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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.

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，它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。<ref name=":45" />

蒙特卡罗方法已经发展成为一种称作'''蒙特卡洛树搜索 Monte-Carlo tree search'''的技术，它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。<ref name=":45" />

The Monte Carlo tree search (MCTS) method has four steps:<ref name=":46">{{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:<ref name=":46">{{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 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.

Monte Carlo Tree Search has been used successfully to play games such as [[Go (game)|Go]],<ref name=":47">Chaslot, Guillaume M. J. -B; Winands, Mark H. M; Van Den Herik, H. Jaap (2008). ''Parallel Monte-Carlo Tree Search''. Lecture Notes in Computer Science. '''5131'''. pp. 60–71. CiteSeerX 10.1.1.159.4373. doi:10.1007/978-3-540-87608-3_6. ISBN <bdi>978-3-540-87607-6</bdi>.</ref> [[Tantrix]],<ref name=":48">Bruns, Pete. Monte-Carlo Tree Search in the game of Tantrix: Cosc490 Final Report (PDF) (Report).</ref> [[Battleship (game)|Battleship]],<ref name=":49">David Silver; Joel Veness. "Monte-Carlo Planning in Large POMDPs" (PDF). ''0.cs.ucl.ac.uk''. Retrieved 28 October 2017.</ref> [[Havannah]],<ref name=":50">Lorentz, Richard J (2011). "Improving Monte–Carlo Tree Search in Havannah". ''Computers and Games''. Lecture Notes in Computer Science. '''6515'''. pp. 105–115. Bibcode:2011LNCS.6515..105L. doi:10.1007/978-3-642-17928-0_10. ISBN <bdi>978-3-642-17927-3</bdi>.</ref> and [[Arimaa]].<ref name=":51">Tomas Jakl. "Arimaa challenge – comparison study of MCTS versus alpha-beta methods" (PDF). ''Arimaa.com''. Retrieved 28 October 2017.</ref>

Monte Carlo Tree Search has been used successfully to play games such as [[Go (game)|Go]],<ref name=":47">Chaslot, Guillaume M. J. -B; Winands, Mark H. M; Van Den Herik, H. Jaap (2008). ''Parallel Monte-Carlo Tree Search''. Lecture Notes in Computer Science. '''5131'''. pp. 60–71. CiteSeerX 10.1.1.159.4373. doi:10.1007/978-3-540-87608-3_6. ISBN <bdi>978-3-540-87607-6</bdi>.</ref> [[Tantrix]],<ref name=":48">Bruns, Pete. Monte-Carlo Tree Search in the game of Tantrix: Cosc490 Final Report (PDF) (Report).</ref> [[Battleship (game)|Battleship]],<ref name=":49">David Silver; Joel Veness. "Monte-Carlo Planning in Large POMDPs" (PDF). ''0.cs.ucl.ac.uk''. Retrieved 28 October 2017.</ref> [[Havannah]],<ref name=":50">Lorentz, Richard J (2011). "Improving Monte–Carlo Tree Search in Havannah". ''Computers and Games''. Lecture Notes in Computer Science. '''6515'''. pp. 105–115. Bibcode:2011LNCS.6515..105L. doi:10.1007/978-3-642-17928-0_10. ISBN <bdi>978-3-642-17927-3</bdi>.</ref> and [[Arimaa]].<ref name=":51">Tomas Jakl. "Arimaa challenge – comparison study of MCTS versus alpha-beta methods" (PDF). ''Arimaa.com''. Retrieved 28 October 2017.</ref>

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

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，<ref name=":47" /> Tantrix，<ref name=":48" /> 战舰，<ref name=":49" /> Havannah，<ref name=":50" />和 Arimaa。<ref name=":51" />

蒙特卡洛树搜索已成功地用于游戏，如围棋，<ref name=":47" /> 《彩虹棋》，<ref name=":48" /> 《海战棋》，<ref name=":49" /> 《三宝棋》，<ref name=":50" />和印度斗兽棋。<ref name=":51" />

{{See also|Computer Go}}

{{See also|Computer Go}}

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.

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。<ref name=":53" />

蒙特卡罗方法在求解辐射场和能量传输耦合积分微分方程方面也很有效，因此这些方法已用于全局光照计算，生成逼真的虚拟三维模型图像，并应用于电子游戏、建筑、设计、计算机生成电影、以及电影特效。<ref name=":53" />

===Search and rescue 搜寻与救援===

===Search and rescue 搜寻与救援===

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.

