A computation is any type of arithmetic or non-arithmetic calculation that is well-defined. Common examples of computation are mathematical equation solving and the execution of computer algorithms .
89-532: Monte Carlo methods , or Monte Carlo experiments , are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. The name comes from the Monte Carlo Casino in Monaco, where the primary developer of the method, mathematician Stanislaw Ulam ,
178-423: A )ln(2/0.01)/ ε ≈ 10.6( b – a )/ ε . Despite its conceptual and algorithmic simplicity, the computational cost associated with a Monte Carlo simulation can be staggeringly high. In general the method requires many samples to get a good approximation, which may incur an arbitrarily large total runtime if the processing time of a single sample is high. Although this is a severe limitation in very complex problems,
267-434: A casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game. Importantly, the law applies (as the name indicates) only when a large number of observations are considered. There is no principle that a small number of observations will coincide with
356-580: 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: Kalos and Whitlock 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,
445-436: A broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The larger the number of repetitions, the better the approximation tends to be. The reason that this method is important is mainly that, sometimes, it is difficult or impossible to use other approaches. The average of the results obtained from a large number of trials may fail to converge in some cases. For instance,
534-567: A collection of independent and identically distributed (iid) samples from a random variable with finite mean, the sample mean converges in probability to the expected value That is, for any positive number ε , lim n → ∞ Pr ( | X ¯ n − μ | < ε ) = 1. {\displaystyle \lim _{n\to \infty }\Pr \!\left(\,|{\overline {X}}_{n}-\mu |<\varepsilon \,\right)=1.} Interpreting this result,
623-558: A computation on each input (test whether it falls within the quadrant). Aggregating the results yields our final result, the approximation of π . There are two important considerations: Uses of Monte Carlo methods require large amounts of random numbers, and their use benefitted greatly from pseudorandom number generators , which are far quicker to use than the tables of random numbers that had been previously used for statistical sampling. Monte Carlo methods are often used in physical and mathematical problems and are most useful when it
712-448: A computational system is a complex object which consists of three parts. First, a mathematical dynamical system D S {\displaystyle DS} with discrete time and discrete state space; second, a computational setup H = ( F , B F ) {\displaystyle H=\left(F,B_{F}\right)} , which is made up of a theoretical part F {\displaystyle F} , and
801-417: A flow of probability distributions with an increasing level of sampling complexity arise (path spaces models with an increasing time horizon, Boltzmann–Gibbs measures associated with decreasing temperature parameters, and many others). These models can also be seen as the evolution of the law of the random states of a nonlinear Markov chain. A natural way to simulate these sophisticated nonlinear Markov processes
890-466: A mean-field particle interpretation of neutron-chain reactions, but the first heuristic-like and genetic type particle algorithm (a.k.a. Resampled or Reconfiguration Monte Carlo methods) for estimating ground state energies of quantum systems (in reduced matrix models) is due to Jack H. Hetherington in 1984. In molecular chemistry, the use of genetic heuristic-like particle methodologies (a.k.a. pruning and enrichment strategies) can be traced back to 1955 with
979-411: A particular pattern: For example, consider a quadrant (circular sector) inscribed in a unit square . Given that the ratio of their areas is π / 4 , the value of π can be approximated using a Monte Carlo method: In this procedure the domain of inputs is the square that circumscribes the quadrant. One can generate random inputs by scattering grains over the square then perform
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#17327873070001068-469: A probabilistic interpretation. By the law of large numbers , integrals described by the expected value of some random variable can be approximated by taking the empirical mean ( a.k.a. the 'sample mean') of independent samples of the variable. When the probability distribution of the variable is parameterized, mathematicians often use a Markov chain Monte Carlo (MCMC) sampler. The central idea
1157-618: A purely physical process occurring inside a closed physical system called a computer . Turing's 1937 proof, On Computable Numbers, with an Application to the Entscheidungsproblem , demonstrated that there is a formal equivalence between computable statements and particular physical systems, commonly called computers . Examples of such physical systems are: Turing machines , human mathematicians following strict rules, digital computers , mechanical computers , analog computers and others. An alternative account of computation
1246-451: A question which occurred to me in 1946 as I was convalescing from an illness and playing solitaires. The question was what are the chances that a Canfield solitaire laid out with 52 cards will come out successfully? After spending a lot of time trying to estimate them by pure combinatorial calculations, I wondered whether a more practical method than "abstract thinking" might not be to lay it out say one hundred times and simply observe and count
1335-451: A random variable that does not have a finite variance under some other weaker assumption, and Khinchin showed in 1929 that if the series consists of independent identically distributed random variables, it suffices that the expected value exists for the weak law of large numbers to be true. These further studies have given rise to two prominent forms of the LLN. One is called the "weak" law and
1424-423: A real part B F {\displaystyle B_{F}} ; third, an interpretation I D S , H {\displaystyle I_{DS,H}} , which links the dynamical system D S {\displaystyle DS} with the setup H {\displaystyle H} . Law of large numbers In probability theory , the law of large numbers ( LLN )
1513-437: A rule. "Medium-independence" requires that the property can be instantiated by multiple realizers and multiple mechanisms, and that the inputs and outputs of the mechanism also be multiply realizable . In short, medium-independence allows for the use of physical variables with properties other than voltage (as in typical digital computers); this is imperative in considering other types of computation, such as that which occurs in
1602-428: A well-defined statement or calculation as any statement that could be expressed in terms of the initialisation parameters of a Turing machine . Other (mathematically equivalent) definitions include Alonzo Church 's lambda-definability , Herbrand - Gödel - Kleene 's general recursiveness and Emil Post 's 1-definability . Today, any formal statement or calculation that exhibits this quality of well-definedness
1691-457: Is a mathematical law that states that the average of the results obtained from a large number of independent random samples converges to the true value, if it exists. More formally, the LLN states that given a sample of independent and identically distributed values, the sample mean converges to the true mean . The LLN is important because it guarantees stable long-term results for the averages of some random events . For example, while
1780-423: Is an academic field that involves the study of computation. The notion that mathematical statements should be 'well-defined' had been argued by mathematicians since at least the 1600s , but agreement on a suitable definition proved elusive. A candidate definition was proposed independently by several mathematicians in the 1930s. The best-known variant was formalised by the mathematician Alan Turing , who defined
1869-633: Is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes: optimization , numerical integration , and generating draws from a probability distribution . In physics-related problems, Monte Carlo methods are useful for simulating systems with many coupled degrees of freedom , such as fluids, disordered materials, strongly coupled solids, and cellular structures (see cellular Potts model , interacting particle systems , McKean–Vlasov processes , kinetic models of gases ). Other examples include modeling phenomena with significant uncertainty in inputs such as
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#17327873070001958-494: 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 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
2047-419: Is found throughout the works of Hilary Putnam and others. Peter Godfrey-Smith has dubbed this the "simple mapping account." Gualtiero Piccinini's summary of this account states that a physical system can be said to perform a specific computation when there is a mapping between the state of that system and the computation such that the "microphysical states [of the system] mirror the state transitions between
2136-411: Is generating draws from a sequence of probability distributions satisfying a nonlinear evolution equation. These flows of probability distributions can always be interpreted as the distributions of the random states of a Markov process whose transition probabilities depend on the distributions of the current random states (see McKean–Vlasov processes , nonlinear filtering equation ). In other instances,
2225-929: Is known as Kolmogorov's strong law , see e.g. Sen & Singer (1993 , Theorem 2.3.10). The weak law states that for a specified large n , the average X ¯ n {\displaystyle {\overline {X}}_{n}} is likely to be near μ . Thus, it leaves open the possibility that | X ¯ n − μ | > ε {\displaystyle |{\overline {X}}_{n}-\mu |>\varepsilon } happens an infinite number of times, although at infrequent intervals. (Not necessarily | X ¯ n − μ | ≠ 0 {\displaystyle |{\overline {X}}_{n}-\mu |\neq 0} for all n ). The strong law shows that this almost surely will not occur. It does not imply that with probability 1, we have that for any ε > 0
2314-399: Is no consensus on how Monte Carlo should be defined. For example, Ripley defines most probabilistic modeling as stochastic simulation , with Monte Carlo being reserved for Monte Carlo integration and Monte Carlo statistical tests. Sawilowsky distinguishes between a simulation , a Monte Carlo method, and a Monte Carlo simulation: a simulation is a fictitious representation of reality,
2403-492: Is termed computable , while the statement or calculation itself is referred to as a computation . Turing's definition apportioned "well-definedness" to a very large class of mathematical statements, including all well-formed algebraic statements , and all statements written in modern computer programming languages. Despite the widespread uptake of this definition, there are some mathematical concepts that have no well-defined characterisation under this definition. This includes
2492-418: Is that the probability that, as the number of trials n goes to infinity, the average of the observations converges to the expected value, is equal to one. The modern proof of the strong law is more complex than that of the weak law, and relies on passing to an appropriate subsequence. The strong law of large numbers can itself be seen as a special case of the pointwise ergodic theorem . This view justifies
2581-454: Is to design a judicious Markov chain model with a prescribed stationary probability distribution . That is, in the limit, the samples being generated by the MCMC method will be samples from the desired (target) distribution. By the ergodic theorem , the stationary distribution is approximated by the empirical measures of the random states of the MCMC sampler. In other problems, the objective
2670-476: Is to sample multiple copies of the process, replacing in the evolution equation the unknown distributions of the random states by the sampled empirical measures . In contrast with traditional Monte Carlo and MCMC methodologies, these mean-field particle techniques rely on sequential interacting samples. The terminology mean field reflects the fact that each of the samples ( a.k.a. particles, individuals, walkers, agents, creatures, or phenotypes) interacts with
2759-446: Is where the random numbers equal the tangent of an angle uniformly distributed between −90° and +90°. The median is zero, but the expected value does not exist, and indeed the average of n such variables have the same distribution as one such variable. It does not converge in probability toward zero (or any other value) as n goes to infinity. And if the trials embed a selection bias , typical in human economic/rational behaviour,
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2848-471: Is within ε of μ . If n > k , then n simulations can be run “from scratch,” or, since k simulations have already been done, one can just run n – k more simulations and add their results into those from the sample simulations: An alternate formula can be used in the special case where all simulation results are bounded above and below. Choose a value for ε that is twice the maximum allowed difference between μ and m. Let 0 < δ < 100 be
2937-539: The brain or in a quantum computer . A rule, in this sense, provides a mapping among inputs, outputs, and internal states of the physical computing system. In the theory of computation , a diversity of mathematical models of computation has been developed. Typical mathematical models of computers are the following: Giunti calls the models studied by computation theory computational systems, and he argues that all of them are mathematical dynamical systems with discrete time and discrete state space. He maintains that
3026-400: The embarrassingly parallel nature of the algorithm allows this large cost to be reduced (perhaps to a feasible level) through parallel computing strategies in local processors, clusters, cloud computing, GPU, FPGA, etc. Before the Monte Carlo method was developed, simulations tested a previously understood deterministic problem, and statistical sampling was used to estimate uncertainties in
3115-683: The LAAS-CNRS in a series of restricted and classified research reports with STCAN (Service Technique des Constructions et Armes Navales), the IT company DIGILOG, and the LAAS-CNRS (the Laboratory for Analysis and Architecture of Systems) on radar/sonar and GPS signal processing problems. These Sequential Monte Carlo methodologies can be interpreted as an acceptance-rejection sampler equipped with an interacting recycling mechanism. From 1950 to 1996, all
3204-467: The LLN is used in many fields including statistics, probability theory, economics, and insurance. For example, a single roll of a fair, six-sided die produces one of the numbers 1, 2, 3, 4, 5, or 6, each with equal probability . Therefore, the expected value of the average of the rolls is: 1 + 2 + 3 + 4 + 5 + 6 6 = 3.5 {\displaystyle {\frac {1+2+3+4+5+6}{6}}=3.5} According to
3293-589: The Monte Carlo method while studying neutron diffusion, but he did not publish this work. In the late 1940s, Stanislaw Ulam invented the modern version of the Markov Chain Monte Carlo method while he was working on nuclear weapons projects at the Los Alamos National Laboratory . In 1946, nuclear weapons physicists at Los Alamos were investigating neutron diffusion in the core of a nuclear weapon. Despite having most of
3382-432: The algorithm to obtain m is Suppose we want to know how many times we should expect to throw three eight-sided dice for the total of the dice throws to be at least T. We know the expected value exists. The dice throws are randomly distributed and independent of each other. So simple Monte Carlo is applicable: If n is large enough, m will be within ε of μ for any ε > 0. Let ε = | μ – m | > 0. Choose
3471-656: The average of n results taken from the Cauchy distribution or some Pareto distributions (α<1) will not converge as n becomes larger; the reason is heavy tails . The Cauchy distribution and the Pareto distribution represent two cases: the Cauchy distribution does not have an expectation, whereas the expectation of the Pareto distribution ( α <1) is infinite. One way to generate the Cauchy-distributed example
3560-437: The average of a set of normally distributed variables). The variance of the sum is equal to the sum of the variances, which is asymptotic to n 2 / log n {\displaystyle n^{2}/\log n} . The variance of the average is therefore asymptotic to 1 / log n {\displaystyle 1/\log n} and goes to zero. There are also examples of
3649-434: The average of the first n values goes to zero as n goes to infinity. As an example, assume that each random variable in the series follows a Gaussian distribution (normal distribution) with mean zero, but with variance equal to 2 n / log ( n + 1 ) {\displaystyle 2n/\log(n+1)} , which is not bounded. At each stage, the average will be normally distributed (as
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3738-664: The average to converge almost surely on something (this can be considered another statement of the strong law), it is necessary that they have an expected value (and then of course the average will converge almost surely on that). If the summands are independent but not identically distributed, then provided that each X k has a finite second moment and ∑ k = 1 ∞ 1 k 2 Var [ X k ] < ∞ . {\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k^{2}}}\operatorname {Var} [X_{k}]<\infty .} This statement
3827-477: The calculation of risk in business and, in mathematics, evaluation of multidimensional definite integrals with complicated boundary conditions . In application to systems engineering problems (space, oil exploration , aircraft design, etc.), Monte Carlo–based predictions of failure, cost overruns and schedule overruns are routinely better than human intuition or alternative "soft" methods. In principle, Monte Carlo methods can be used to solve any problem having
3916-568: 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. Monte Carlo simulation methods do not always require truly random numbers to be useful (although, for some applications such as primality testing , unpredictability is vital). Many of the most useful techniques use deterministic, pseudorandom sequences, making it easy to test and re-run simulations. The only quality usually necessary to make good simulations
4005-512: The case where X 1 , X 2 , ... is an infinite sequence of independent and identically distributed (i.i.d.) Lebesgue integrable random variables with expected value E( X 1 ) = E( X 2 ) = ... = μ , both versions of the law state that the sample average X ¯ n = 1 n ( X 1 + ⋯ + X n ) {\displaystyle {\overline {X}}_{n}={\frac {1}{n}}(X_{1}+\cdots +X_{n})} converges to
4094-714: The characteristics of a high-quality Monte Carlo simulation: 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 methods . Computation Mechanical or electronic devices (or, historically , people) that perform computations are known as computers . Computer science
4183-403: The computational states." Philosophers such as Jerry Fodor have suggested various accounts of computation with the restriction that semantic content be a necessary condition for computation (that is, what differentiates an arbitrary physical system from a computing system is that the operands of the computation represent something). This notion attempts to prevent the logical abstraction of
4272-414: The convergence is only weak (in probability). See differences between the weak law and the strong law . The strong law applies to independent identically distributed random variables having an expected value (like the weak law). This was proved by Kolmogorov in 1930. It can also apply in other cases. Kolmogorov also showed, in 1933, that if the variables are independent and identically distributed, then for
4361-514: The design of the H-bomb, though severely limited by the computational tools at the time. Von Neumann, Nicholas Metropolis and others programmed the ENIAC computer to perform the first fully automated Monte Carlo calculations, of a fission weapon core, in the spring of 1948. In the 1950s Monte Carlo methods were used at Los Alamos for the development of the hydrogen bomb , and became popularized in
4450-476: The desired confidence level – the percent chance that, when the Monte Carlo algorithm completes, m is indeed within ε of μ . Let z be the z -score corresponding to that confidence level. Let s be the estimated variance, sometimes called the “sample” variance; it is the variance of the results obtained from a relatively small number k of “sample” simulations. Choose a k ; Driels and Shin observe that “even for sample sizes an order of magnitude lower than
4539-636: The desired confidence level, expressed as a percentage. Let every simulation result r 1 , r 2 , … r i , … r n be such that a ≤ r i ≤ b for finite a and b . To have confidence of at least δ that | μ – m | < ε /2, use a value for n such that n ≥ 2 ( b − a ) 2 ln ( 2 / ( 1 − ( δ / 100 ) ) ) / ϵ 2 {\displaystyle n\geq 2(b-a)^{2}\ln(2/(1-(\delta /100)))/\epsilon ^{2}} For example, if δ = 99%, then n ≥ 2( b –
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#17327873070004628-526: The empirical measures of the process. When the size of the system tends to infinity, these random empirical measures converge to the deterministic distribution of the random states of the nonlinear Markov chain, so that the statistical interaction between particles vanishes. Suppose one wants to know the expected value μ of a population (and knows that μ exists), but does not have a formula available to compute it. The simple Monte Carlo method gives an estimate for μ by running n simulations and averaging
4717-409: The expected value is the theoretical probability of success, and the average of n such variables (assuming they are independent and identically distributed (i.i.d.) ) is precisely the relative frequency. For example, a fair coin toss is a Bernoulli trial. When a fair coin is flipped once, the theoretical probability that the outcome will be heads is equal to 1 ⁄ 2 . Therefore, according to
4806-456: The expected value or that a streak of one value will immediately be "balanced" by the others (see the gambler's fallacy ). The LLN only applies to the average of the results obtained from repeated trials and claims that this average converges to the expected value; it does not claim that the sum of n results gets close to the expected value times n as n increases. Throughout its history, many mathematicians have refined this law. Today,
4895-639: The expected value: (Lebesgue integrability of X j means that the expected value E( X j ) exists according to Lebesgue integration and is finite. It does not mean that the associated probability measure is absolutely continuous with respect to Lebesgue measure .) Introductory probability texts often additionally assume identical finite variance Var ( X i ) = σ 2 {\displaystyle \operatorname {Var} (X_{i})=\sigma ^{2}} (for all i {\displaystyle i} ) and no correlation between random variables. In that case,
4984-425: The fields of physics , physical chemistry , and operations research . The Rand Corporation and the U.S. Air Force were two of the major organizations responsible for funding and disseminating information on Monte Carlo methods during this time, and they began to find a wide application in many different fields. The theory of more sophisticated mean-field type particle Monte Carlo methods had certainly started by
5073-429: The first rigorous analysis of these particle algorithms were written by Pierre Del Moral in 1996. Branching type particle methodologies with varying population sizes were also developed in the end of the 1990s by Dan Crisan, Jessica Gaines and Terry Lyons, and by Dan Crisan, Pierre Del Moral and Terry Lyons. Further developments in this field were described in 1999 to 2001 by P. Del Moral, A. Guionnet and L. Miclo. There
