Automatic PCA Dimension Selection for High Dimensional Data and Small Sample Sizes
Bayesian inference from high-dimensional data involves the integration over a large number of model parameters. Accurate evaluation of such high-dimensional integrals raises a unique set of issues. These issues are illustrated using the exemplar of model selection for principal component analysis (PCA). A Bayesian model selection criterion, based on a Laplace approximation to the model evidence for determining the number of signal principal components present in a data set, has previously been show to perform well on various test data sets. Using simulated data we show that for d-dimensional data and small sample sizes, N, the accuracy of this model selection method is strongly affected by increasing values of d. By taking proper account of the contribution to the evidence from the large number of model parameters we show that model selection accuracy is substantially improved. The accuracy of the improved model evidence is studied in the asymptotic limit d -> infinity at fixed ratio alpha = N/d, with alpha < 1. In this limit, model selection based upon the improved model evidence agrees with a frequentist hypothesis testing approach.
pca bayesian model selection random matrix theory high dimensional inference