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，以便在搜索和救援行动中计算可能的船只位置。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。<ref name=":54" /> 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。<ref name=":55" />

美国海岸警卫队在其计算机建模软件—'''搜救最优规划系统 Search and Rescue Optimal Planning System (SAROPS)''' 中使用蒙特卡罗方法，以便在搜索和救援行动中计算可能的船只位置。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。<ref name=":54" /> 然后根据这些数据推断生成搜索模式，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，作为概率分布的一个实际应用，以最迅速和最便捷的救援方法，拯救生命和资源。<ref name=":55" />

===Finance and business 金融与商业===

===Finance and business 金融与商业===

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.

{{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}}

{{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 methods in finance]] are often used to [[Corporate finance#Quantifying uncertainty|evaluate investments in projects]] at a business unit or corporate level, or other financial valuations. 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 name=":56">Carmona, René; Del Moral, Pierre; Hu, Peng; Oudjane, Nadia (2012). Carmona, René A.; Moral, Pierre Del; Hu, Peng; et al. (eds.). ''An Introduction to Particle Methods with Financial Applications''. ''Numerical Methods in Finance''. Springer Proceedings in Mathematics. '''12'''. Springer Berlin Heidelberg. pp. 3–49. CiteSeerX 10.1.1.359.7957. doi:10.1007/978-3-642-25746-9_1. ISBN <bdi>978-3-642-25745-2</bdi>.</ref><ref name=":57">Carmona, René; Del Moral, Pierre; Hu, Peng; Oudjane, Nadia (2012). ''Numerical Methods in Finance''. Springer Proceedings in Mathematics. '''12'''. doi:10.1007/978-3-642-25746-9. ISBN <bdi>978-3-642-25745-2</bdi>.</ref><ref name="kr11">Kroese, D. P.; Taimre, T.; Botev, Z. I. (2011). ''Handbook of Monte Carlo Methods''. John Wiley & Sons.</ref> Additionally, they can be used to estimate the financial impact of medical interventions.<ref name=":58">Arenas, Daniel J.; Lett, Lanair A.; Klusaritz, Heather; Teitelman, Anne M. (2017). "A Monte Carlo simulation approach for estimating the health and economic impact of interventions provided at a student-run clinic". ''PLOS ONE''. '''12''' (12): e0189718. Bibcode:2017PLoSO..1289718A. doi:10.1371/journal.pone.0189718. PMC 5746244. <nowiki>PMID 29284026</nowiki>.</ref>

[[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 other financial valuations. 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 name=":56">Carmona, René; Del Moral, Pierre; Hu, Peng; Oudjane, Nadia (2012). Carmona, René A.; Moral, Pierre Del; Hu, Peng; et al. (eds.). ''An Introduction to Particle Methods with Financial Applications''. ''Numerical Methods in Finance''. Springer Proceedings in Mathematics. '''12'''. Springer Berlin Heidelberg. pp. 3–49. CiteSeerX 10.1.1.359.7957. doi:10.1007/978-3-642-25746-9_1. ISBN <bdi>978-3-642-25745-2</bdi>.</ref><ref name=":57">Carmona, René; Del Moral, Pierre; Hu, Peng; Oudjane, Nadia (2012). ''Numerical Methods in Finance''. Springer Proceedings in Mathematics. '''12'''. doi:10.1007/978-3-642-25746-9. ISBN <bdi>978-3-642-25745-2</bdi>.</ref><ref name="kr11">Kroese, D. P.; Taimre, T.; Botev, Z. I. (2011). ''Handbook of Monte Carlo Methods''. John Wiley & Sons.</ref> Additionally, they can be used to estimate the financial impact of medical interventions.<ref name=":58">Arenas, Daniel J.; Lett, Lanair A.; Klusaritz, Heather; Teitelman, Anne M. (2017). "A Monte Carlo simulation approach for estimating the health and economic impact of interventions provided at a student-run clinic". ''PLOS ONE''. '''12''' (12): e0189718. Bibcode:2017PLoSO..1289718A. doi:10.1371/journal.pone.0189718. PMC 5746244. <nowiki>PMID 29284026</nowiki>.</ref>

蒙特卡罗方法在金融经常被用于评估在一个业务单位或公司层面的项目投资，或其他金融估值。它们可以用来模拟项目进度，其中模拟汇总了对最坏情况、最好情况和每个任务最可能持续时间的估计，以确定整个项目的结果[https://risk.octigo.pl/]蒙特卡罗方法也被用于期权定价，违约风险分析。<ref name=":56" /><ref name=":57" /><ref name="kr11" /> 此外，它们还可以用来估计医疗干预的财务影响。<ref name=":58" />