5162-402: The halting problem and the busy beaver game . It remains an open question as to whether there exists a more powerful definition of 'well-defined' that is able to capture both computable and 'non-computable' statements. Some examples of mathematical statements that are computable include: Some examples of mathematical statements that are not computable include: Computation can be seen as
5251-537: The idea to John von Neumann , and we began to plan actual calculations. Being secret, the work of von Neumann and Ulam required a code name. A colleague of von Neumann and Ulam, Nicholas Metropolis , suggested using the name Monte Carlo , which refers to the Monte Carlo Casino in Monaco where Ulam's uncle would borrow money from relatives to gamble. Monte Carlo methods were central to the simulations required for further postwar development of nuclear weapons, including
5340-415: The inequality | X ¯ n − μ | < ε {\displaystyle |{\overline {X}}_{n}-\mu |<\varepsilon } holds for all large enough n , since the convergence is not necessarily uniform on the set where it holds. The strong law does not hold in the following cases, but the weak law does. There are extensions of
5429-411: The intuitive interpretation of the expected value (for Lebesgue integration only) of a random variable when sampled repeatedly as the "long-term average". Law 3 is called the strong law because random variables which converge strongly (almost surely) are guaranteed to converge weakly (in probability). However the weak law is known to hold in certain conditions where the strong law does not hold and then
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#17327873070005518-407: The law of large numbers does not help in solving the bias. Even if the number of trials is increased the selection bias remains. The Italian mathematician Gerolamo Cardano (1501–1576) stated without proof that the accuracies of empirical statistics tend to improve with the number of trials. This was then formalized as a law of large numbers. A special form of the LLN (for a binary random variable)
5607-411: The law of large numbers to collections of estimators, where the convergence is uniform over the collection; thus the name uniform law of large numbers . Suppose f ( x , θ ) is some function defined for θ ∈ Θ, and continuous in θ . Then for any fixed θ , the sequence { f ( X 1 , θ ), f ( X 2 , θ ), ...} will be a sequence of independent and identically distributed random variables, such that
5696-414: The law of large numbers, if a large number of six-sided dice are rolled, the average of their values (sometimes called the sample mean ) will approach 3.5, with the precision increasing as more dice are rolled. It follows from the law of large numbers that the empirical probability of success in a series of Bernoulli trials will converge to the theoretical probability. For a Bernoulli random variable ,
5785-426: The law of large numbers, the proportion of heads in a "large" number of coin flips "should be" roughly 1 ⁄ 2 . In particular, the proportion of heads after n flips will almost surely converge to 1 ⁄ 2 as n approaches infinity. Although the proportion of heads (and tails) approaches 1 ⁄ 2 , almost surely the absolute difference in the number of heads and tails will become large as
5874-411: The mapping account of pancomputationalism , the idea that everything can be said to be computing everything. Gualtiero Piccinini proposes an account of computation based on mechanical philosophy . It states that physical computing systems are types of mechanisms that, by design, perform physical computation, or the manipulation (by a functional mechanism) of a "medium-independent" vehicle according to
5963-608: The mid-1960s, with the work of Henry P. McKean Jr. on Markov interpretations of a class of nonlinear parabolic partial differential equations arising in fluid mechanics. An earlier pioneering article by Theodore E. Harris and Herman Kahn, published in 1951, used mean-field genetic -type Monte Carlo methods for estimating particle transmission energies. Mean-field genetic type Monte Carlo methodologies are also used as heuristic natural search algorithms (a.k.a. metaheuristic ) in evolutionary computing. The origins of these mean-field computational techniques can be traced to 1950 and 1954 with
6052-433: The most important and influential ideas of the 20th century, and they have enabled many scientific and technological breakthroughs. Monte Carlo methods also have some limitations and challenges, such as the trade-off between accuracy and computational cost, the curse of dimensionality, the reliability of random number generators, and the verification and validation of the results. Monte Carlo methods vary, but tend to follow
