It is argued that the posterior predictive distribution for the binomial and multinomial distributions, when viewed via a hypergeometric-like representation, suggests the uniform prior on the parameters for these models. The argument is supported by studying variations on an example by Fisher, and complements Bayes' original argument for a uniform prior predictive distribution for the binomial. The fact that both arguments lead to invariance under transformation is also discussed.
"Posterior predictive arguments in favor of the Bayes-Laplace prior as the consensus prior for binomial and multinomial parameters." Bayesian Anal. 4 (1) 151 - 158, March 2009. https://doi.org/10.1214/09-BA405