Prior Parameters in Multivariate Normal Model

DESCRIPTION:

Constructs a class priorGauss object, which represents the hyperparameters of the normal inverted-Wishart distribution used as a prior distribution for the Gaussian model. Below, the prior may be specified either by a character string, or by explicitly setting the values of the parameters.

USAGE:

priorGauss(type = "general", tau = 0, mean = 0, df = 1, scale = df) 

OPTIONAL ARGUMENTS:

type
character, the type of the prior. The possible priors are:
"general"

a general prior. In this case you must specify all hyperparameters, including a vector for argument mean, a matrix for argument scale, and the tau and df parameters. This is the default.

"noninformative"
a noninformative prior. This yields the usual estimate of the mean and the unbiased estimate of the covariance matrix. All other arguments to priorGauss are ignored.
"ml"
maximum likelihood estimation, no prior. Not available for data augmentation. In this case all other arguments to priorGauss are ignored.
"ridge"
a ridge prior. In this case, the tau and mean arguments to priorGauss are ignored, and the default for the scale and df arguments are both 1. When a scalar value of scale is input, it is used to define the final scale matrix, see the description of the scale argument.

tau
the "degrees of freedom" in the multivariate normal prior distribution for the mean. This can be thought of as the equivalent number of observations to be used for mean in the Bayes estimates. The larger the degrees of freedom, the stronger the belief in the prior distribution of the mean. For both "noninformative" and "ridge" priors, tau is zero, the minimum possible value for tau. This gives an improper uniform prior on the means. For maximum likelihood estimation ( "ml"), tau is set to Machine$double.xmax.
mean
a vector containing the mean of the prior distribution. These are usually subjective estimates of the multivariate normal mean. mean is set to zero for "ml", "noninformative", and "ridge" priors.
df
the "degrees of freedom" in the scale parameter. This is the number of degrees of freedom used in inverse Wishart prior distribution for the covariance matrix parameters. The larger the degrees of freedom, the stronger the belief in the prior distribution. For a noninformative prior, df is set to -1, the minimum value. This means that the mode for the posterior distribution of the covariance matrix given the data will be the usual unbiased estimate of the multivariate normal covariance matrix. For "ridge" estimates, df = 1 by default, but it is also possible to specify any value greater than -1. Parameter df is ignored for maximum likelihood estimation.
scale
the scale parameter in the inverse Wishart prior. This is the expected sums of squares and cross-products for the multivariate normal data when there are df degrees of freedom. For a "noninformative" prior, scale is set to zero. Argument scale is not used in maximum likelihood estimation. For type = "ridge", you can give any scale matrix of the correct dimensionality for your problem. Alternatively, if a scalar value is input for scale, then the final scale matrix is taken as the diagonal matrix containing the scalar value times the variances of the observations for the nonmissing data (i.e., for each variable, variances are computed on the observed values). In this case, the usual biased maximum likelihood estimates are computed, see . For type = "general", any positive definite square matrix of the appropriate dimension may be used.

VALUE:

an object of class "priorGauss". This is a list containing the components type, tau, mean, df, and scale.

DETAILS:

For "ridge" priors, the prior can be computed using routine dataDepPrior. See .

REFERENCES:

Schafer, J. L. (1997), Analysis of Incomplete Multivariate Data, Chapman & Hall, London.

SEE ALSO:

, , , , .

EXAMPLES:

priorGauss(type = "noninformative")