del Moral|first1 = Pierre|last2 = Doucet|first2 = Arnaud|last3 = Jasra|first3 = Ajay|arxiv = cond-mat/0212648}}</ref>

Another class of methods for sampling points in a volume is to simulate random walks over it (Markov chain Monte Carlo). Such methods include the Metropolis–Hastings algorithm, Gibbs sampling, Wang and Landau algorithm, and interacting type MCMC methodologies such as the sequential Monte Carlo samplers.

另一类方法是模拟体积上的随机游动(马尔科夫蒙特卡洛)。这些方法包括 Metropolis-Hastings 算法、 Gibbs 抽样、 Wang 和 Landau 算法以及交互式 MCMC 方法，如序贯蒙特卡罗抽样。

=== Simulation and optimization ===

{{Main|Stochastic optimization}}

Another powerful and very popular application for random numbers in numerical simulation is in [[Optimization (mathematics)|numerical optimization]]. The problem is to minimize (or maximize) functions of some vector that often has many dimensions. Many problems can be phrased in this way: for example, a [[computer chess]] program could be seen as trying to find the set of, say, 10 moves that produces the best evaluation function at the end. In the [[traveling salesman problem]] the goal is to minimize distance traveled. There are also applications to engineering design, such as [[multidisciplinary design optimization]]. It has been applied with quasi-one-dimensional models to solve particle dynamics problems by efficiently exploring large configuration space. Reference <ref>Spall, J. C. (2003), ''Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control'', Wiley, Hoboken, NJ. http://www.jhuapl.edu/ISSO</ref> is a comprehensive review of many issues related to simulation and optimization.

Another powerful and very popular application for random numbers in numerical simulation is in numerical optimization. The problem is to minimize (or maximize) functions of some vector that often has many dimensions. Many problems can be phrased in this way: for example, a computer chess program could be seen as trying to find the set of, say, 10 moves that produces the best evaluation function at the end. In the traveling salesman problem the goal is to minimize distance traveled. There are also applications to engineering design, such as multidisciplinary design optimization. It has been applied with quasi-one-dimensional models to solve particle dynamics problems by efficiently exploring large configuration space. Reference is a comprehensive review of many issues related to simulation and optimization.

另一个强大的和非常流行的应用随机数在数值模拟是在数值优化。问题在于如何最小化(或最大化)某些向量的函数，这些向量通常具有多个维度。许多问题可以这样表述: 例如，一个计算机国际象棋程序可以被视为试图找到一组，比如说，10步棋，最终产生最好的评价函数。在旅行商问题中，目标是使旅行距离最小。在工程设计中也有一些应用，如多学科设计优化。它已被应用于准一维模型，以解决粒子动力学问题，有效地探索大型位形空间。参考文献是对许多与模拟和优化有关的问题的全面回顾。

The [[traveling salesman problem]] is what is called a conventional optimization problem. That is, all the facts (distances between each destination point) needed to determine the optimal path to follow are known with certainty and the goal is to run through the possible travel choices to come up with the one with the lowest total distance. However, let's assume that instead of wanting to minimize the total distance traveled to visit each desired destination, we wanted to minimize the total time needed to reach each destination. This goes beyond conventional optimization since travel time is inherently uncertain (traffic jams, time of day, etc.). As a result, to determine our optimal path we would want to use simulation - optimization to first understand the range of potential times it could take to go from one point to another (represented by a probability distribution in this case rather than a specific distance) and then optimize our travel decisions to identify the best path to follow taking that uncertainty into account.

The traveling salesman problem is what is called a conventional optimization problem. That is, all the facts (distances between each destination point) needed to determine the optimal path to follow are known with certainty and the goal is to run through the possible travel choices to come up with the one with the lowest total distance. However, let's assume that instead of wanting to minimize the total distance traveled to visit each desired destination, we wanted to minimize the total time needed to reach each destination. This goes beyond conventional optimization since travel time is inherently uncertain (traffic jams, time of day, etc.). As a result, to determine our optimal path we would want to use simulation - optimization to first understand the range of potential times it could take to go from one point to another (represented by a probability distribution in this case rather than a specific distance) and then optimize our travel decisions to identify the best path to follow taking that uncertainty into account.

