Comment on Article by Polson and Scott

We congratulate the authors for a very interesting article. The key contribution of this paper, as we see it and as suggested in the title, is the introduction of latent parameters to carry out Bayesian analysis with support vector machines. The basic identities (4) and (6) are particularly useful in this regard, which enable one to overcome much of the complexities of a non smooth loss resulting in a non smooth likelihood. As an anecdote, from a Bayesian angle, it is extremely convenient to view the loss as the negative of the loglikelihood (for example associating squared error loss with the normal likelihood) and the penalty part with the prior. This is the approach taken in this paper, and also earlier in Mallick et al. (2005). Based on a loss function, we can obtain the normalized or the non-normalized (or pseudo) likelihood. The authors considered the non-normalized likelihood and that way obtained the pseudo posterior distribution. This pseudo posterior distribution may not be suitable to make probabilistic inference. Mallick et al. (2005) considered both the normalized and the non-normalized likelihoods and the classification performances were compatible. It will be interesting to see how the proposed method can be adapted for the model with the normalized likelihood.