Where each \ xi _ { k } is an independent draw from a multivariate normal distribution on \ mathbb { R } ^ { d } with mean 0 and covariance matrix equal to the d \ times d identity matrix.
42.
By properties of multivariate normal distribution, this means that " P? " and " M? " are independent, and therefore estimators \ hat \ beta and \ hat \ sigma ^ 2 will be independent as well.
43.
Later work for the multivariate normal distribution allowed the classifier to be nonlinear : several classification rules can be derived based on slight different adjustments of the Mahalanobis distance, with a new observation being assigned to the group whose centre has the lowest adjusted distance from the observation.
44.
Where we used the fact that \ hat { \ beta }-\ beta is just an affine transformation of \ varepsilon by the matrix ( X'X ) ^ {-1 } X'( see article on the multivariate normal distribution under the affine transformation section ).
45.
One particular use of the precision matrix is in the context of Bayesian analysis of the multivariate normal distribution : for example, Bernardo & Smith prefer to parameterise the multivariate normal distribution in terms of the precision matrix, rather than the covariance matrix, because of certain simplifications that then arise.
46.
One particular use of the precision matrix is in the context of Bayesian analysis of the multivariate normal distribution : for example, Bernardo & Smith prefer to parameterise the multivariate normal distribution in terms of the precision matrix, rather than the covariance matrix, because of certain simplifications that then arise.
47.
For example, in attempting to find the maximum likelihood estimate of a multivariate normal distribution using matrix calculus, if the domain is a " k " x1 column vector, then the result using the numerator layout will be in the form of a 1x " k " row vector.
48.
The main difference between EDAs and most conventional evolutionary algorithms is that evolutionary algorithms generate new candidate solutions using an " implicit " distribution defined by one or more variation operators, whereas EDAs use an " explicit " probability distribution encoded by a Bayesian network, a multivariate normal distribution, or another model class.
49.
In that case, the joint distribution of " w " is a multivariate normal distribution; the independence between the variables then implies that the distribution has Hartley, the transform " W " of " w " will be a Gaussian white noise vector, too; that is, the " n"
50.
If all parameters are scalar values, then this means that there will be one more hyperparameter than parameter; but this also applies to vector-valued and matrix-valued parameters . ( See the general article on the exponential family, and consider also the Wishart distribution, conjugate prior of the covariance matrix of a multivariate normal distribution, for an example where a large dimensionality is involved .)