旅行推销员问题被称为传统的最佳化问题问题。也就是说，确定最佳路径所需的所有事实(每个目的地之间的距离)都是确定无疑的，目标是通过可能的旅行选择得出总距离最小的路径。然而，让我们假设，我们不想最小化访问每个想要的目的地所需的总距离，而是想最小化到达每个目的地所需的总时间。这超越了传统的优化，因为旅行时间是固有的不确定性(交通堵塞，一天的时间，等)。因此，为了确定我们的最佳路径，我们需要使用模拟优化来首先了解从一个点到另一个点所需的潜在时间范围(在这个例子中用概率分布表示，而不是特定的距离) ，然后优化我们的旅行决策，以确定最佳路径遵循考虑到这种不确定性。

===Inverse problems===

Probabilistic formulation of [[inverse problem]]s leads to the definition of a [[probability distribution]] in the model space. This probability distribution combines [[prior probability|prior]] information with new information obtained by measuring some observable parameters (data).

Probabilistic formulation of inverse problems leads to the definition of a probability distribution in the model space. This probability distribution combines prior information with new information obtained by measuring some observable parameters (data).

反问题的概率公式导致了模型空间中概率分布的定义。该概率分布将先前的信息与通过测量一些可观测的参数(数据)获得的新信息结合起来。

As, in the general case, the theory linking data with model parameters is nonlinear, the posterior probability in the model space may not be easy to describe (it may be multimodal, some moments may not be defined, etc.).

As, in the general case, the theory linking data with model parameters is nonlinear, the posterior probability in the model space may not be easy to describe (it may be multimodal, some moments may not be defined, etc.).

因为，在一般情况下，连接数据和模型参数的理论是非线性的，模型空间中的后验概率可能不容易描述(它可能是多模态的，一些矩可能没有定义，等等。).

When analyzing an inverse problem, obtaining a maximum likelihood model is usually not sufficient, as we normally also wish to have information on the resolution power of the data. In the general case we may have many model parameters, and an inspection of the [[marginal probability]] densities of interest may be impractical, or even useless. But it is possible to pseudorandomly generate a large collection of models according to the [[posterior probability distribution]] and to analyze and display the models in such a way that information on the relative likelihoods of model properties is conveyed to the spectator. This can be accomplished by means of an efficient Monte Carlo method, even in cases where no explicit formula for the ''a priori'' distribution is available.

When analyzing an inverse problem, obtaining a maximum likelihood model is usually not sufficient, as we normally also wish to have information on the resolution power of the data. In the general case we may have many model parameters, and an inspection of the marginal probability densities of interest may be impractical, or even useless. But it is possible to pseudorandomly generate a large collection of models according to the posterior probability distribution and to analyze and display the models in such a way that information on the relative likelihoods of model properties is conveyed to the spectator. This can be accomplished by means of an efficient Monte Carlo method, even in cases where no explicit formula for the a priori distribution is available.

当分析一个反问题时，获得一个最大似然模型通常是不够的，因为我们通常也希望有关于数据的分辨率的信息。在一般情况下，我们可能有许多模型参数，检查的边际概率密度的兴趣可能是不切实际的，甚至无用的。但是，根据美国后验概率协会，可以伪随机地生成大量的模型集合，并以这样一种方式分析和显示模型，模型属性的相对可能性信息被传达给观众。这可以通过一个有效的蒙特卡罗方法安全委员会来实现，即使在没有黎曼显式公式安全先验概率的情况下也是如此。

The best-known importance sampling method, the Metropolis algorithm, can be generalized, and this gives a method that allows analysis of (possibly highly nonlinear) inverse problems with complex ''a priori'' information and data with an arbitrary noise distribution.<ref>{{harvnb|Mosegaard|Tarantola|1995}}</ref><ref>{{harvnb|Tarantola|2005}}</ref>

The best-known importance sampling method, the Metropolis algorithm, can be generalized, and this gives a method that allows analysis of (possibly highly nonlinear) inverse problems with complex a priori information and data with an arbitrary noise distribution.

