  We consider the fixed-confidence best arm identification (FC-BAI) problem in
the Bayesian setting. This problem aims to find the arm of the largest mean
with a fixed confidence level when the bandit model has been sampled from the
known prior. Most studies on the FC-BAI problem have been conducted in the
frequentist setting, where the bandit model is predetermined before the game
starts. We show that the traditional FC-BAI algorithms studied in the
frequentist setting, such as track-and-stop and top-two algorithms, result in
arbitrarily suboptimal performances in the Bayesian setting. We also obtain a
lower bound of the expected number of samples in the Bayesian setting and
introduce a variant of successive elimination that has a matching performance
with the lower bound up to a logarithmic factor. Simulations verify the
theoretical results.
