Integration, method Markov chains

The difficulty arises from the fact that the one-step transition probabilities of the Markov chain involve only ratios of probability densities, in which Z(N,V,T) cancels out. This way, the Metropolis Markov chain procedure intentionally avoids the calculation of the configurational integral, the Monte Carlo method not being able to directly apply equation (31). 

MCMC methods are essentially Monte Carlo numerical integration that is wrapped around a purpose built Markov chain. Both Markov chains and Monte Carlo integration may exist without reference to the other. A Markov chain is any chain where the current state of the chain is conditional on the immediate past state only—this is a so-called first-order Markov chain higher order chains are also possible. The chain refers to a sequence of realizations from a stochastic process. The nature of the Markov process is illustrated in the description of the MH algorithm (see Section 5.1.3.1). 

In the Monte Carlo method to estimate a many-dimensional integral by sampling the integrand. Metropolis Monte Carlo or, more generally, Markov chain Monte Carlo (MCMC), to which this volume is mainly devoted, is a sophisticated version of this where one uses properties of random walks to solve problems in high-dimensional spaces, particularly those arising in statistical mechanics. 

Bayesian statistical theory had been published, imtil 80 in twenty century Bayesian statistical theory has been in theory research phase, integral calculation is a big barrier in his development and application. However, Markov Chain Monte Carlo (MCMC) has been used to Bayesian statistical inference in recently, a main characteristic of this method is Metropolis-Hastings updating and Gibbs sampling, it can solve well the problem of numerical integration and sampling in multi dimensional distribution, which is convenient for posterior inference of parameters and accelerate the application of Bayesian theory. 

Typically, the required integrals are analytically insoluble. Consequently, Markov Chain Monte Carlo (MCMC) methods are used to sample distributions in a way that focuses the sampling in areas of high probability thus providing a means of efficient approximation to the desired integrals. This is the set of different classifiers that delivers the set of classification probabilities. 

ABSTRACT In most cases, Model Based Safety Analysis (MBSA) of critical systems focuses only on the process and not on the control system of this process. For instance, to assess the dependability attributes of power plants, only a model (Fault Tree, Markov chain. ..) of the physical components of the plant (pumps, steam generator, turbine, alternator. ..) is used. In this paper, we claim that for repairable and/or phased-mission systems, not only the process but the whole closed-loop system Proc-ess/Control must be considered to perform a relevant MBSA. Indeed, a part of the control functions aims to handle the dynamical mechanisms that change the mission phase as well as manage repairs and redundancies in the process. Therefore, the achievement of these mechanisms depends on the functional/dysfunctional status of the control components, on which these functions are implemented. A qualitative or quantitative analysis method which considers both the process and the control provides consequently more realistic results by integrating the failures of the control components that may lead to the non-achievement of these mechanisms. This claim is exemplified on an industrial study case issued from a power plant. The system is modeled by a BDMP (Boolean logic Driven Markov Process), assuming first that the control components are faultless, i.e. only the faults in the process are considered, and afterwards that they may fail. The minimal cut sequences of the system are computed in both cases. The comparison of these two sets of minimal cut sequences shows the benefit of the second approach. 

For general unidentifiable cases, the evidence integral in Eq. 40 can be computed using the transitional Markov chain Monte Carlo (TMCMC) method (Ching and Chen 2007). 

The most complex systems of kinetic equations cannot be solved analytically. In addition, when two of the differential equations of these systems describe processes that occur on drastically different timescales, their numerical integration using methods involving finite increments is unstable and unreliable. These methods are inherently deterministic, since their time evolution is continuous and dictated by the system of differential equations. Alternatively, we can apply stochastic methods to determine the rates of these reactions. These methods are based on the probabihty of a reaction occurring within an ensanble of molecules. This prob-abihstic formulation is a reflection either of the random nature of the coUisions that are responsible for bimolecular reactions or of the random decay of molecules undergoing unimolecular processes. Stochastic methods allow us to study complex reactions without either solving differential equations or supplying closed-form rate equations. The method of Markov chains... 

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