Discussion of “ Least Angle Regression ”

Being able to reliably, and automatically, select variables in linear regression models is a notoriously difficult problem. This research attacks this question head on, introducing not only a computationally efficient algorithm and method, LARS (and its derivatives), but at the same time introducing comprehensive theory explaining the intricate details of the procedure as well as theory to guide its practical implementation. This is a fascinating paper and I commend the authors for this important work. Automatic variable selection, the main theme of this paper, has many goals. So before embarking upon a discussion of the paper it is important to first sit down and clearly identify what the objectives are. The authors make it clear in their introduction that, while often the goal in variable selection is to select a " good " linear model, where goodness is measured in terms of prediction accuracy performance, it is also important at the same time to choose models which lean toward the parsimonious side. So here the goals are pretty clear: we want good prediction error performance but also simpler models. These are certainly reasonable objectives and quite justifiable in many scientific settings. At the same, however, one should recognize the difficulty of the task, as the two goals, low prediction error and smaller models, can be diametrically opposed. By this I mean that certainly from an oracle point of view it is true that minimizing prediction error will identify the true model, and thus, by going after prediction error (in a perfect world), we will also get smaller models by default. However, in practice, what happens is that small gains in prediction error often translate into larger models and less dimension reduction. So as procedures get better at reducing prediction error, they can also get worse at picking out variables accurately. Unfortunately, I have some misgivings that LARS might be falling into this trap. Mostly my concern is fueled by the fact that Mallows' C p is the criterion used for determining the optimal LARS model. The use of C p