Probability vectors

Thus, if an ensemble can be prepared that is at equilibrium, then one Metropolis Moni Carlo step should return an ensemble that is still at equilibrium. A consequence of this that the elements of the probability vector for the limiting distribution must satisfy ... 

Unidirectional kinetic processes cannot be immediately interpreted as Markov chains, since only the (1,1) element of the /- -matrix would differ from zero, violating the stochastic matrix constraints (Section II. 1). An artificial Markov matrix complying with this constraint can be visualized, however, with the understanding that no other element of this imbedded P-matrix, past the (1,1) element, will have a physical meaning. It follows that the initial state probability vector is non-zero only in its (1,1)... 

Mean-field theory can be used to predict the effects of mutation rate and parent fitness on the moments of the mutant fitness distribution (Voigt et al, 2000a). In this analysis, only the portion of the mutant distribution that is not dead (zero fitness) or parent (unmutated) is considered. The mutant effects are averaged over the transition probabilities without the cases of mutations to stop codons or when no mutations are made on a sequence. In order to obtain the fitness distribution, two probabilities are required (1) the probability pi(a) that a particular amino-acid identity a exists at a residue i, and (2) the transition probability that one amino acid will mutate into another Q = 1 — (1 — pm)3. The probability vectors p a) can be determined through a mean-held approach (Lee, 1994 Koehl and Delarue, 1996 Saven and Wolynes, 1997). The amino acid transition probabilities Q are calculated based on the special connectivity of the genetic code and the per-nucleotide mutation rate. Removing transitions to stop codons and unmutated sequences only requires the proper normalization of the probabilities pi and the moments. For example, the first moment of the fitness improvement w of the uncoupled fitness function is written as... 

The CBO matrix reflects the promoted, valence state of AO in the molecule, with the diagonal elements measuring the effective electron occupations of the basis functions, yiii = Ni = Npi. The AO-probability vector in this state, p = pi = Ni/N], groups the probabilities of the basis functions being occupied in the molecule. 

The spectral models may be regarded as a series of 20 probabilities of absorbance at each wavelength. Hence if the total absorbance over 20 wavelengths is summed to x, then the probability at each wavelength is simply the absorbance divided by x. Convert the three models into three probability vectors. 

The quadratic probability measure is related to the Brier quadratic score, which is a loss function for comparing two probability vectors, and is used for the elucidation of probabilities [3,4,5]. The QPM ranges from 0 to 1 with values closer to 1 being preferred, since this implies the classes can be differentiated with a higher degree of certainty. 

Condition (1) provides a clear distinction between the molecular dynamics and Monte Carlo methods, for in a molecular dynamics simulation aU of the states are connected in time. Suppose the system is in a state m. We denote the probability of moving to state n as 7t , . The various 7r , can be considered to constitute an N x N matrix Jt (the transition matrix), where N is the number of possible states. Each row of the transition matrix sums to 1 (i.e. the sum of the probabilities 7r , for a given m equals 1). The probability that the system is in a particular state is represented by a probability vector p ... 

We can illustrate the use of this transition matrix as follows. Suppose the initial probability vector is (1,0) and so the system starts with a 100% probability of being in state 1 and no probability of being in state 2. Then the second state is given by ... 

Earlier in the book, the five most probable vectors of contamination were identified as ... 

Let Hh denote the total material balances in ascending order Vhm nj/h) and ph, the associated probability vector. For a given total safety stock level along the pipeline r, the a-service level of the pipeline storage system is defined by Ph(,) with / determined such that r > and r < To define a safety stock level satisfying a desired... 

Because of these connections to probability vectors, scalar products of two distinct compatible probability distributions are always positive definite, so we have ... 

The first passage time D(/) and the stationary probabilities vector rr (and consequently 4) can be estimated from Xt = (X, i > 0) as follows. A Markov process Xj (initial state / and transition matrix M) is observed with the two time sequences /j and L so that ... 

The probability vector of staying in the macro-states given in (3) is for this particular case... 

