更改

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(见麦肯-弗拉索夫 McKean-Vlasov过程，非线性滤波方程)。在其他情况下，我们给出了采样复杂度不断增加的概率分布流(如时间范围不断增加的路径空间模型，与温度参数降低有联系的'''玻尔兹曼—吉布斯 Boltzmann-Gibbs'''测度，以及许多其他例子)。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。模拟这些复杂非线性马尔可夫过程的一个自然的方法是对过程的多个副本进行抽样，用抽样的经验测度替代演化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。平均场一词反映了每个样本(也就是粒子、个体、步行者、媒介、生物或表现型)与过程的经验测量相互作用的事实。当系统的大小趋近于无穷时，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。

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(见麦肯-弗拉索夫 McKean-Vlasov过程，非线性滤波方程)。在其他情况下，我们给出了采样复杂度不断增加的概率分布流(如时间范围不断增加的路径空间模型，与温度参数降低有联系的'''玻尔兹曼—吉布斯 Boltzmann-Gibbs'''测度，以及许多其他例子)。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。模拟这些复杂非线性马尔可夫过程的一个自然的方法是对过程的多个副本进行抽样，用抽样的经验测度替代演化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。平均场一词反映了每个样本(也就是粒子、个体、步行者、媒介、生物或表现型)与过程的经验测量相互作用的事实。当系统的大小趋近于无穷时，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 ==

== Overview（下面的步骤格式和维基原文差别很大，序号都是1，不知道为啥） ==

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

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

  Define a domain of possible inputs

  Define a domain of possible inputs

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

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

  Generate inputs randomly from a probability distribution over the domain

  Generate inputs randomly from a probability distribution over the domain

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

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

  Perform a deterministic computation on the inputs

  Perform a deterministic computation on the inputs

# Aggregate the results

# Aggregate the results

  Aggregate the results

  Aggregate the results

4、汇总结果[[File:Pi 30K.gif|thumb|right| Monte Carlo method applied to approximating the value of {{pi}}.|链接=Special:FilePath/Pi_30K.gif]]

 

 

 

[[File:Pi 30K.gif|thumb|right| Monte Carlo method applied to approximating the value of {{pi}}.|链接=Special:FilePath/Pi_30K.gif]]

  Monte Carlo method applied to approximating the value of .

  Monte Carlo method applied to approximating the value of .

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:

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:

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

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

  Draw a square, then inscribe a quadrant within it

  Draw a square, then inscribe a quadrant within it

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

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

  Uniformly scatter a given number of points over the square

  Uniformly scatter a given number of points over the square

# 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

  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

# 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, {{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 .

  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 .

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 {{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 .

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)。聚合这些结果会产生最终的结果，近似为。

在这个过程中，输入域是限定四分之一圆的正方形。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(测试它是否在四分之一圆内)。汇总这些结果会产生最终的结果—π的近似值。

 

 

 

There are two important considerations:

There are two important considerations: