Matrix Gyrations and Model Selection

J.A. Stark and W.J. Fitzgerald

In Proceedings of the Section on Bayesian Statistical Science, Annual Meeting of the American Statistical Association (pp 134-139, August 1996)

Recent research has shown that Markov chain Monte Carlo (MCMC) methods can be effective tools in variable selection. In this type of problem the model comprises a subset of a candidate set of variables, and the number of possible models is often too large for exhaustive searching. In Bayesian model selection the concept of model probabilities allows for the use of numerical approaches such as MCMC. The most computationally expensive task is evaluating possible model changes. In the linear marginalised case (parameters integrated out), algorithms generally require order p-squared operations for each candidate variable, where p is the size of the trial model.

The gyration is a matrix transformation that is useful in analysing many statistical problems. The complementary descriptions of a multivariate Gaussian (correlations, and mean and covariance) can be brought together in augmented matrices that are gyrations of each other. The gyration also encapsulates the task of evaluating changes to a model, yielding the determinant of the correlation matrix as well as the mean-squared-error. Hence the task of evaluating models when incorporating or removing variables can be analysed. This leads to an algorithm for performing these evaluations with order p operations.