On the choice of flattening constants for estimating multinomial probabilities

Bayesian estimation of the cell probabilities for the multinomial distribution (under a symmetric Dirichlet prior) leads to the use of a flattening constant [alpha] to smooth the raw cell proportions. The unsmoothed estimator corresponds to [alpha] = 0. The risk functions (under quadratic loss) of the Bayesian estimators for [alpha] > 0 are compared to that for [alpha] = 0 and this leads to an interpretation of any given choice of [alpha] > 0 in terms of the maximum number of "small" cell probabilities for which the corresponding smoothed estimator has smaller risk than the unsmoothed estimator. A real set of data is used to illustrate our interpretation of three a priori and three empirically determined choices of [alpha] that have appeared in the literature.