Probabilities, joint

If this criterion is based on the maximum-likelihood principle, it leads to those parameter values that make the experimental observations appear most likely when taken as a whole. The likelihood function is defined as the joint probability of the observed values of the variables for any set of true values of the variables, model parameters, and error variances. The best estimates of the model parameters and of the true values of the measured variables are those which maximize this likelihood function with a normal distribution assumed for the experimental errors. 

Consider a diffusion process in which a molecule is initially in a conformation defined by the coordinates rj,. A transition probability can be constructed using a product of joint probabilities /5(r(,+i rj,) for moving between intermediate positions rj, and rj,+ i integrated over all intennediate states,... 

Before setting up priors and likelihoods, we can factor the joint probability of the core structure choice and the alignment t by using Bayes rule ... 

The cancer risk equation described below estimates tlie incremental individual lifetime cancer risk for simultaneous exposure to several carcinogens and is based on EPA s risk assessment guidelines. Tliis equation represents an approximation of the precise equation for combining risks wliich accounts for tlie joint probabilities of tlie same individual developing cancer as a consequence of exposure to two or more carcinogens. The difference between tlie precise equation and tlie approximation described is negligible for total cancer risks less tlian 0.1. Thus, tlie simple additive equation is appropriate for most risk assessments. The cancer risk equation for multiple substances is given by ... 

Mutual information, effectively measures the degree to which two probability distributions or, in the context of CA, two sites or blocks - are correlated. Given probability distributions pi and pj and the joint probability distribution py, 1 is defined by ... 

The properties of joint distribution functions can be stated most easily in terms of their associated probability density functions. The n + mth order joint probability density function px. . , ( > ) is defined by the equation... 

The physical interpretation of these joint moments is similar in every respect to the interpretation already given for moments of the form ak = E[k]. Thus, a . .. provides a measure of the center of mass of the joint probability density function p 1,...,second order central moments provide a measure of the spread of this density function about its center of mass.30... 

The joint characteristic function is thus seen to be the -dimensional Fourier transform of the joint probability density function The -dimensional Fourier transform, like its one-dimensional counterpart, can be inverted by means of the formula... 

In other words, knowledge of the joint characteristic function of a family of random variables is tantamount to knowledge of their joint probability density function and vice versa. 

This conditional probability can also be written in terms of the joint probability density function for lt , n+m as follows ... 

The conditional probability density functions defined by Eq. (3-170) are joint probability density functions for fixed values of xn... 

It can be shown that the right-hand side of Eq. (3-208) is the -dimensional characteristic function of a -dimensional distribution function, and that the -dimensional distribution function of afn, , s n approaches this distribution function. Under suitable additional hypothesis, it can also be shown that the joint probability density function of s , , sjn approaches the joint probability density function whose characteristic function is given by the right-hand side of Eq. (3-208). To preserve the analogy with the one-dimensional case, this distribution (density) function is called the -dimensional, zero mean gaussian distribution (density) function. The explicit form of this density function can be obtained by taking the i-dimensional Fourier transform of e HsA, with the result.45... 

In terms of these functions, we define all possible joint probability density functions for a time function X(t) by writing... 

This ensemble over J7 ,Fn can also be considered as a triple product ensemble, UnVnZn, where Zn is an ensemble consisting of the events zt corresponding to error, and ze corresponding to no error. The joint probabilities in this triple product ensemble are... 

Consequences While this may still appear reasonable, lower accepted impurity limits AIL quickly demand either very high m or then target levels TL below the LOQ, as is demonstrated in Fig. 4.7. If several impurities are involved, each with its own TL and AIL, the risk of at least one exceeding its AIL rapidly increases (joint probabilities, see Section 4.24). For k impurities, the risk is [1 - (1 - 0.05) ], that is for k = 13, every other batch would fail ... 

