Expectation Propagation for t-Exponential Family Using Q-Algebra

TL;DR

This paper introduces an expectation propagation algorithm tailored for the t-exponential family by leveraging q-algebra, enabling efficient approximation in models involving Student-t distributions and noisy data.

Contribution

It develops the first EP algorithm for the t-exponential family using q-algebra, facilitating natural parameter calculations and improved inference for Student-t based models.

Findings

The proposed EP algorithm effectively approximates posteriors in t-exponential models.

Numerical experiments show improved performance in Student-t process classification.

The method handles noisy data more robustly than traditional approaches.

Abstract

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 denied by the deformed exponential, we cannot derive an efficient learning algorithm for the t-exponential family such as expectation propagation (EP). 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 classication, and demonstrate their performance numerically.