  In data sets with many predictors, algorithms for identifying a good subset
of predictors are often used. Most such algorithms do not account for any
relationships between predictors. For example, stepwise regression might select
a model containing an interaction AB but neither main effect A or B. This paper
develops mathematical representations of this and other relations between
predictors, which may then be incorporated in a model selection procedure. A
Bayesian approach that goes beyond the standard independence prior for variable
selection is adopted, and preference for certain models is interpreted as prior
information. Priors relevant to arbitrary interactions and polynomials, dummy
variables for categorical factors, competing predictors, and restrictions on
the size of the models are developed. Since the relations developed are for
priors, they may be incorporated in any Bayesian variable selection algorithm
for any type of linear model. The application of the methods here is
illustrated via the Stochastic Search Variable Selection algorithm of George
and McCulloch (1993), which is modified to utilize the new priors. The
performance of the approach is illustrated with two constructed examples and a
computer performance dataset. Keywords: Model Selection, Prior Distributions,
Interaction, Dummy Variable
