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- A Gentle Introduction to Markov Chain Monte Carlo for Probability
- Markov Chain Monte Carlo Simulations and Their Statistical Analysis
- Markov Chain Monte Carlo Methods for Simulations of Biomolecules

*Metrics details. In quantitative biology, mathematical models are used to describe and analyze biological processes. The parameters of these models are usually unknown and need to be estimated from experimental data using statistical methods.*

Rugged Free Energy Landscapes pp Cite as. The computer revolution has been driven by a sustained increase of computational speed of approximately one order of magnitude a factor of ten every 5 years since about In natural sciences, this has led to a continuous increase of the importance of computer simulations.

## A Gentle Introduction to Markov Chain Monte Carlo for Probability

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. 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 mainly used in three problem classes: [1] 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.

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Important decisions in the oil industry rely on reservoir simulation predictions. Unfortunately, most of the information available to build the necessary reservoir simulation models are uncertain, and one must quantify how this uncertainty propagates to the reservoir predictions. Recently, ensemble methods based on the Kalman filter have become very popular due to its relatively easy implementation and computational efficiency. However, ensemble methods based on the Kalman filter are developed based on an assumption of a linear relationship between reservoir parameters and reservoir simulation predictions as well as the assumption that the reservoir parameters follows a Gaussian distribution, and these assumptions do not hold for most practical applications. When these assumptions do not hold, ensemble methods only provide a rough approximation of the posterior probability density functions pdf 's for model parameters and predictions of future reservoir performance. The Markov chain Monte Carlos MCMC method provides the means to sample the posterior pdf, although with an extremely high computational cost because, for each new state proposed in the Markov chain, the evaluation of the acceptance probability requires one reservoir simulation run.

## Markov Chain Monte Carlo Simulations and Their Statistical Analysis

Model personalization requires the estimation of patient-specific tissue properties in the form of model parameters from indirect and sparse measurement data. Moreover, a low-dimensional representation of the parameter space is needed, which often has a limited ability to reveal the underlying tissue heterogeneity. As a result, significant uncertainty can be associated with the estimated values of the model parameters which, if left unquantified, will lead to unknown variability in model outputs that will hinder their reliable clinical adoption. Probabilistic estimation of model parameters, however, remains an unresolved challenge. Direct Markov Chain Monte Carlo MCMC sampling of the posterior distribution function pdf of the parameters is infeasible because it involves repeated evaluations of the computationally expensive simulation model. To accelerate this inference, one popular approach is to construct a computationally efficient surrogate and sample from this approximation. However, by sampling from an approximation, efficiency is gained at the expense of sampling accuracy.

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By constructing a Markov chain that has the desired distribution as its equilibrium distribution , one can obtain a sample of the desired distribution by recording states from the chain. The more steps are included, the more closely the distribution of the sample matches the actual desired distribution. Various algorithms exist for constructing chains, including the Metropolis—Hastings algorithm. MCMC methods are primarily used for calculating numerical approximations of multi-dimensional integrals , for example in Bayesian statistics , computational physics , [1] computational biology [2] and computational linguistics. In Bayesian statistics, the recent development of MCMC methods has made it possible to compute large hierarchical models that require integrations over hundreds to thousands of unknown parameters. In rare event sampling , they are also used for generating samples that gradually populate the rare failure region. Markov chain Monte Carlo methods create samples from a continuous random variable , with probability density proportional to a known function.

This book teaches modern Markov chain Monte Carlo (MC) simulation techniques step by step. The material should be accessible to advanced undergraduate.

## Markov Chain Monte Carlo Methods for Simulations of Biomolecules

Probabilistic inference involves estimating an expected value or density using a probabilistic model. Often, directly inferring values is not tractable with probabilistic models, and instead, approximation methods must be used. Markov Chain Monte Carlo sampling provides a class of algorithms for systematic random sampling from high-dimensional probability distributions. Unlike Monte Carlo sampling methods that are able to draw independent samples from the distribution, Markov Chain Monte Carlo methods draw samples where the next sample is dependent on the existing sample, called a Markov Chain. This allows the algorithms to narrow in on the quantity that is being approximated from the distribution, even with a large number of random variables.

This article provides an introduction to Markov chain Monte Carlo methods in statistical inference. Over the past twelve years or so, these have revolutionized what can be achieved computationally, especially in the Bayesian paradigm. Markov chain Monte Carlo has exactly the same goals as ordinary Monte Carlo and both are intended to exploit the fact that one can learn about a complex probability distribution if one can sample from it. Although the ordinary version can only rarely be implemented, it is convenient initially to presume otherwise and to focus on the rationale of the sampling approach, rather than computational details. The article then moves on to describe implementation via Markov chains, especially the Hastings algorithm, including the Metropolis method and the Gibbs sampler as special cases.

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Может случиться так, что компьютер, найдя нужный ключ, продолжает поиски, как бы не понимая, что нашел то, что искал. - Стратмор смотрел на нее отсутствующим взглядом. - Я полагаю, у этого алгоритма меняющийся открытый текст. Сьюзан затаила дыхание. Первое упоминание о меняющемся открытом тексте впервые появилось в забытом докладе венгерского математика Джозефа Харне, сделанном в 1987 году.

Кто он. - Понятия не имею. - Похож на китайца.

*Хейл с перепачканным кровью лицом быстро приближался к. Его руки снова обхватили ее - одна сдавила левую грудь, другая - талию - и оторвали от двери.*

[Book] Markov Chain Monte Carlo Simulations And Their Statistical Analysis: applied Markov chain Monte Carlo methods to produce more accurate emphasizing techniques, such as the PDF decomposition, that arise in a.

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