  As an automatic method of determining model complexity using the training
data alone, Bayesian linear regression provides us a principled way to select
hyperparameters. But one often needs approximation inference if distribution
assumption is beyond Gaussian distribution. In this paper, we propose a
Bayesian linear regression model with Student-t assumptions (BLRS), which can
be inferred exactly. In this framework, both conjugate prior and expectation
maximization (EM) algorithm are generalized. Meanwhile, we prove that the
maximum likelihood solution is equivalent to the standard Bayesian linear
regression with Gaussian assumptions (BLRG). The $q$-EM algorithm for BLRS is
nearly identical to the EM algorithm for BLRG. It is showed that $q$-EM for
BLRS can converge faster than EM for BLRG for the task of predicting online
news popularity.