6141-417: The name "la loi des grands nombres" ("the law of large numbers"). Thereafter, it was known under both names, but the "law of large numbers" is most frequently used. After Bernoulli and Poisson published their efforts, other mathematicians also contributed to refinement of the law, including Chebyshev , Markov , Borel , Cantelli , Kolmogorov and Khinchin . Markov showed that the law can apply to
6230-625: The necessary data, such as the average distance a neutron would travel in a substance before it collided with an atomic nucleus and how much energy the neutron was likely to give off following a collision, the Los Alamos physicists were unable to solve the problem using conventional, deterministic mathematical methods. Ulam proposed using random experiments. He recounts his inspiration as follows: The first thoughts and attempts I made to practice [the Monte Carlo Method] were suggested by
6319-403: The noise of the system. Another pioneering article in this field was Genshiro Kitagawa's, on a related "Monte Carlo filter", and the ones by Pierre Del Moral and Himilcon Carvalho, Pierre Del Moral, André Monin and Gérard Salut on particle filters published in the mid-1990s. Particle filters were also developed in signal processing in 1989–1992 by P. Del Moral, J. C. Noyer, G. Rigal, and G. Salut in
6408-491: The number of flips becomes large. That is, the probability that the absolute difference is a small number approaches zero as the number of flips becomes large. Also, almost surely the ratio of the absolute difference to the number of flips will approach zero. Intuitively, the expected difference grows, but at a slower rate than the number of flips. Another good example of the LLN is the Monte Carlo method . These methods are
6497-418: The number of successful plays. This was already possible to envisage with the beginning of the new era of fast computers, and I immediately thought of problems of neutron diffusion and other questions of mathematical physics, and more generally how to change processes described by certain differential equations into an equivalent form interpretable as a succession of random operations. Later [in 1946], I described
6586-586: The number required, the calculation of that number is quite stable." The following algorithm computes s in one pass while minimizing the possibility that accumulated numerical error produces erroneous results: Note that, when the algorithm completes, m k is the mean of the k results. n is sufficiently large when n ≥ s 2 / ( z ϵ ) 2 {\displaystyle n\geq s^{2}/(z\epsilon )^{2}} If n ≤ k , then m k = m ; sufficient sample simulations were done to ensure that m k
6675-399: The other the "strong" law, in reference to two different modes of convergence of the cumulative sample means to the expected value; in particular, as explained below, the strong form implies the weak. There are two different versions of the law of large numbers that are described below. They are called the strong law of large numbers and the weak law of large numbers . Stated for
6764-573: The proofs. This assumption of finite variance is not necessary . Large or infinite variance will make the convergence slower, but the LLN holds anyway. Mutual independence of the random variables can be replaced by pairwise independence or exchangeability in both versions of the law. The difference between the strong and the weak version is concerned with the mode of convergence being asserted. For interpretation of these modes, see Convergence of random variables . The weak law of large numbers (also called Khinchin 's law) states that given
6853-432: The publications on Sequential Monte Carlo methodologies, including the pruning and resample Monte Carlo methods introduced in computational physics and molecular chemistry, present natural and heuristic-like algorithms applied to different situations without a single proof of their consistency, nor a discussion on the bias of the estimates and on genealogical and ancestral tree based algorithms. The mathematical foundations and
6942-483: 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 . 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
7031-857: The sample mean of this sequence converges in probability to E[ f ( X , θ )]. This is the pointwise (in θ ) convergence. A particular example of a uniform law of large numbers states the conditions under which the convergence happens uniformly in θ . If Then E[ f ( X , θ )] is continuous in θ , and sup θ ∈ Θ ‖ 1 n ∑ i = 1 n f ( X i , θ ) − E [ f ( X , θ ) ] ‖ → P 0. {\displaystyle \sup _{\theta \in \Theta }\left\|{\frac {1}{n}}\sum _{i=1}^{n}f(X_{i},\theta )-\operatorname {E} [f(X,\theta )]\right\|{\overset {\mathrm {P} }{\rightarrow }}\ 0.} This result
