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Monte Carlo methods 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 mainly used in three problem classes: [2] optimization, numerical integration, and generating draws from a probability distribution.
In statistics, Markov chain Monte Carlo (MCMC) is a class of algorithms used to draw samples from a probability distribution.Given a probability distribution, one can construct a Markov chain whose elements' distribution approximates it – that is, the Markov chain's equilibrium distribution matches the target distribution.
The antithetic variates technique consists, for every sample path obtained, in taking its antithetic path — that is given a path {, …,} to also take {, …,}.The advantage of this technique is twofold: it reduces the number of normal samples to be taken to generate N paths, and it reduces the variance of the sample paths, improving the precision.
The general motivation to use the Monte Carlo method in statistical physics is to evaluate a multivariable integral. The typical problem begins with a system for which the Hamiltonian is known, it is at a given temperature and it follows the Boltzmann statistics .
This method, also known as Monte Carlo cross-validation, [21] [22] creates multiple random splits of the dataset into training and validation data. [23] For each such split, the model is fit to the training data, and predictive accuracy is assessed using the validation data. The results are then averaged over the splits.
In computational statistics, the Metropolis-adjusted Langevin algorithm (MALA) or Langevin Monte Carlo (LMC) is a Markov chain Monte Carlo (MCMC) method for obtaining random samples – sequences of random observations – from a probability distribution for which direct sampling is difficult.
The control variates method is a variance reduction technique used in Monte Carlo methods. It exploits information about the errors in estimates of known quantities ...
Hamiltonian Monte Carlo sampling a two-dimensional probability distribution The Hamiltonian Monte Carlo algorithm (originally known as hybrid Monte Carlo ) is a Markov chain Monte Carlo method for obtaining a sequence of random samples whose distribution converges to a target probability distribution that is difficult to sample directly.