Dirichlet distribution

Because we are dealing with count data and proportions for the values qi, the appropriate conjugate prior distribution for the q s is the Dirichlet distribution,... 

A prior distribution for sequence profiles can be derived from mixtures of Dirichlet distributions [16,51-54]. The idea is simple Each position in a multiple alignment represents one of a limited number of possible distributions that reflect the important physical forces that determine protein structure and function. In certain core positions, we expect to get a distribution restricted to Val, He, Met, and Leu. Other core positions may include these amino acids plus the large hydrophobic aromatic amino acids Phe and Trp. There will also be positions that are completely conserved, including catalytic residues (often Lys, GIu, Asp, Arg, Ser, and other polar amino acids) and Gly and Pro residues that are important in achieving certain backbone conformations in coil regions. Cys residues that form disulfide bonds or coordinate metal ions are also usually well conserved. 

A prior distribution of the probabilities of the 20 amino acids at a particular position in a multiple alignment can be represented by a Dirichlet distribution, described in Section lI.E. That is, it is an expression of the values of the probabilities of each residue type r, where r ranges from 1 to 20, and E( i0,. = 1 ... 

CX.0 = Z(=iCx.r represents the total number of counts that the prior distribution represents, and the a, the counts for each type of amino acid (not necessarily integers). Because different distributions will occur in multiple sequence alignments, the prior distribution for any position should be represented as a mixture of N Dirichlet distributions ... 

In the previous sections our discussions has concentrated on just two r.v. s. We refer to their joint distribution as a bivariate distribution. When more than two r.v. s, say Xi, X2,..., Xk, are jointly distributed, we can similarly define their joint p.m.f. (or p.d.f.), denoted by/(xi, X2,..., x ), referred to as a multivariate distribution. The properties of a multivariate distribution are namral extensions of the bivariate distribution properties. Examples of a multivariate distribution are multinomial, multivariate, normal, and Dirichlet distributions. 

In this section, we introduce a multinomial distribution, an important family of discrete multivariate distributions. This family generalized the binomial family to the situation in which each trial has n (rather than two) distinct possible outcomes. Then multivariate normal and Dirichlet distributions wUl be discussed. 

The Dirichlet distribution, often denoted Dir(a), is a family of continuous multivariate probability distributions parameterized by the vector a of positive real numbers. It is the multivariate generalization of the beta distribution and conjugate prior of the... 

In the case study, we considered the weight vector with a uniform distribution, that is, no prior preference information available, and with a Dirichlet distribution with parameter vector Wg = (0.48,0.29,0.04,0.19) so that the average weights are the same as their relative importance. Evaluation was based... 

Keywords Reliabihty growth Model Bayesian analysis Dirichlet distribution MCMC simulation Gibbs... 

Order Dirichlet distribution The expression of order Dirithlet distribution is following. 

Inspection of above analysis of order Dirichlet distribution shows that in whole period of product development there is only a shape parameter p describing the variance of product reliability in various test phases, however experts usually need to know test results and correct approach of last phase so that they can give accurate reliability assessment of next one. 

A disadvantage of the order Dirichlet distribution is that it only has one shape parameter p to describe the variance of product rehabdity in various test periods, and is not consistent with present practice. Based on general Bate distribution of various test phases Reference [14] established a new prior distribution family by conditional distribution modahty, which is fit for rehabdity growth model of new product. The distribution eliminates some of disadvantage of order Dirichlet distribution, and can express wed the prior information. 

The reference [15] analyzed the reliability growth of product by order Dirichlet distribution, and assiun-ing value of parameter p is 44. Niunber of Rehabdity growth test K , success number Si as well as point estimation value of Ri based on prior information have been shown in the second column, the third column and the fourth colimm of table 1. 

Notice in table 3 that variance of posterior estimation value based on a new prior distribution is bigger than variance of order Dirichlet distribution, especially in the early two phases. Comparing the lower bound Ric,l(0.90) of 90% confidence limit, the posterior estimation value based on a new prior distribution is smaller. Whereas the posterior estimation value based on order Dirichlet distribution is evidently effected by prior information, i.e. order Dirichlet distribution has a mainly influence on posterior estimation value. 

Inspection of the table 4 shows that if employing order Dirichlet distribution to depict the expert experience when the test sample is smaller, the prior distribution will be whip hand so that posterior estimation is mainly determined by prior information. It is not consistent with present practice. 

Dirichlet distribution can take full advantage of history information, expert experience and test date of various phases. 

This paper presents a new method for the application of Bayesian theory and technology to product reliability growth during product development phase. The research work mainly focuses on how to determent prior distribution parameters of a new Dirichlet distribution and presents the relevant optimization model and method, finally demonstrate the validity of Bayesian reliability growth model by WinBUGS software. The conclusions as following ... 

Another important problem in the atmospheric chemistiy models of Titan is the handling of the uncertainty of reaction branching ratios. This can be an important issue for the uncertainty analysis of many other reaction kinetic models. Chemical kinetic databases provide the uncertainty of rate coefficients independently of each other. Yet, for multichannel reactions using a direct method, it is easier to measure the overall rate coefficient than the rate coefficients of the constituent reaction steps. The branching ratios are then determined in other measurements. However, it is important to note that the branching ratios are correlated, since their sum has a unit value. Carrasco et al. (Carrasco and Pemot 2007 Plessis et al. 2010) demonstrated that the correlated branching ratios follow a Dirichlet distribution. The method was applied to the case of Titan ionospheric chemistry and used for the estimation of the effect of branching ratio correlations on the uncertainty of calculated concentrations. 

Carrasco, N Pemot, P. Modeling of branching ratio uncratarnty in chemical networks by Dirichlet distributions. J. Phys. Chem. A 111, 3507-3512 (2007)... 

The unknown parameters 0i,...,0 are assumed to have a Dirichlet distribution with parameters ai,...,a , that is,... 

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