Editing Monte Carlo method (section)

==Definitions==
There is no consensus on how ''Monte Carlo'' should be defined. For example, Ripley<ref name=Ripley>{{harvnb|Ripley|1987}}</ref> defines most probabilistic modeling as ''[[stochastic simulation]]'', with ''Monte Carlo'' being reserved for [[Monte Carlo integration]] and Monte Carlo statistical tests. [[Shlomo Sawilowsky|Sawilowsky]]<ref name=Sawilowsky>{{harvnb|Sawilowsky|2003}}</ref> 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).

Here are some examples:
* Simulation: Drawing ''one'' pseudo-random uniform variable from the interval [0,1] can be used to simulate the tossing of a coin: If the value is less than or equal to 0.50 designate the outcome as heads, but if the value is greater than 0.50 designate the outcome as tails. This is a simulation, but not a Monte Carlo simulation.
* Monte Carlo method: Pouring out a box of coins on a table, and then computing the ratio of coins that land heads versus tails is a Monte Carlo method of determining the behavior of repeated coin tosses, but it is not a simulation.
* Monte Carlo simulation: Drawing ''a large number'' of pseudo-random uniform variables from the interval [0,1] at one time, or once at many different times, and assigning values less than or equal to 0.50 as heads and greater than 0.50 as tails, is a ''Monte Carlo simulation'' of the behavior of repeatedly tossing a coin.

Kalos and Whitlock<ref name="Kalos">{{harvnb|Kalos|Whitlock|2008}}</ref> point out that such distinctions are not always easy to maintain. For example, the emission of radiation from atoms is a natural stochastic process. It can be simulated directly, or its average behavior can be described by stochastic equations that can themselves be solved using Monte Carlo methods. "Indeed, the same computer code can be viewed simultaneously as a 'natural simulation' or as a solution of the equations by natural sampling."

Convergence of the Monte Carlo simulation can be checked with the [[Gelman-Rubin statistic]].

===Monte Carlo and random numbers===

The main idea behind this method is that the results are computed based on repeated random sampling and statistical analysis. The Monte Carlo simulation is, in fact, random experimentations, in the case that, the results of these experiments are not well known.
Monte Carlo simulations are typically characterized by many unknown parameters, many of which are difficult to obtain experimentally.<ref name="usaus">{{cite journal|author-last1=Shojaeefard |author-first1=M.H. |author-last2=Khalkhali |author-first2=A. |author-last3=Yarmohammadisatri |first3=Sadegh |title=An efficient sensitivity analysis method for modified geometry of Macpherson suspension based on Pearson Correlation Coefficient |journal=Vehicle System Dynamics |volume=55 |issue=6 |pages=827–852 |doi=10.1080/00423114.2017.1283046 |year=2017 |bibcode = 2017VSD....55..827S |s2cid=114260173}}</ref> Monte Carlo simulation methods do not always require [[Random number generation#True vs. pseudo-random numbers|truly random number]]s to be useful (although, for some applications such as [[primality testing]], unpredictability is vital).<ref>{{harvnb|Davenport|1992}}</ref> Many of the most useful techniques use deterministic, [[pseudorandom number generator|pseudorandom]] sequences, making it easy to test and re-run simulations. The only quality usually necessary to make good [[simulation]]s is for the pseudo-random sequence to appear "random enough" in a certain sense.

What this means depends on the application, but typically they should pass a series of statistical tests. Testing that the numbers are [[Uniform distribution (continuous)|uniformly distributed]] or follow another desired distribution when a large enough number of elements of the sequence are considered is one of the simplest and most common ones. Weak correlations between successive samples are also often desirable/necessary.

Sawilowsky lists the characteristics of a high-quality Monte Carlo simulation:<ref name=Sawilowsky/>
* the (pseudo-random) number generator has certain characteristics (e.g. a long "period" before the sequence repeats)
* the (pseudo-random) number generator produces values that pass tests for randomness
* there are enough samples to ensure accurate results
* the proper sampling technique is used
* the algorithm used is valid for what is being modeled
* it simulates the phenomenon in question.

[[Pseudo-random number sampling]] algorithms are used to transform uniformly distributed pseudo-random numbers into numbers that are distributed according to a given [[probability distribution]].

[[Low-discrepancy sequences]] are often used instead of random sampling from a space as they ensure even coverage and normally have a faster order of convergence than Monte Carlo simulations using random or pseudorandom sequences. Methods based on their use are called [[quasi-Monte Carlo method]]s.

In an effort to assess the impact of random number quality on Monte Carlo simulation outcomes, astrophysical researchers tested cryptographically secure pseudorandom numbers generated via Intel's [[RDRAND]] instruction set, as compared to those derived from algorithms, like the [[Mersenne Twister]], in Monte Carlo simulations of radio flares from [[brown dwarfs]]. No statistically significant difference was found between models generated with typical pseudorandom number generators and RDRAND for trials consisting of the generation of 10<sup>7</sup> random numbers.<ref>{{cite journal|author-last1=Route |author-first1=Matthew |title=Radio-flaring Ultracool Dwarf Population Synthesis |journal=The Astrophysical Journal |date=August 10, 2017 |volume=845 |issue=1 |page=66 |doi=10.3847/1538-4357/aa7ede |arxiv=1707.02212 |bibcode=2017ApJ...845...66R |s2cid=118895524 |doi-access=free }}</ref>

=== Monte Carlo simulation versus "what if" scenarios ===
There are ways of using probabilities that are definitely not Monte Carlo simulations – for example, deterministic modeling using single-point estimates. Each uncertain variable within a model is assigned a "best guess" estimate. Scenarios (such as best, worst, or most likely case) for each input variable are chosen and the results recorded.{{sfn|Vose|2008|page=13}}

By contrast, Monte Carlo simulations sample from a [[probability distribution]] for each variable to produce hundreds or thousands of possible outcomes. The results are analyzed to get probabilities of different outcomes occurring.{{sfn|Vose|2008|page=16}} For example, a comparison of a spreadsheet cost construction model run using traditional "what if" scenarios, and then running the comparison again with Monte Carlo simulation and [[triangular distribution|triangular probability distribution]]s shows that the Monte Carlo analysis has a narrower range than the "what if" analysis.{{Example needed|date=May 2012}} This is because the "what if" analysis gives equal weight to all scenarios (see [[Corporate finance#Quantifying uncertainty|quantifying uncertainty in corporate finance]]), while the Monte Carlo method hardly samples in the very low probability regions. The samples in such regions are called "rare events".