The independence between source vectors is approximated by a multivariate, joint probability density function (pdf) of the components from each source vector, and the joint pdf is maximized rather than the individual independencies between each frequency bin.
Propagation of uncertainty in multidimensional dynamical systems, in the presence of parametric uncertainties, can be quantified by the solution of the underlying Liouville equation that governs the evolution of a multivariate joint probability density function of random variables associated with states and parameters.
We show in File S1 that if the genotypes of some individuals are unobserved, the joint density of (g 1, g 2 ) is not multivariate normal; rather, it is a mixture of multivariate normal densities (see File S1).
Given marginal distribution functions F1, …, F d and their densities f1, …, f d, the joint density with the copula defined by this multivariate T-distribution is For this copula, the relationships between the θ ij' s and the τ ij' s are the same as for the multivariate normal copula.
Suppose that the elements of the random vector x have a multivariate Laplacian distribution with mean 0 and covariance matrix R. The joint density is given by [15] (22).
To evaluate (9), one needs to compute the joint density.
We observe zones of densely welded ignimbrite with high joint density, while less-welded zones have lower joint density.
As all the multivariate joint distributions can be written in terms of a copula and univariate marginal pdfs [29], copulas are used as a popular statistical tool for modeling multivariate distributions.
For K loci the joint density of the allelic frequencies takes the form The marginal distribution of allelic counts is obtained by mixing a multinomial distribution over the multivariate beta density above, but results are not available in closed form, so a numerical solution is needed again.
Using a Bayesian technique, the highly multivariate joint posterior distribution of all transcript concentrations is estimated.
