  In recent years, Ising prior with the network information for the "in" or
"out" binary random variable in Bayesian variable selections has received more
and more attentions. In this paper, we discover that even without the
informative prior a Bayesian variable selection problem itself can be
considered as a complete graph and described by a Ising model with random
interactions. There are many advantages of treating variable selection as a
graphical model, such as it is easy to employ the single site updating as well
as the cluster updating algorithm, suitable for problems with small sample size
and larger variable number, easy to extend to nonparametric regression models
and incorporate graphical prior information and so on. In a Bayesian variable
selection Ising model the interactions are determined by the linear model
coefficients, so we systematically study the performance of different scale
normal mixture priors for the model coefficients by adopting the global-local
shrinkage strategy. Our results prove that the best prior of the model
coefficients in terms of variable selection should maintain substantial weight
on small shrinkage instead of large shrinkage. We also discuss the connection
between the tempering algorithms for Ising models and the global-local
shrinkage approach, showing that the shrinkage parameter plays a tempering
role. The methods are illustrated with simulated and real data.
