  We consider the fixed-budget best arm identification problem with rewards
following normal distributions. In this problem, the forecaster is given $K$
arms (or treatments) and $T$ time steps. The forecaster attempts to find the
arm with the largest mean, via an adaptive experiment conducted using an
algorithm. The algorithm's performance is evaluated by simple regret,
reflecting the quality of the estimated best arm. While frequentist simple
regret can decrease exponentially with respect to $T$, Bayesian simple regret
decreases polynomially. This paper demonstrates that the Bayes optimal
algorithm, which minimizes the Bayesian simple regret, does not yield an
exponential decrease in simple regret under certain parameter settings. This
contrasts with the numerous findings that suggest the asymptotic equivalence of
Bayesian and frequentist approaches in fixed sampling regimes. Although the
Bayes optimal algorithm is formulated as a recursive equation that is virtually
impossible to compute exactly, we lay the groundwork for future research by
introducing a novel concept termed the expected Bellman improvement.
