How do Bayesians justify using conjugate priors on grounds other than mathematical convenience? In the 1920's the Cambridge philosopher William Ernest Johnson in effect characterized symmetric Dirichlet priors for multinomial sampling in terms of a natural and easily assessed subjective condition. Johnson's proof can be generalized to include asymmetric Dirichlet priors and those finitely exchangeable sequences with linear posterior expectation of success. Some interesting open problems that Johnson's result raises, and its historical and philosophical background, are also discussed.
Sandy L. Zabell. "W. E. Johnson's "Sufficientness" Postulate." Ann. Statist. 10 (4) 1090 - 1099, December, 1982. https://doi.org/10.1214/aos/1176345975