
These functions provide the density of the hyper-g prior (Liang et al., 2008), and both the density and random generation of Zellner's g-prior (Zellner, 1986).
dhyperg(g, alpha=3, log=FALSE)
dzellner(beta, g, sigma, X, log=FALSE)
rzellner(n, g, sigma, X)
This is a positive scale hyperhyperparameter that is
proper when alpha=3
.
This is regression effects
This is hyperparameter
This is the number of random deviates to generate.
This is the residual standard deviation
This is a full-rank
Logical. If log=TRUE
, then the logarithm of the
density is returned.
dhyperg
gives the density of the hyper-g prior of Liang et
al. (2008), dzellner
gives the density of Zellner's g-prior,
and rzellner
generates random deviates.
Application: Continuous Multivariate
Density:
Inventor: Zellner, A. (1986)
Notation 1:
Notation 2:
Parameter 1: location parameter
Parameter 2: scale parameter
Parameter 3: scale parameter
Mean:
Variance:
Mode:
Zellner's g-prior is a popular, data-dependent, elliptical, improper,
least-informative prior distribution on regression effects
One of many nice properties of Zellner's g-prior is that it adapts
automatically to near-collinearity between different
predictors. Zellner's g-prior 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 g-prior 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
For variable selection, large values of
The design matrix
Hyperparameter
Bove, D.S. and Held, L. (2011). "Hyper-g 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. 2665-2674..
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 g-Prior". Annals of Statistics, 39, p. 2740--2765.
Zellner, A. (1986). "On Assessing Prior Distributions and Bayesian Regression Analysis with g-Prior Distributions". In Bayesian Inference and Decision Techniques: Essays in Honor of Bruno de Finetti, p. 233--243. Elsevier: Amsterdam, North Holland.
BayesFactor
and
dmvt
# NOT RUN {
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)
# }
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