![]() ![]() particles, individuals, walkers, agents, creatures, or phenotypes) interacts with the empirical measures of the process. The terminology mean field reflects the fact that each of the samples ( a.k.a. In contrast with traditional Monte Carlo and MCMC methodologies, these mean-field particle techniques rely on sequential interacting samples. A natural way to simulate these sophisticated nonlinear Markov processes 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. These models can also be seen as the evolution of the law of the random states of a nonlinear Markov chain. In other instances we are given a flow of probability distributions with an increasing level of sampling complexity (path spaces models with an increasing time horizon, Boltzmann–Gibbs measures associated with decreasing temperature parameters, and many others). 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 problems, the objective is generating draws from a sequence of probability distributions satisfying a nonlinear evolution equation. ![]() By the ergodic theorem, the stationary distribution is approximated by the empirical measures of the random states of the MCMC sampler. That is, in the limit, the samples being generated by the MCMC method will be samples from the desired (target) distribution. The central idea is to design a judicious Markov chain model with a prescribed stationary probability distribution. When the probability distribution of the variable is parameterized, mathematicians often use a Markov chain Monte Carlo (MCMC) sampler. the 'sample mean') of independent samples of the variable. 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. In principle, Monte Carlo methods can be used to solve any problem having a probabilistic interpretation. 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. ![]() Other examples include modeling phenomena with significant uncertainty in inputs such as the calculation of risk in business and, in mathematics, evaluation of multidimensional definite integrals with complicated boundary conditions. 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). Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. The underlying concept is to use randomness to solve problems that might be deterministic in principle. Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. Please edit it to move non-essential details to the body, or discuss this on the talk page. This article's lead section may be too long. ![]()
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