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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 Metropolis-Hastings algorithm sampling a normal one-dimensional posterior probability distribution.. In statistics and statistical physics, the Metropolis–Hastings algorithm is a Markov chain Monte Carlo (MCMC) method for obtaining a sequence of random samples from a probability distribution from which direct sampling is difficult.
When that happens, a random point from the path is chosen for the MCMC sample and the process is repeated from that new point. In detail, a binary tree is constructed to trace the path of the leap frog steps. To produce a MCMC sample, an iterative procedure is conducted.
The deviance information criterion (DIC) is a hierarchical modeling generalization of the Akaike information criterion (AIC). It is particularly useful in Bayesian model selection problems where the posterior distributions of the models have been obtained by Markov chain Monte Carlo (MCMC) simulation.
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.
In computational statistics, the preconditioned Crank–Nicolson algorithm (pCN) is a Markov chain Monte Carlo (MCMC) method for obtaining random samples – sequences of random observations – from a target probability distribution for which direct sampling is difficult.
Figure 1. Probabilistic parameters of a hidden Markov model (example) X — states y — possible observations a — state transition probabilities b — output probabilities. In its discrete form, a hidden Markov process can be visualized as a generalization of the urn problem with replacement (where each item from the urn is returned to the original urn before the next step). [7]
The basic idea of importance sampling is to sample the states from a different distribution to lower the variance of the estimation of E[X;P], or when sampling from P is difficult. This is accomplished by first choosing a random variable L ≥ 0 {\displaystyle L\geq 0} such that E [ L ; P ] = 1 and that P - almost everywhere L ( ω ) ≠ 0 ...