  Motivated by examples from genetic association studies, this paper considers
the model selection problem in a general complex linear model system and in a
Bayesian framework. We discuss formulating model selection problems and
incorporating context-dependent {\it a priori} information through different
levels of prior specifications. We also derive analytic Bayes factors and their
approximations to facilitate model selection and discuss their theoretical and
computational properties. We demonstrate our Bayesian approach based on an
implemented Markov Chain Monte Carlo (MCMC) algorithm in simulations and a real
data application of mapping tissue-specific eQTLs. Our novel results on Bayes
factors provide a general framework to perform efficient model comparisons in
complex linear model systems.
