Probabilities of states

Experimentally, phases can be obtained by measurements of occupation probabilities of states using Eq. (9). (We have calculationally verified this for the case treated in [264].)... 

Figures 3.24 and 3.25, for example, show the probabilities for the 2 ° pos.sible configurations on an = 10 lattice obtained at time t = 10 for evolutions by rules R126 and R30 (both in class c3), respectively. The figures show that the evolution modifies the probabilities of states from initially being equally likely to distributions in which individual configurations can have widely differing final probabilities. The properties of the more probable states will then tend to dominate the statistical averages over the enseiiible.

P -order LST The l Lorder LST - which is identical to the mean field theory -is a refinement of the 0 -order LST in that it uses the probabilities of states of sites within blocks to estimate the probabilities of those blocks. Consider, for example, the set of LST equations for r = 1 given in eejuation 5,76. 

Then, letting Pa and pg be the probabilities of states a and /3, respectively, the probability that the overlap between a and (3 is equal to q is given by... 

Note that for both FEP and TI, the umbrella restraint introduces a term that depends on e+VblaJRT, which may (since Vbias is always > 0) fluctuate widely, especially if the biasing function is attempting to restrain the system to a conformation far from a local minimum. As a result, use of the umbrella term Equations 19 and 20 is often problematic. This has led to the development of alternate (but related) approaches to Umbrella Sampling.41,51,52 Many of these derive from the following equation, which relates the work function W to the probability of states, corrected for use of the biasing potential ... 

A multicanonical MC simulation is performed, for instance, with the usual Metropolis criterion [118] The transition probability of state x with potential energy E to state x with potential energy E is given by... 

With either formulation, the state probability

random effect for interindividual variability. With zero-inflated Poisson-based analyses, covariates can be evaluated as predictors of the probability of state, the mean number of events, or both. Under typically employed canonical links. 

Figure 7.21. Probabilities of states 44, 36 and 6 for steam valve actuator fault.

Figure 7.23. Probabilities of state 43, 35, and 13 for holding tube temperature sensor fault.

Consider M gas molecules first contained in one half of a vessel, as illustrated in Fig. 3.4, and then allowed to occupy the whole volume. The probability that state A could occur by chance is ( )M, that is the same as the chance that M objects will all be in one of two boxes between which they have been randomly distributed. We can write the relative probability of state A with respect to state B as... 

In agreement with the binding hypothesis it is observed that agents which suppress inactivation of Na+ channels also enhance the veratridine action, obviously by increasing the probability of state O which in turn feeds O. This is true for pronase (Scruggs and Narahashi 1982),... 

Recent applications of the Markov chain theory in geology have introduced continuous-lag modelling of spatial variability (Carle and Fogg, 1997 Fogg et al., 1998). The continuous-lag approach extends the probability of state transitions recorded at fixed intervals (discrete-lag) to any desired interval by considering conditional rates of... 

Equations 15.19 and 15.20 are the Baum-Welch re-estimation formulae for the means and co-variances of a HMM. A similar but slightly more complex formula can be derived for the transition probabilities. Of course, to apply equations 15.19 and 15.20, the probability of state occupation Lj t) must be calculated. This is done efficiently using the forward-backward algorithm. Let the forward probability tty(t) for some modelMwith A states be defined as... 

Notice that in the definitions above, the forward probability is a joint probability whereas the backward probability is a conditional probability. This somewhat asymmetric definition is deliberate since it allows the probability of state occupation to be determined by taking the product of the two probabilities. From the definitions. 

Considering different component states leads to the determination of probabilities of state transitions based on fadure injection and logical dependencies. The state transition probabdities used in equation (3) are defined as follows (16) ... 

W probability of state (I chemical potential e energy levels p vibration frequency S grand partition function a symmetry factor... 

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