A general method has been developed for the estimation of model parameters from experimental observations when the model relating the parameters and input variables to the output responses is a Monte Carlo simulation. The method provides point estimates as well as joint probability regions of the parameters. In comparison to methods based on analytical models, this approach can prove to be more flexible and gives the investigator a more quantitative insight into the effects of parameter values on the model. The parameter estimation technique has been applied to three examples in polymer science, all of which concern sequence distributions in polymer chains. The first is the estimation of binary reactivity ratios for the terminal or Mayo-Lewis copolymerization model from both composition and sequence distribution data. Next a procedure for discriminating between the penultimate and the terminal copolymerization models on the basis of sequence distribution data is described. Finally, the estimation of a parameter required to model the epimerization of isotactic polystyrene is discussed. 

Therefore to make meaningful inferences from experiments such as those reported by Yamashita et al. either the error structure must be known or sufficient data must be provided, preferably in the form of optimally designed replicates. This analysis confirms that it is generally insufficient to evaluate only point estimates. In fact these are secondary to evaluating and reporting joint probability regions. 

Applications of the method to the estimation of reactivity ratios from diad sequence data obtained by NMR indicates that sequence distribution is more informative than composition data. The analysis of the data reported by Yamashita et al. shows that the joint 95% probability region is dependent upon the error structure. Hence this information should be reported and integrated into the analysis of the data. Furthermore reporting only point estimates is generally insufficient and joint probability regions are required. 

To apply the above method, we must decide the distribution of parameter values to explore. One immediate answer would be to impose on the parameters an appropriate joint probability distribution, but this would require us to know it, or at least to have a reasonable idea of what it might be. 

If the mathematical model of the process under consideration is adequate, it is very reasonable to assume that the measured responses from the i,h experiment are normally distributed. In particular the joint probability density function conditional on the value of the parameters (k and ,) is of the form,... 

If we now further assume that measurements from different experiments are independent, the joint probability density function for the all the measured responses is simply the product,... 

The Loglikelihood function is the log of the joint probability density function and is regarded as a function of the parameters conditional on the observed responses. Hence, we have... 

One more quantitative way to characterize the chemical structure of copolymers is based on the consideration of chemical correlation functions (correlators) [2]. The simplest of these, Ya k), describes the joint probability of finding two randomly chosen monomeric units divided along the macromolecule by an arbitrary sequence Uk ... 

The correlator (6) is of the utmost importance because its generating function enters into an expression which describes the angular dependence of intensity of scattering of light or neutrons [3]. It is natural to extend expression (6) for the two-point chemical correlation function by introducing the w-point correlator ya1... (kl...,kn l) which equals the joint probability of finding in a macromolecule n monomeric units Maj.Ma> divided by (n-1) arbitrary sequences... 

In brief, the Bayesian approach uses PDFs of pattern classes to establish class membership. As shown in Fig. 22, feature extraction corresponds to calculation of the a posteriori conditional probability or joint probability using the Bayes formula that expresses the probability that a particular pattern label can be associated with a particular pattern. 

The knowledge required to implement Bayes formula is daunting in that a priori as well as class conditional probabilities must be known. Some reduction in requirements can be accomplished by using joint probability distributions in place of the a priori and class conditional probabilities. Even with this simplification, few interpretation problems are so well posed that the information needed is available. It is possible to employ the Bayesian approach by estimating the unknown probabilities and probability density functions from exemplar patterns that are believed to be representative of the problem under investigation. This approach, however, implies supervised learning where the correct class label for each exemplar is known. The ability to perform data interpretation is determined by the quality of the estimates of the underlying probability distributions. 

Bricogne, G. (1988) A Bayesian statistical theory ofthe phase problem. I. A multichannel maximum-entropy formalism for constructing generalized joint probability distributions of structure factors, Acta Cryst., A44, 517-545. 

We can, therefore, let /cx be the subject of our calculations (which we approximate via an array in the computer). Post-simulation, we desire to examine the joint probability distribution p(N, U) at normal thermodynamic conditions. The reweighting ensemble which is appropriate to fluctuations in N and U is the grand-canonical ensemble consequently, we must specify a chemical potential and temperature to determine p. Assuming -7CX has converged upon the true function In f2ex, the state probabilities are given by... 

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