Rejoinder on Causal Inference Through Potential Outcomes and Principal Stratification: Application to Studies with ``Censoring'' Due to Death by J. M. Moguerza and A. Munoz

We are very grateful to the Executive Editors George Casella and Edward George for their active interest in our paper and for organizing this challenging discussion. We also thank all the discussants for their insightful and stimulating comments. When we submitted the original manuscript in 2003, we were tempted to go for a more general paper on kernel methods. We decided to focus on support vector machines (SVMs), waiting for a mature development of the new and exciting ideas related to kernel methods, such as manifold learning and other related topics. We will refer to some of these methods below. Let us begin, first, with some general considerations. Regarding the question of the dimensionality induced by the feature space, Hastie and Zhu remark in their comment that usual kernels do not automatically lead to infinite-dimensional feature spaces. They give a nice example that involves the radial (Gaussian) kernel function. This agrees with results in Keerthi and Lin [8], where an explanation of the performance of the Gaussian kernel is given when, according to the notation in the comment by Hastie and Zhu, γ → 0 and λ is chosen in the appropriate way. In this case, the SVM classifier converges to a linear SVM classifier, and the effective dimension of the kernel is finite, agreeing with the empirical conclusion provided by the discussants. We also agree with the assertions of some of the discussants regarding the probabilistic interpretability of the SVM output (the sign of some estimated function). Our comment was rather along the line of Sollich [18], who proposed to make Bayesian methods available for the support vector methodology, while leaving as much as possible of the standard SVM framework intact. This is not an easy task. In fact, as Bartlett, Jordan and McAuliffe remark,