Editing Monte Carlo method (section)

===Applied statistics===
The standards for Monte Carlo experiments in statistics were set by Sawilowsky.<ref>{{cite journal |author-last1=Cassey |author-last2=Smith |year=2014 |title=Simulating confidence for the Ellison-Glaeser Index |journal=Journal of Urban Economics |volume=81 |page=93 |doi=10.1016/j.jue.2014.02.005}}</ref> In applied statistics, Monte Carlo methods may be used for at least four purposes:
# 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 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 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 |author-last1=Spall |author-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 |author-last1=Das |author-first1=Sonjoy |author-last2=Spall |author-first2=James C. |author-last3=Ghanem |author-first3=Roger}}</ref>

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).

{{anchor|Monte Carlo tree search}}