最著名的重要性抽样方法，Metropolis–Hastings 演算法，可以推广，这提供了一种方法，允许分析(可能是高度非线性)与复杂的先验信息和数据与任意噪声分布的反问题。

== See also ==

{{Portal|Mathematics}}

{{Div col|colwidth=30em}}

* [[Auxiliary field Monte Carlo]]

* [[Biology Monte Carlo method]]

* [[Comparison of risk analysis Microsoft Excel add-ins]]

* [[Direct simulation Monte Carlo]]

* [[Dynamic Monte Carlo method]]

* [[Genetic algorithms]]

* [[Kinetic Monte Carlo]]

* [[List of software for Monte Carlo molecular modeling]]

* [[Mean field particle methods]]

* [[Monte Carlo method for photon transport]]

* [[Monte Carlo methods for electron transport]]

* [[Morris method]]

* [[Multilevel Monte Carlo method]]

* [[Particle filter]]

* [[Quasi-Monte Carlo method]]

* [[Sobol sequence]]

* [[Temporal difference learning]]

{{div col end}}

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* {{cite journal |last = Metropolis |first = N. |author-link = Nicholas Metropolis |url = http://library.lanl.gov/cgi-bin/getfile?15-12.pdf |title=The beginning of the Monte Carlo method |journal = Los Alamos Science |issue=1987 Special Issue dedicated to Stanislaw Ulam |pages=125–130 |year=1987 }}

* {{cite journal |last1 = Metropolis |first1 = N. |author1-link = Nicholas Metropolis |last2=Rosenbluth |first2=Arianna W.|last3=Rosenbluth |first3=Marshall N. |last4=Teller |first4=Augusta H. |last5=Teller |first5=Edward |year=1953 |title=Equation of State Calculations by Fast Computing Machines |journal=Journal of Chemical Physics |volume=21 |issue=6 |page=1087 |doi = 10.1063/1.1699114 |url =

|bibcode = 1953JChPh..21.1087M |title-link = Equation of State Calculations by Fast Computing Machines }}

|bibcode = 1953JChPh..21.1087M |title-link = Equation of State Calculations by Fast Computing Machines }}

| bibcode 1953JChPh. . 21.1087 m | title-link / 状态方程计算由快速计算机计算}

* {{cite journal |last1 = Metropolis |first1 = N. |author1-link = Nicholas Metropolis |last2 = Ulam |first2 = S. |authorlink2=Stanislaw Ulam |year=1949 |title = The Monte Carlo Method |journal = Journal of the American Statistical Association |volume=44 |issue=247 |pages=335–341 |doi = 10.2307/2280232 |pmid=18139350 |jstor=2280232 }}

* {{cite journal |doi = 10.1002/prot.340150104 |title = Insertion of peptide chains into lipid membranes: an off-lattice Monte Carlo dynamics model |first1 = M. |last1 = Milik |first2 = J. |last2 = Skolnick |journal = Proteins |volume = 15 |issue = 1 |pages = 10–25 |date=Jan 1993 |pmid = 8451235 |url = https://semanticscholar.org/paper/793bd7ab0e505ef5d12ed2f0798b22675e088407 }}

* {{cite journal |last1 = Mosegaard |first1 = Klaus |last2 = Tarantola |first2 = Albert |year = 1995 |title = Monte Carlo sampling of solutions to inverse problems |journal = J. Geophys. Res. |volume = 100 |number =B7 |pages = 12431–12447 |doi = 10.1029/94JB03097 |url = http://www.math.pitt.edu/~cbsg/Materials/MonteCarlo_latex.pdf |bibcode = 1995JGR...10012431M }}

* {{cite journal |title = Monte Carlo Simulations of Proteins in Cages: Influence of Confinement on the Stability of Intermediate States |author=P. Ojeda |author2=M. Garcia |author3=A. Londono |author4=N.Y. Chen |journal = Biophys. J. |volume = 96 |issue = 3 |pages = 1076–1082 |date=Feb 2009 |doi = 10.1529/biophysj.107.125369 |pmid = 18849410 |pmc = 2716574 |bibcode = 2009BpJ....96.1076O }}

* {{cite journal |doi = 10.1504/IJVD.2001.001963 |last1 = Int Panis |first1 = L. |last2 = de Nocker |first2 = L. |last3 = De Vlieger |first3 = I. |last4= Torfs |first4= R. |year=2001 |title = Trends and uncertainty in air pollution impacts and external costs of Belgian passenger car traffic|journal= International Journal of Vehicle Design|volume = 27 |issue = 1–4 |pages = 183–194 }}

* {{cite journal |last1 = Int Panis |first1 = L. |last2 = Rabl |first2 = A. |last3 = de Nocker |first3 = L. |last4 = Torfs |first4 = R. |year = 2002 |title = Diesel or Petrol ? An environmental comparison hampered by uncertainty |journal = Mitteilungen Institut für Verbrennungskraftmaschinen und Thermodynamik |editor-first = P. |editor-last = Sturm |publisher = Technische Universität Graz Austria |volume = Heft 81 Vol 1 |pages = 48–54 }}

