"The underlying concept is to use randomness to solve problems that might be deterministic in principle."
A probabilistic modeling technique that uses random sampling to obtain numerical results.
Probability Theory: The foundation of Monte Carlo simulation, probability theory is the study of random events and their likelihood of occurrence.
Random Variables: Random variables are variables whose values depend on chance. They are used to model random natural phenomena such as the weather or the outcome of a coin toss.
Probability Distributions: Probability distributions explain how likely it is that different values of a random variable will occur. The most commonly used are the normal and uniform distributions.
Monte Carlo Simulation Basics: Monte Carlo simulation is a computational technique used to estimate the probability of outcomes in situations where there are many variables with uncertain values.
Generating Random Numbers: Random numbers are essential for Monte Carlo simulation. They can be generated using various algorithms, including the linear congruential generator and the Mersenne Twister.
Sampling Techniques: Sampling is the process of selecting a subset of data from a larger set. It is a crucial element of Monte Carlo simulation because it allows the modeler to estimate the probability of an event occurring.
Random Walks: A random walk is a mathematical model used to simulate random behavior such as stock prices, temperature fluctuations, or other unpredictable events.
Markov Chains: A Markov chain is a mathematical model that describes a sequence of events where the probability of each event depends only on the state of the previous event.
Monte Carlo Integration: Monte Carlo integration is a computational technique used to estimate the value of an integral. It is based on the idea of random sampling.
Variance Reduction Techniques: Variance reduction techniques are used to improve the accuracy and efficiency of Monte Carlo simulations. They include importance sampling, stratified sampling, and control variates.
Applications of Monte Carlo Simulation: Monte Carlo simulations are used in many fields, including finance, engineering, physics, and biology. They are especially useful in situations where traditional analytical methods are impractical or impossible.
Standard Monte Carlo simulation: This is the most basic and widely used form of Monte Carlo simulation, where random variables are generated based on certain probability distributions, and then used to simulate a range of different outcomes.
Markov Chain Monte Carlo (MCMC) simulation: This is a type of Monte Carlo simulation used to sample from complex probability distributions, where the probabilities for each event depend on the previous event in the chain.
Gibbs sampling: This is a type of MCMC simulation where the variables in the distribution are updated one by one, rather than all at once.
Hit-and-Run sampling: This is a type of random walk Monte Carlo simulation used when the probability distribution cannot be easily sampled from directly, and instead involves randomly walking through the distribution using a series of linear steps.
Importance sampling: This is a Monte Carlo simulation technique that involves sampling from an easier-to-sample distribution, then using weights to adjust the probabilities to estimate values of a harder-to-sample distribution.
Rejection sampling: This is a Monte Carlo simulation technique that involves generating samples from an alternate distribution, then accepting or rejecting them based on whether they fall within the desired distribution.
Stratified sampling: This is a Monte Carlo simulation technique that involves partitioning the parameter space into sub-domains, and then generating random samples from each sub-domain.
Latin hypercube sampling: This is a Monte Carlo simulation technique that involves partitioning the parameter space into equal sub-domains, and then drawing a random sample from each sub-domain.
Cross-validation: This is a Monte Carlo simulation technique used for model validation, where data is split into training and testing sets multiple times and the results are averaged to estimate how well the model generalizes to new data.
Bayesian Monte Carlo simulation: This is a Monte Carlo simulation technique that incorporates prior information about the variables being modeled, based on Bayesian probability theory.
"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."
"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."
"Monte Carlo–based predictions of failure, cost overruns and schedule overruns are routinely better than human intuition or alternative 'soft' methods."
"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."
"Evaluation of multidimensional definite integrals with complicated boundary conditions."
"When the probability distribution of the variable is parameterized, mathematicians often use a Markov chain Monte Carlo (MCMC) sampler. The central idea is to design a judicious Markov chain model with a prescribed stationary probability distribution."
"The samples being generated by the MCMC method will be samples from the desired (target) distribution."
"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."
"These mean-field particle techniques rely on sequential interacting samples."
"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."
"The computational cost associated with a Monte Carlo simulation can be staggeringly high."
"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."