dist.Zellner {LaplacesDemon}  R Documentation 
Hyperpriorg Prior and Zellner's gPrior
Description
These functions provide the density of the hyperg prior (Liang et al., 2008), and both the density and random generation of Zellner's gprior (Zellner, 1986).
Usage
dhyperg(g, alpha=3, log=FALSE)
dzellner(beta, g, sigma, X, log=FALSE)
rzellner(n, g, sigma, X)
Arguments
alpha 
This is a positive scale hyperhyperparameter that is
proper when 
beta 
This is regression effects 
g 
This is hyperparameter 
n 
This is the number of random deviates to generate. 
sigma 
This is the residual standard deviation

X 
This is a fullrank 
log 
Logical. If 
Details
Application: Continuous Multivariate
Density:
p(\theta) = \frac{1}{(2\pi)^{J/2}(g \sigma^2(\textbf{X}^T \textbf{X})^{1})^{1}^{1/2}} \exp(\frac{1}{2}(\theta  \mu)'(g \sigma^2(\textbf{X}^T \textbf{X})^{1})^{1}(\theta  \mu))
Inventor: Zellner, A. (1986)
Notation 1:
\theta \sim \mathrm{N}_J(0, g \sigma^2(\textbf{X}^T \textbf{X})^{1})
Notation 2:
p(\theta) = \mathrm{N}_J(\theta  g, \sigma^2, \textbf{X})
Parameter 1: location parameter
\beta
Parameter 2: scale parameter
g > 0
Parameter 3: scale parameter
\sigma^2 > 0
Mean:
Variance:
Mode:
Zellner's gprior is a popular, datadependent, elliptical, improper,
leastinformative prior distribution on regression effects
\beta
in a Gaussian regression model. It is a particular
form in the conjugate NormalGamma family. Zellner's gprior is also
used for estimating Bayes factors (for hypothesis testing) with a
simpler form, as well as in model selection and variable selection. The
marginal posterior distribution of regression effects \beta
is multivariate t.
One of many nice properties of Zellner's gprior is that it adapts
automatically to nearcollinearity between different
predictors. Zellner's gprior puts most of its prior mass in the
direction that causes the regression coefficients of correlated
predictors to be smoothed away from each other. When coupled with model
selection, Zellner's gprior discourages highly collinear predictors
from entering the models simultaneously by inducing a negative
correlation between the coefficients. However, when it is desirable for
collinear predictors to enter simultaneously, a modification has been
proposed (though not included here) in which
(\textbf{X}^T \textbf{X})^{1}
is replaced with
(\textbf{X}^T \textbf{X})^\lambda
. For more
information, see Krishna et al. (2009).
For variable selection, large values of g
, with a prior mean of
zero for \beta
, encourage models with few, large
coefficients. Conversely, small values of g
encourage saturated
models with many, small coefficients.
The design matrix \textbf{X}
is converted to Fisher's
information matrix, which is used as a covariance matrix for
\beta
. This is computationally efficient, because each
element of the covariance matrix does not need to be estimated as a
parameter. When \textbf{X}
is nearly singular, regression
effects \beta
may be poorly estimated.
Hyperparameter g
acts as an inverse relative prior sample size, or
as a dimensionality penalty. Zellner (1986) recommended that a
hyperprior distribution is assigned to g
so that it is estimated
from the data, although in practice g
has often been fixed, usually
to N
when no information is available, since it has the
interpretation of adding prior information equivalent to one
observation. A variety of hyperpriors have been suggested for g
,
such as in Bove and Held (2011), Liang et al. (2008), and Maruyama and
George (2011). g
becomes diffuse as it approaches infinity, and
the Bayes factor approaches zero. The hyperg prior of Liang et al.
(2008) is proper when \alpha > 2
, and any value in
the interval (2,4]
may be reasonable.
Value
dhyperg
gives the density of the hyperg prior of Liang et
al. (2008), dzellner
gives the density of Zellner's gprior,
and rzellner
generates random deviates.
References
Bove, D.S. and Held, L. (2011). "Hyperg Priors for Generalized Linear Models". Bayesian Analysis, 6(3), p. 387–410.
Krishna, A., Bondell, H.D., and Ghosh, S.K. (2009). "Bayesian Variable Selection Using an Adaptive Powered Correlation Prior". Journal of Statistical Planning Inference, 139(8), p. 26652674..
Liang, F., Paulo, R., Molina, G., Clyde, M.A., and Berger, J.O. (2008). "Mixtures of g Priors for Bayesian Variable Selection". Journal of the American Statistical Association, 103, p. 410–423.
Maruyama, Y. and George, E.I. (2011). "Fully Bayes Factors with a Generalised gPrior". Annals of Statistics, 39, p. 2740–2765.
Zellner, A. (1986). "On Assessing Prior Distributions and Bayesian Regression Analysis with gPrior Distributions". In Bayesian Inference and Decision Techniques: Essays in Honor of Bruno de Finetti, p. 233–243. Elsevier: Amsterdam, North Holland.
See Also
BayesFactor
and
dmvt
Examples
library(LaplacesDemon)
set.seed(667)
beta < rnorm(10)
g < 100
sigma < 2
X < cbind(1,matrix(rnorm(100*9),100,9))
dhyperg(g, alpha=3)
dzellner(beta, g, sigma, X)
rzellner(1, g, sigma, X)