* {{cite book |last1 = Press |first1 = William H. |last2 = Teukolsky |first2 = Saul A. |last3 = Vetterling |first3 = William T. |last4 = Flannery |first4 = Brian P. |title = Numerical Recipes in Fortran 77: The Art of Scientific Computing |edition = 2nd |series = Fortran Numerical Recipes |volume = 1 |year = 1996 |orig-year = 1986 |publisher = [[Cambridge University Press]] |isbn=978-0-521-43064-7 }}

* {{cite book |last = Ripley |first = B. D. |title = Stochastic Simulation |publisher = [[Wiley & Sons]] |year=1987 }}

* {{cite book |title = Monte Carlo Statistical Methods |last1=Robert |first1 = C. |last2 = Casella |first2 = G. |year=2004 |edition=2nd |publisher=Springer |location=New York |isbn=978-0-387-21239-5 |url =https://archive.org/details/springer_10.1007-978-1-4757-4145-2}}

* {{cite book |title = Simulation and the Monte Carlo Method |last1 = Rubinstein |first1 = R. Y. |last2 = Kroese |first2 = D. P. |year=2007 |edition=2nd |publisher = John Wiley & Sons |location=New York |isbn = 978-0-470-17793-8 |url = }}

* {{cite journal |last = Savvides |first = Savvakis C. |title = Risk Analysis in Investment Appraisal |journal= Project Appraisal Journal |year= 1994 |volume = 9 |issue= 1 |doi = 10.2139/ssrn.265905 |url = https://mpra.ub.uni-muenchen.de/10035/1/MPRA_paper_10035.pdf }}

* {{cite book |last1 = Sawilowsky |first1 = Shlomo S. |last2 = Fahoome |first2 = Gail C. |year = 2003 |title = Statistics via Monte Carlo Simulation with Fortran |location = Rochester Hills, MI |publisher = JMASM |isbn=978-0-9740236-0-1 }}

* {{cite journal |last = Sawilowsky |first = Shlomo S. |title = You think you've got trivials? |journal=[[Journal of Modern Applied Statistical Methods]] |volume=2 |issue=1 |pages=218–225 |year=2003 |url = https://digitalcommons.wayne.edu/cgi/viewcontent.cgi?article=1744&context=jmasm |doi = 10.22237/jmasm/1051748460 |doi-access = free }}

* {{cite conference |conference = Neural Information Processing Systems 2010 |last1 = Silver |first1 = David |last2 = Veness |first2 = Joel |year = 2010 |title = Monte-Carlo Planning in Large POMDPs |url = http://books.nips.cc/papers/files/nips23/NIPS2010_0740.pdf |editor1-last = Lafferty |editor1-first = J. |editor2-last = Williams |editor2-first = C. K. I. |editor3-last = Shawe-Taylor |editor3-first = J. |editor4-last = Zemel |editor4-first = R. S. |editor5-last = Culotta |editor5-first = A. |book-title = Advances in Neural Information Processing Systems 23 |publisher = Neural Information Processing Systems Foundation }}

* {{cite book |first = László |last = Szirmay-Kalos |title = Monte Carlo Methods in Global Illumination - Photo-realistic Rendering with Randomization |publisher = VDM Verlag Dr. Mueller e.K. |year = 2008 |isbn = 978-3-8364-7919-6 }}

* {{cite book |title = Inverse Problem Theory |last = Tarantola |first = Albert |author-link = Albert Tarantola |year = 2005 |publisher = Society for Industrial and Applied Mathematics |location = Philadelphia |isbn = 978-0-89871-572-9 |url = http://www.ipgp.jussieu.fr/~tarantola/Files/Professional/SIAM/index.html }}

* {{cite book |first = David |last = Vose |title = Risk Analysis, A Quantitative Guide |edition = 3rd |publisher =[[John Wiley & Sons]] |year = 2008 }}

* {{cite book |last1 = Mazhdrakov |first1 = Metodi |last2 = Benov |first2 = Dobriyan |last3 = Valkanov |first3 = Nikolai |year = 2018 |title = The Monte Carlo Method. Engineering Applications |publisher = ACMO Academic Press |page = 250 |isbn = 978-619-90684-3-4 |url = https://books.google.com/?id=t0BqDwAAQBAJ&dq=the+monte+carlo+method+engineering+applications+mazhdrakov }}

{{refend}}

== External links ==

* {{Commons category-inline|Monte Carlo method}}



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