在金融领域中，蒙特卡罗方法经常被用于评估业务单位或公司层面的项目投资，或其他金融估值。它们可以用来模拟项目进度，其中模拟汇总了对最坏情况、最好情况和每个任务最可能持续时间的估计，以确定整个项目的结果[https://risk.octigo.pl/]蒙特卡罗方法也用于期权定价，违约风险分析。<ref name=":56" /><ref name=":57" /><ref name="kr11" /> 此外，它们还可以用来估计医疗干预的财务影响。<ref name=":58" />

===Law===

===Law 法律===

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 name="montecarloanalysis">Elwart, Liz; Emerson, Nina; Enders, Christina; Fumia, Dani; Murphy, Kevin (December 2006). "Increasing Access to Restraining Orders for Low Income Victims of Domestic Violence: A Cost-Benefit Analysis of the Proposed Domestic Abuse Grant Program" (PDF). State Bar of Wisconsin. Archived from the original (PDF) on 6 November 2018. Retrieved 2016-12-12.</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 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 name="montecarloanalysis">Elwart, Liz; Emerson, Nina; Enders, Christina; Fumia, Dani; Murphy, Kevin (December 2006). "Increasing Access to Restraining Orders for Low Income Victims of Domestic Violence: A Cost-Benefit Analysis of the Proposed Domestic Abuse Grant Program" (PDF). State Bar of Wisconsin. Archived from the original (PDF) on 6 November 2018. Retrieved 2016-12-12.</ref>

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，还有许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。<ref name="montecarloanalysis" />

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，还有许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。<ref name="montecarloanalysis" />

==Use in mathematics==

==Use in mathematics 数学应用==

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。

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 ===

=== Integration 积分 ===

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

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

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。<ref name=":61" />

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。<ref name=":61" />

=== Simulation and optimization ===

=== Simulation and optimization 模拟与优化 ===

{{Main|Stochastic optimization}}

{{Main|Stochastic optimization}}

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可能需要的时间范围(在这个例子中用概率分布代表，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。

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可能需要的时间范围(在这个例子中用概率分布代表，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===

===Inverse problems 反问题===

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). 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.).

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). 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.).

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。<ref name=":63" /><ref name=":64" />

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。<ref name=":63" /><ref name=":64" />

===Philosophy===

===Philosophy 哲学===

Popular exposition of the Monte Carlo Method was conducted by McCracken<ref name=":65">McCracken, D. D., (1955) The Monte Carlo Method, Scientific American, 192(5), pp. 90-97</ref>. Method's general philosophy was discussed by [[Elishakoff]]<ref name=":66">Elishakoff, I., (2003) Notes on Philosophy of the Monte Carlo Method, International Applied Mechanics, 39(7), pp.753-762</ref> and Grüne-Yanoff and Weirich<ref name=":67">Grüne-Yanoff, T., & Weirich, P. (2010). The philosophy and epistemology of simulation: A review, Simulation & Gaming, 41(1), pp. 20-50</ref>.

Popular exposition of the Monte Carlo Method was conducted by McCracken<ref name=":65">McCracken, D. D., (1955) The Monte Carlo Method, Scientific American, 192(5), pp. 90-97</ref>. Method's general philosophy was discussed by [[Elishakoff]]<ref name=":66">Elishakoff, I., (2003) Notes on Philosophy of the Monte Carlo Method, International Applied Mechanics, 39(7), pp.753-762</ref> and Grüne-Yanoff and Weirich<ref name=":67">Grüne-Yanoff, T., & Weirich, P. (2010). The philosophy and epistemology of simulation: A review, Simulation & Gaming, 41(1), pp. 20-50</ref>.

由 McCracken 主持的蒙特卡罗方法博览会的普及展览。<ref name=":65" />方法的一般哲学由 Elishakoff、<ref name=":66" /> Grüne-Yanoff 和 weurich 讨论。<ref name=":67" />

由 McCracken 主持的蒙特卡罗方法博览会的普及展览。<ref name=":65" />方法的一般哲学由 Elishakoff、<ref name=":66" /> Grüne-Yanoff 和 weurich 讨论。<ref name=":67" />

== See also ==

== See also 另见 ==

{{Portal|Mathematics}}

{{Portal|Mathematics}}

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== References ==

== References 参考文献 ==

=== Citations ===

=== Citations ===