Markov chains Metropolis algorithm

The MC method can be implemented by a modification of the classic Metropolis scheme [25,67]. The Markov chain is generated by a three-step sequence. The first step is identical to the classic Metropolis algorithm a randomly selected molecule i is displaced within a small cube of side length 26r centered on its original position... 

This way the generated configurations are included with probability which is obtained from the transition probabilities The Metropolis algorithm is a subset of the use of Markov chains to sample configuration space. 

The Metropolis-Hastings algorithm is the most general form of the MCMC processes. It is also the easiest to conceptualize and code. An example of pseudocode is given in the five-step process below. The Markov chain process is clearly shown in the code, where samples that are generated from the prior distribution are accepted as arising from the posterior distribution at the ratio of the probability of the joint... 

Because step 2 of the adapted Metropolis algorithm for GCEMC simulations involves a change in density by 1/V between members — 1 and n of the Markov chain, some care has to be taken in computing U if the interaction potential is short-range, that is, if it decays sufficiently rapidly but does not go to zero at any finite separation between molecules. An example is the Leonard-Jones (12,6) (LJ) potential defined by... 

Thus, we take advantage of the accuracy, robustness and efficiency of the direct problem solution, to tackle the associated inverse heat transfer problem analysis [26, 27] towards the simultaneous estimation of momentum and thermal accommodation coefficients in micro-channel flows with velocity slip and temperature jump. A Bayesian inference approach is adopted in the solution of the identification problem, based on the Monte Carlo Markov Chain method (MCMC) and the Metropolis-Hastings algorithm [28-30]. Only simulated temperature measurements at the external faces of the channel walls, obtained for instance via infrared thermography [30], are used in the inverse analysis in order to demonstrate the capabilities of the proposed approach. A sensitivity analysis allows for the inspection of the identification problem behavior when the external wall Biot number is also included among the parameters to be estimated. 

The most commonly used Monte Carlo method with Markov Chain algorithms are the Metropolis-Hastings, here employed, and the Gibbs sampler [28, 29]. 

The Metropolis algorithm generates a Markov chain of states. A Markov chain satisfies the following two conditions ... 

Adaptive Markov Chain Monte Carlo Simulation 2.5.3.1 Metropolis-Hastings Algorithm... 

The practical problem is how to generate a set X, / = 1,2,..., M according to the desired probability density P(X). In general, the complicated probability density is not completely known in advance. Especially the normalization is often unknown. The Metropolis algorithm offers an efficient way of generating such a set as a sequence of weakly correlated position vectors (the Markov chain) without any knowledge of normalization. The Metropolis method consists of the following steps ... 

In Chapter 7 we develop a method for finding a Markov chain that has good mixing properties. We will use the Metropolis-Hastings algorithm with heavy-tailed independent candidate density. We then discuss the problem of statistical inference on the sample from the Markov chain. We want to base our inferences on an approximately random sample from the posterior. This requires that we determine... 

In Section 6.1 we show how the Metropolis-Hastings algorithm can be used to find a Markov chain that has the posterior as its long-run distribution in the case of a single parameter. There are two kinds of candidate densities we can use, random-walk candidate densities, or independent candidate densities. We see how the chain... 

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