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第473行： − − 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。 第483行：
第478行： − − 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. − − 提供比精确检验更有效的假设检验的实现，例如排列检验(通常无法计算) ，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. − − 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. − − 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。 − − 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)。 第519行：
第498行： − + − − − 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。 − The Monte Carlo tree search (MCTS) method has four steps: − 蒙特卡罗树搜索(MCTS)方法有四个步骤: − − 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. − 展开叶节点并选择其中一个子节点。 − − Play a simulated game starting with that node. − − 以该节点开始玩一个模拟游戏。 第583行：
第547行： + + + + − + − 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改变。+ 第611行：
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第659行： − − 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.
→Physical sciences
A ''Monte Carlo method'' simulation is defined as any method that utilizes sequences of random numbers to perform the simulation. Monte Carlo simulations are applied to many topics including [[quantum chromodynamics]], cancer radiation therapy, traffic flow, [[stellar evolution]] and VLSI design. All these simulations require the use of random numbers and therefore [[pseudorandom number generator]]s, which makes creating random-like numbers very important.
A ''Monte Carlo method'' simulation is defined as any method that utilizes sequences of random numbers to perform the simulation. Monte Carlo simulations are applied to many topics including [[quantum chromodynamics]], cancer radiation therapy, traffic flow, [[stellar evolution]] and VLSI design. All these simulations require the use of random numbers and therefore [[pseudorandom number generator]]s, which makes creating random-like numbers very important.
A simple example of how a computer would perform a Monte Carlo simulation is the calculation of [[Pi|π]]. If a square enclosed a circle and a point were randomly chosen inside the square the point would either lie inside the circle or outside it. If the process were repeated many times, the ratio of the random points that lie inside the circle to the total number of random points in the square would approximate the ratio of the area of the circle to the area of the square. From this we can estimate pi, as shown in the [[Python (programming language)|Python]] code below utilizing a [[SciPy]] package to generate pseudorandom numbers with the [[Mersenne twister|MT19937]] algorithm. Note that this method is a computationally inefficient way to [[Numerical approximations of π|numerically approximate π]].
A simple example of how a computer would perform a Monte Carlo simulation is the calculation of [[Pi|π]]. If a square enclosed a circle and a point were randomly chosen inside the square the point would either lie inside the circle or outside it. If the process were repeated many times, the ratio of the random points that lie inside the circle to the total number of random points in the square would approximate the ratio of the area of the circle to the area of the square. From this we can estimate pi, as shown in the [[Python (programming language)|Python]] code below utilizing a [[SciPy]] package to generate pseudorandom numbers with the [[Mersenne twister|MT19937]] algorithm. Note that this method is a computationally inefficient way to [[Numerical approximations of π|numerically approximate π]].
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:
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:
===Engineering===
===Engineering===
蒙特卡罗方法被广泛应用于工程设计中的敏感度分析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,
蒙特卡罗方法被广泛应用于工程设计中的敏感度分析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,
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 (chemical engineering)|process design]]. The need arises from the interactive, co-linear and non-linear behavior of typical process simulations. For example,
* 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 [[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 risk analysis]].<ref name="mbv01">{{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 | volume = | pages = 250| isbn =978-619-90684-3-4 | doi = |url=https://books.google.com/books?id=t0BqDwAAQBAJ&q=the+monte+carlo+method+engineering+applications+mazhdrakov}}</ref>
* In [[geostatistics]] and [[geometallurgy]], Monte Carlo methods underpin the design of [[mineral processing]] [[process flow diagram|flowsheets]] and contribute to [[quantitative risk analysis]].<ref name="mbv01">{{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 | volume = | pages = 250| isbn =978-619-90684-3-4 | doi = |url=https://books.google.com/books?id=t0BqDwAAQBAJ&q=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.)
* 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>
* 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 [[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]]s.
* 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]]s.
* In groundwater modeling, Monte Carlo methods are utilized to generate a large number of realizations of heterogeneous parameter field for model uncertainty quantification or parameter inversion.
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.
===Climate change and radiative forcing===
===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 [[IPCC|Intergovernmental Panel on Climate Change]] relies on Monte Carlo methods in [[probability density function]] analysis of [[radiative forcing]].
{{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>}}
{{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>}}
#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.<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。
#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 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. 提供比精确检验更有效的假设检验的实现，例如排列检验(通常无法计算) ，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.
#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|s2cid = 16090098}}</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.<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|s2cid = 16090098}}</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，这些估计的平均值可以形成费雪资讯矩阵的估计。
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 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)。
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 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.<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，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。
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.
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:<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(MCTS)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.
#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.
#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.
#Play a simulated game starting with that node. 以该节点开始玩一个模拟游戏。
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
===Finance and business===
===Finance and business===
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改变。
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.
{{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}}
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).
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).