On the choice of flattening constants for estimating multinomial probabilities

AbstractBayesian estimation of the cell probabilities for the multinomial distribution (under a symmetric Dirichlet prior) leads to the use of a flattening constant α to smooth the raw cell proportions. The unsmoothed estimator corresponds to α = 0. The risk functions (under quadratic loss) of the Bayesian estimators for α > 0 are compared to that for α = 0 and this leads to an interpretation of any given choice of α > 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 α that have appeared in the literature