Expectation Propagation for t-Exponential Family Using q-Algebra
Exponential family distributions are highly useful in machine learning since their calculation can be performed efficiently through natural parameters. The exponential family has recently been extended to the t-exponential family, which contains Student-t distributions as family members and thus allows us to handle noisy data well. However, since the t-exponential family is defined by the deformed exponential, an efficient learning algorithm for the t-exponential family such as expectation propagation (EP) cannot be derived in the same way as the ordinary exponential family. In this paper, we borrow the mathematical tools of q-algebra from statistical physics and show that the pseudo additivity of distributions allows us to perform calculation of t-exponential family distributions through natural parameters. We then develop an expectation propagation (EP) algorithm for the t-exponential family, which provides a deterministic approximation to the posterior or predictive distribution with simple moment matching. We finally apply the proposed EP algorithm to the Bayes point machine and Student-t process classification, and demonstrate their performance numerically.
The authors proposed an expectation propagation algorithm that can work with distributions in the t-exponential family, a generalization of the exponential family that contains the t-distribution as a particular case. The proposed approach is based on using q-algebra operations to work with the pseudo additivity of t-exponential distributions. The gains of the proposed EP algorithm with respect to assumed density filtering are illustrated in experiments with a Bayes point machine. The authors also illustrate the proposed EP algorithms in experiments with a t-process for classification in the presence of outliers.
Quality
The proposed method seems technically sound, although I did not check the derivations in detail. The experiments illustrate that the proposed method works and it is useful.
Clarity
The paper is clearly written and easy to follow and understand. I only miss an explanation of how the authors compute the integrals in equation (33) in the experiments. I assume that they have to integrate likelihood factors with respect to Student t distributions. How is this done in practice?
Originality
The paper is original. Up to my knowledge, this is the first generalization of EP to work with t-exponential families.
Significance
The proposed method seems to be highly significant, extending the applicability of EP with analytic operations to a wider family of distributions, besides the exponential one.