更改

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

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

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，或者用于研究生物系统，例如基因组、蛋白质<ref name=":39" /> 或膜<ref name=":40" />。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。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，例如。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)。

蒙特卡罗方法被用于'''计算生物学 Computational Biology'''的各个领域，例如在系统发育学中的贝叶斯推断，或者用于研究生物系统，例如基因组、蛋白质<ref name=":39" /> 或膜<ref name=":40" />。该系统可以在粗粒度或从头计算框架中研究，这取决于所需的准确性。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，例如。在无法进行物理实验的情况下，可以进行思维实验(例如：断键，在特定位置引入杂质，改变局部/整体结构，或引入外部场)。

===Computer graphics 计算机图形学===

===Computer graphics 计算机图形学===

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.

===Applied statistics 应用统计学===

===Applied statistics 应用统计学===

#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 name=":43">Sawilowsky, Shlomo S.; Fahoome, Gail C. (2003). ''Statistics via Monte Carlo Simulation with Fortran''. Rochester Hills, MI: JMASM. ISBN <bdi>978-0-9740236-0-1</bdi>.</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.<ref name=":43">Sawilowsky, Shlomo S.; Fahoome, Gail C. (2003). ''Statistics via Monte Carlo Simulation with Fortran''. Rochester Hills, MI: JMASM. ISBN <bdi>978-0-9740236-0-1</bdi>.</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，无限大的样本量和无限小的处理效果) ，实际数据往往没有这样的分布。<ref name=":43" />

#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。虽然根据经典理论分布(例如，正态曲线，'''柯西分布 Cauchy distribution''')数据的渐近条件(即，无限大的样本量和无限小的处理效果) ， i 型误差和统计的幂次特性可以进行计算，但是实际数据往往没有这样的分布。<ref name=":43" />

#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 [[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 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.

#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 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。然后基于这个样本进行近似和总结后验的所有基本特征。

#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 name=":44">Spall, James C. (2005). "Monte Carlo Computation of the Fisher Information Matrix in Nonstandard Settings". ''Journal of Computational and Graphical Statistics''. '''14''' (4): 889–909. CiteSeerX 10.1.1.142.738. doi:10.1198/106186005X78800. S2CID 16090098.</ref><ref name=":45">{{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.<ref name=":44">Spall, James C. (2005). "Monte Carlo Computation of the Fisher Information Matrix in Nonstandard Settings". ''Journal of Computational and Graphical Statistics''. '''14''' (4): 889–909. CiteSeerX 10.1.1.142.738. doi:10.1198/106186005X78800. S2CID 16090098.</ref><ref name=":45">{{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，这些估计的平均值可以形成费雪资讯矩阵的估计。<ref name=":44" /><ref name=":45" />

#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海赛矩阵的有效随机估计，这些估计的平均值可以形成'''费雪信息量 Fisher Information'''矩阵的估计。<ref name=":44" /><ref name=":45" />

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).

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)。

蒙特卡罗方法也是近似随机和排列检验之间的折衷。近似随机化测试是基于所有排列的特定子集(这可能需要大量的内务处理，其中的排列经过充分考虑)。蒙特卡罗方法是基于指定数量的随机排列(如果一个排列被绘制两次或更频繁，则在精度上有较小的损失，因为不必跟踪哪些排列已经被选择)。

{{anchor|Monte Carlo tree search}}

{{anchor|Monte Carlo tree search}}