We derive the asymptotic approximation for the posterior distribution when the data are multinomial and the prior is Dirichlet conditioned on satisfying a finite set of linear equality and inequality constraints so the posterior is also Dirichlet conditioned on satisfying these same constraints. When only equality constraints are imposed, the asymptotic approximation is normal. Otherwise it is normal conditioned on satisfying the inequality constraints. In both cases the posterior is a root--consistent estimator of the parameter vector of the multinomial distribution. As an application we consider the constrained Polya posterior which is a non-informative stepwise Bayes posterior for finite population sampling which incorporates prior information involving auxiliary variables. The constrained Polya posterior is a root--consistent estimator of the population distribution, hence yields a root--consistent estimator of the population mean or any other differentiable function of the vector of population probabilities.
"Asymptotics for Constrained Dirichlet Distributions." Bayesian Anal. 8 (1) 89 - 110, March 2013. https://doi.org/10.1214/13-BA804