更改

量子蒙特卡罗法，更具体地说，扩散蒙特卡罗方法也可以解释为费曼-卡克路径积分的平均场粒子蒙特卡罗近似。量子蒙特卡罗法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，但是第一个启发式和遗传类型粒子算法(简称为 a。用于量子系统基态能量估计的重构蒙特卡罗方法(简化矩阵模型)是1984年杰克 · 赫瑟林顿提出的

量子蒙特卡罗法，更具体地说，扩散蒙特卡罗方法也可以解释为费曼-卡克路径积分的平均场粒子蒙特卡罗近似。量子蒙特卡罗法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，但是第一个启发式和遗传类型粒子算法(简称为 a。用于量子系统基态能量估计的重构蒙特卡罗方法(简化矩阵模型)是1984年杰克 · 赫瑟林顿提出的

| last1 = Del Moral | first1 = P.

<nowiki> </nowiki><nowiki>| last1 =Del Moral |</nowiki> first1 =P.

| last2 = Miclo | first2 = L.

<nowiki> </nowiki><nowiki>| last2 =Miclo |</nowiki> first2 =L.

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", 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. These Sequential Monte Carlo methodologies can be interpreted as an acceptance-rejection sampler equipped with an interacting recycling mechanism.

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", 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. These Sequential Monte Carlo methodologies can be interpreted as an acceptance-rejection sampler equipped with an interacting recycling mechanism.

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。1993年，Gordon 等人在他们的开创性工作中发表了蒙特卡罗重采样算法在贝叶斯推论统计学中的首次应用。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。我们还引用了北川真志郎关于相关的”蒙特卡洛过滤器”的另一篇开创性文章，以及皮埃尔 · 德尔 · 莫勒尔和希米尔康 · 卡瓦略、皮埃尔 · 德尔 · 莫勒尔、安德烈 · 莫宁和杰拉德 · 萨鲁特在1990年代中期发表的关于粒子过滤器的文章。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 的 LAAS-CNRS (the Laboratory for Analysis and Architecture of Systems)共同完成的一系列关于雷达/声纳和 GPS 信号处理问题的限制性和机密性研究报告。这些序贯蒙特卡罗方法可以解释为一个接受拒绝采样器配备了相互作用的回收机制。

在高级信号处理和贝叶斯推断中使用 序列蒙特卡罗方法是最近才出现的。1993年，高登等人在他们的开创性工作中发表了蒙特卡罗重采样算法在贝叶斯推论统计学中的首次应用。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。我们还引用了北川源四郎关于相关的”蒙特卡洛过滤器”的另一篇开创性文章，以及皮埃尔 · 德尔 · 莫勒尔和希米尔康 · 卡瓦略、皮埃尔 · 德尔 · 莫勒尔、安德烈 · 莫宁和杰拉德 · 萨鲁特在1990年代中期发表的关于粒子过滤器的文章。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 的 LAAS-CNRS (the Laboratory for Analysis and Architecture of Systems)共同完成的一系列关于雷达/声纳和 GPS 信号处理问题的限制性和机密性研究报告。这些序贯蒙特卡罗方法可以解释为一个接受拒绝采样器配备了相互作用的回收机制。

  | 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

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

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

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 were written by Pierre Del Moral in 1996.

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 were written by Pierre Del Moral in 1996.

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皮埃尔 · 德尔 · 莫勒尔在1996年写的。

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皮埃尔 · 德尔 · 莫勒尔在1996年写的。

  | doi = 10.1007/BFb0103798

  | doi =10.1007/BFb0103798

  | mr = 1768060

  | mr =1768060

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.

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.

20世纪90年代末，Dan Crisan，Jessica Gaines 和 Terry Lyons，以及 Dan Crisan，Pierre Del Moral 和 Terry Lyons 也开发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 进一步发展了这一领域。

20世纪90年代末，Dan Crisan，Jessica Gaines 和 Terry Lyons，以及 Dan Crisan，Pierre Del Moral 和 Terry Lyons 也开发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 进一步发展了这一领域。

  | pages = 1–145

  | pages =1–145

  | publisher = Springer |location = Berlin

  | publisher =Springer |location =Berlin

  | series = Lecture Notes in Mathematics

  | series =Lecture Notes in Mathematics

  | title = Séminaire de Probabilités, XXXIV

  | title =Séminaire de Probabilités, XXXIV

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:

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，而蒙特卡洛模型则保留给蒙地卡罗积分和蒙特卡洛统计检验。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:

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，而蒙特卡洛模型则保留给蒙地卡罗积分和蒙特卡洛统计检验。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:

  | volume = 1729

  | volume =1729

  | year = 2000

  | year =2000

  |isbn = 978-3-540-67314-9

  |isbn =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/

  }}</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 |s2cid = 89611599|url = https://semanticscholar.org/paper/1570c85ba9aca1cb413ada31e215e0917c3ccba7}}</ref>

  <nowiki>}}</nowiki></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 |s2cid = 89611599|url = https://semanticscholar.org/paper/1570c85ba9aca1cb413ada31e215e0917c3ccba7}}</ref>

量子蒙特卡罗方法，更具体地说，扩散蒙特卡罗方法也可以解释为费曼—卡茨路径积分的平均场粒子蒙特卡罗近似。量子蒙特卡罗方法的起源通常归功于'''恩里科·费米Enrico Fermi'''和'''罗伯特·里希特迈耶 Robert Richtmyer'''于1948年开发了中子链式反应的平均场粒子解释，但是用于估计量子系统的基态能量(在简化矩阵模型中)的第一个类启发式和遗传型粒子算法(也称为重取样或重构蒙特卡洛方法)则是由杰克·H·海瑟林顿在1984年提出。在分子化学中，使用遗传类启发式的粒子方法(又名删减和富集策略)可以追溯到1955年—'''马歇尔·罗森布鲁斯 Marshall Rosenbluth'''和'''阿里安娜·罗森布鲁斯Arianna Rosenbluth'''的开创性工作。

量子蒙特卡罗方法，更具体地说，扩散蒙特卡罗方法也可以解释为费曼—卡茨路径积分的平均场粒子蒙特卡罗近似。量子蒙特卡罗方法的起源通常归功于'''恩里科·费米Enrico Fermi'''和'''罗伯特·里希特迈耶 Robert Richtmyer'''于1948年开发了中子链式反应的平均场粒子解释，但是用于估计量子系统的基态能量(在简化矩阵模型中)的第一个类启发式和遗传型粒子算法(也称为重取样或重构蒙特卡洛方法)则是由杰克·H·海瑟林顿在1984年提出。在分子化学中，使用遗传类启发式的粒子方法(又名删减和富集策略)可以追溯到1955年—'''马歇尔·罗森布鲁斯 Marshall Rosenbluth'''和'''阿里安娜·罗森布鲁斯Arianna Rosenbluth'''的开创性工作。

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><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.

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.

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.

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