To use the Bayesian model, the joint prior distribution on the probability vector Px j,f Px i) needs to be determined. This can be achieved by quantifying expert uncertainty on the random variables px, j. The dependency existing betweenpxi j may be then determined by computing rank correlations. Considering a rank correlation of 1 simplifies considerably subsequent calculations and is also concepmally consistent, as it impMes that large probabilities in one location are associated with large probabilities in another location. 

The state probability vector of the system evolves according to the differential equation ... 

Effect of management decision is emulated adopting a stochastic binary matrix R, named reset matrix, which operates directly on the state probability vector in the fhllness of time. This matrix is used to force diagonal transitions in Fig. 5 to occur instantaneously. Reset matrix R is formulated setting, for every i and k, the element r, to 1 if transition from state i to state k is one of those forced by the management actions, and setting rn to 0 otherwise. 

High dimension probability vectors are difficult to interpret. More insight is obtained by aggregating the probability vector. Typical aggregated reliability states are Functioning system. Failed system (.Aggregation of a probability vector can be defined by... 

There is still a need to find an expression for the initial transient state probability vector q, which requires the fraction of unreacted group to be calculated. We do this by using the following inventory. 

Note that aperiodicity is often a strong assumption in Markov chains. One can still carry out the steady state probability computations even if a Markov chain has a period greater than one. Components of the steady state probability vector n can then be interpreted as long run proportions of time that the underlying stochastic process would be in a given state. 

Shannon entropy [3] in the (normalized) discrete probability vector/ = p or in the continuous probability density p(r). 

Entropy deficiency thus provides a measure of information resemblance between two probability vectors or densities. The more the two distributions differ from each other, the larger the information distance becomes. For individual events, the logarithm of probability ratio / = log(p /p°) or 7(r) = log[p(r)/p°(r)], called surprisal, provides a measure of the event information in the current distribution relative to that in the reference distribution. The equality in Equation 8.2 takes place only for vanishing surprisal for all events (i.e., when the two probability distributions are identical). 

This survey of IT probes of chemical bonds continues with some rudiments on the entropic characteristics of the dependent probability distributions and information descriptors of a transmission of signals in communication systems [3,4,7,8]. For two mutually dependent (discrete) probability vectors of two separate sets of events a and b, P(a) = P(fl) =Pi) =p and P(b) = P(pp = qj = q, one decomposes the joint probabilities of the simultaneous events a/ b = [a,Air) into these two schemes, P afdb) = P(fli/ b) = Jt,y] = 31, as products of the margin probabilities of events in one set, say a, and the corresponding conditional probabilities P( la) = [P(/li) = 3t,y/Pj] of outcomes in set b, given that events a have already occurred [jty=p (j i). The relevant normalization conditions for the joint and conditional probabilities then read ... 

FIGURE 8.12 Entropy for two dependent probability distributions p and q. Two circles enclose areas representing the entropies S(p) and S(q) of two separate probability vectors, while their common (overlap) area corresponds to the mutual information I(p. q) in these two distributions. The remaining part of each circle represents the corresponding conditional entropy S(p q) or 5(g p), measuring the residual uncertainty about events in one set, when one has full knowledge of the occurrence of events in the other set of outcomes. The area enclosed by the envelope of two circles then represents the entropy of the product (joint) distribution S(n) = S( (aAb)) = S(p) + S(q) - I(p q) = S(p) + S q p) = S(q) + S(plq). 

The reader can now recall completely what he knows about vectors. If not particularly interested in theoretical linear algebra, but mastering vectors rather operationally as objects of a computation routine, he imagines probably vectors just as certain ordered iV-tuples . That s what they are in common practice. 

In this paper we propose a commutative version of the (Olbrich 1965) representation. We then use this version to define a hypothesis test to test for differences in contingency tables. The test statistic is the angle between the two multinomial realisations. We use the variance stabilising property of the transformation in order to approximate the distribution under the null hypothesis that the two realisations come from the same underlying probability vector. The test does not require intensive numerical methods. 

Suppose we have two multinomial reaUsations x and y with underlying probability vectors p and q respectively. We wish to test whether the two vec-Fors come from the same underlying multinomial distribution. The hypothesis test will take the form... 

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