Markov theorem

The matrix IzmxN containing the model parameters can then be directly estimated using the Gauss-Markov theorem to find the least squares solution of generic linear problems written in matrix form [37] ... 

The probability density functions of the observations are generally unknown, but the Gauss-Markov theorem ensures that least-squares is always an acceptable estimator. However, the results of least squares are strongly influenced by discordant observations, so-called outliers. The robust-resistant techniques use weight-modification functions of O—Cy which progressively down-weight outliers. Tnese functions implicitly define probability functions p. They may alternatively be interpreted as an appreciation of the reliability of certain measurements. This approaches the frequently used option to simply omit discordant observations because they are judged to be unreliable. 

It should be emphasized that these results do not require an assumption of the type of the error distribution function. Moreover, it can be shown (the Gauss-Markov theorem Bard, 1974, p. 59 Hamilton, 1964, Chap. 4 Hudson, 1963, Chap. 5 Seber, 1977, Chap. 3) that... 

Also, one can be interested in the probability of reaching the boundary by a Markov process, having random initial distribution. In this case, one should first solve the task with the fixed initial value xo and after that, averaging for all possible values of xo should be performed. If an initial value xo is distributed in the interval (c, d ) D (c, d) with the probability Vko(x o). then, following the theorem about the sum of probabilities, the complete probability to reach... 

Proof with the Escape-Rate Theory Markov Chains and Information Theoretic Aspects Eluctuation Theorem for the Currents... 

Doob s theorem states that a Gaussian process is Markovian if and only if its time correlation function is exponential. It thus follows that V is a Gaussian-Markov Process. From this it follows that the probability distribution, P(V, t), in velocity space satisfies the Fokker-Planck equation,... 

If the random force has a delta function correlation function then K(t) is a delta function and the classical Langevin theory results. The next obvious approximation to make is that F is a Gaussian-Markov process. Then is exponential by Doob s theorem and K t) is an exponential. The velocity autocorrelation function can then be found. This approximation will be discussed at length in a subsequent section. The main thing to note here is that the second fluctuation dissipation theorem provides an intuitive understanding of the memory function. ... 

Some restrictions are imposed when we start the application of limit theorems to the transformation of a stochastic model into its asymptotic form. The most important restriction is given by the rule where the past and future of the stochastic processes are mixed. In this rule it is considered that the probability that a fact or event C occurs will depend on the difference between the current process (P(C) = P(X(t)e A/V(X(t))) and the preceding process (P (C/e)). Indeed, if, for the values of the group (x,e), we compute = max[P (C/e) — P(C)], then we have a measure of the influence of the process history on the future of the process evolution. Here, t defines the beginning of a new random process evolution and tIt- gives the combination between the past and the future of the investigated process. If a Markov connection process is homogenous with respect to time, we have = 1 or Tt O after an exponential evolution. If Tt O when t increases, the influence of the history on the process evolution decreases rapidly and then we can apply the first type limit theorems to transform the model into an asymptotic... 

Several reviews have focussed on stochastic dynamics. Harris and Schiitz discuss a range of fluctuation theorems (the Jarzynski equality, the ES FR and the GC FR) in the context of stochastic, Markov systems, but in a widely applicable context. They investigate the conditions under which they apply, including an analysis of the conditions under which the GC FR is valid. In 2006, Gaspard reviewed studies where a stochastic approach to the treatment of boundaries is used to obtain FRs for the current in nanosystems, with a focus on work from his group, and also published a review on Hamiltonian systems that includes a discussion on FRs and the JE. 

Population models describe the relationship between individuals and a population. Individual parameter sets are considered to arise from a joint population distribution described by a set of means and variances. The conditional dependencies among individual data sets, individual variables, and population variables can be represented by a graphical model, which can then be translated into the probability distributions in Bayes theorem. For most cases of practical interest, the posterior distribution is obtained via numerical simulation. It is also the case that the complexity of the posterior distribution for most PBPK models is such that standard MC sampling is inadequate leading instead to the use of Markov Chain Monte Carlo (MCMC) methods... 

Some sufficient conditions for a finite Markov chain to be ergodic are based on the following theorems, given without proof [17, pp.247-255]. The first one states that A finite irreducible aperiodic Markov chain is ergodic. Let ... 

On the basis of the following theorem, i.e., if the transition matrix P for a finite irreducible aperiodic Markov chain with Z states is doubly stochastic, then the stationary probabilities are given by... 

Automation of inference sometimes produces fundamental tensions. For example, some automated procedures have no guarantees of reliability—no theorems about their error rates, no confidence intervals—but in practice work much better than procedures that do have such guarantees. Which are to be preferred, and why In some cases, theoretically well-founded procedures produce results that are inferior to stupid procedirres. For example, in determining protein homologue families, a simple matching procedure appears to work as well or better than procedures using sophisticated Hidden Markov models. Which sort of procedure is to be preferred, and why ... 

Application of continuous-lag Markov chains requires the knowledge of lag-dependent transition probabilities. Through Sylvester s theorem (Agterberg, 1974 Carle and Fogg, 1997) these can be derived from ... 

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