Orthogonalizations and Prior Distributions for Orthogonalized Model Mixing

Prediction methods based on mixing over a set of plausible models can help alleviate the sensitivity of inference and decisions to modeling assumptions. One important application area is prediction in linear models. Computing techniques for model mixing in linear models include Markov chain Monte Carlo methods as well as importance sampling. Clyde, DeSimone and Parmigiani (1996) developed an importance sampling strategy based on expressing the space of predictors in terms of an orthogonal basis. This leads both to a better identified problem and to simple approximations to the posterior model probabilities. Such approximations can be used to construct efficient importance samplers. For brevity, we call this strategy orthogonalized model mixing. Two key elements of orthogonalized model mixing are: a) the orthogonalization method and b) the prior probability distributions assigned to the models and the coefficients. In this paper we consider in further detail the specification of these two elements. In particular, after identifying the aspects of these specifications that are essential to the success of the importance sampler, we list and briefly discuss a number of different alternatives for both a) and b). We highlight the features that may make each one of the options attractive in specific situations and we illustrate some important points via a simulated data set.