  We consider the problem of sequential learning from categorical observations
bounded in [0,1]. We establish an ordering between the Dirichlet posterior over
categorical outcomes and a Gaussian posterior under observations with N(0,1)
noise. We establish that, conditioned upon identical data with at least two
observations, the posterior mean of the categorical distribution will always
second-order stochastically dominate the posterior mean of the Gaussian
distribution. These results provide a useful tool for the analysis of
sequential learning under categorical outcomes.
