  We consider Thompson sampling for linear bandit problems with finitely many
independent arms, where rewards are sampled from normal distributions that are
linearly dependent on unknown parameter vectors and with unknown variance.
Specifically, with a Bayesian formulation we consider multivariate normal-gamma
priors to represent environment uncertainty for all involved parameters. We
show that our chosen sampling prior is a conjugate prior to the reward model
and derive a Bayesian regret bound for Thompson sampling under the condition
that the 5/2-moment of the variance distribution exist.
