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In stochastic bandit problems, a Bayesian policy called Thompson sampling(TS) has recently attracted much attention for its excellent empiricalperformance. However, the theoretical analysis of this policy is difficult andits asymptotic optimality is only proved for one-parameter models. In thispaper we discuss the optimality of TS for the model of normal distributionswith unknown means and variances as one of the most fundamental example ofmultiparameter models. First we prove that the expected regret of TS with theuniform prior achieves the theoretical bound, which is the first result to showthat the asymptotic bound is achievable for the normal distribution model. Nextwe prove that TS with Jeffreys prior and reference prior cannot achieve thetheoretical bound. Therefore the choice of priors is important for TS andnon-informative priors are sometimes risky in cases of multiparameter models.

Optimality of Thompson Sampling for Gaussian Bandits Depends on Priors