7120-606: The seminal work of Marshall N. Rosenbluth and Arianna W. Rosenbluth . The use of Sequential Monte Carlo in advanced signal processing and Bayesian inference is more recent. It was in 1993, that Gordon et al., published in their seminal work the first application of a Monte Carlo resampling algorithm in Bayesian statistical inference. The authors named their algorithm 'the bootstrap filter', and demonstrated that compared to other filtering methods, their bootstrap algorithm does not require any assumption about that state-space or
7209-412: The series, keeping the expected value constant. If the variances are bounded, then the law applies, as shown by Chebyshev as early as 1867. (If the expected values change during the series, then we can simply apply the law to the average deviation from the respective expected values. The law then states that this converges in probability to zero.) In fact, Chebyshev's proof works so long as the variance of
7298-461: The simulations. Monte Carlo simulations invert this approach, solving deterministic problems using probabilistic metaheuristics (see simulated annealing ). An early variant of the Monte Carlo method was devised to solve the Buffon's needle problem , in which π can be estimated by dropping needles on a floor made of parallel equidistant strips. In the 1930s, Enrico Fermi first experimented with
7387-404: The simulations’ results. It has no restrictions on the probability distribution of the inputs to the simulations, requiring only that the inputs are randomly generated and are independent of each other and that μ exists. A sufficiently large n will produce a value for m that is arbitrarily close to μ ; more formally, it will be the case that, for any ε > 0, | μ – m | ≤ ε . Typically,
7476-789: The variance of the average of n random variables is Var ( X ¯ n ) = Var ( 1 n ( X 1 + ⋯ + X n ) ) = 1 n 2 Var ( X 1 + ⋯ + X n ) = n σ 2 n 2 = σ 2 n . {\displaystyle \operatorname {Var} ({\overline {X}}_{n})=\operatorname {Var} ({\tfrac {1}{n}}(X_{1}+\cdots +X_{n}))={\frac {1}{n^{2}}}\operatorname {Var} (X_{1}+\cdots +X_{n})={\frac {n\sigma ^{2}}{n^{2}}}={\frac {\sigma ^{2}}{n}}.} which can be used to shorten and simplify
7565-494: The weak law applying even though the expected value does not exist. The strong law of large numbers (also called Kolmogorov 's law) states that the sample average converges almost surely to the expected value That is, Pr ( lim n → ∞ X ¯ n = μ ) = 1. {\displaystyle \Pr \!\left(\lim _{n\to \infty }{\overline {X}}_{n}=\mu \right)=1.} What this means
7654-455: The weak law states that for any nonzero margin specified ( ε ), no matter how small, with a sufficiently large sample there will be a very high probability that the average of the observations will be close to the expected value; that is, within the margin. As mentioned earlier, the weak law applies in the case of i.i.d. random variables, but it also applies in some other cases. For example, the variance may be different for each random variable in
7743-613: The work of Alan Turing on genetic type mutation-selection learning machines and the articles by Nils Aall Barricelli at the Institute for Advanced Study in Princeton, New Jersey . Quantum Monte Carlo , and more specifically diffusion Monte Carlo methods can also be interpreted as a mean-field particle Monte Carlo approximation of Feynman – Kac path integrals. The origins of Quantum Monte Carlo methods are often attributed to Enrico Fermi and Robert Richtmyer who developed in 1948
7832-467: Was first proved by Jacob Bernoulli . It took him over 20 years to develop a sufficiently rigorous mathematical proof which was published in his Ars Conjectandi ( The Art of Conjecturing ) in 1713. He named this his "Golden Theorem" but it became generally known as " Bernoulli's theorem ". This should not be confused with Bernoulli's principle , named after Jacob Bernoulli's nephew Daniel Bernoulli . In 1837, S. D. Poisson further described it under
7921-902: Was inspired by his uncle's gambling habits. Monte Carlo methods are mainly used in three distinct problem classes: optimization, numerical integration, and generating draws from a probability distribution. They can also be used to model phenomena with significant uncertainty in inputs, such as calculating the risk of a nuclear power plant failure. Monte Carlo methods are often implemented using computer simulations, and they can provide approximate solutions to problems that are otherwise intractable or too complex to analyze mathematically. Monte Carlo methods are widely used in various fields of science, engineering, and mathematics, such as physics, chemistry, biology, statistics, artificial intelligence, finance, and cryptography. They have also been applied to social sciences, such as sociology, psychology, and political science. Monte Carlo methods have been recognized as one of
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