  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.
