# Expectation Propagation for t-Exponential Family Using Q-Algebra

arXiv: https://arxiv.org/abs/1705.09046 · HTML: https://mlsanity.com/paper/1705.09046/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 May 2017.

## Built on by (most cited first, 2 of 2)

- [q-Paths: Generalizing the Geometric Annealing Path using Power Means](https://mlsanity.com/paper/2107.00745/) (2021)
- [Fat-Tailed Variational Inference with Anisotropic Tail Adaptive Flows](https://mlsanity.com/paper/2205.07918/) (2022)

## Abstract (arXiv)

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.