how can a posterior be uniform?

A bemusing question from X validated:

How can we have a posterior distribution that is a uniform distribution?

With the underlying message that a uniform distribution does not depend on the data, since it is uniform! While it is always possible to pick the parameterisation a posteriori so that the posterior is uniform, by simply using the inverse cdf transform, or to pick the prior a posteriori so that the prior cancels the likelihood function, there exist more authentic discrete examples of a data realisation leading to a uniform distribution, as eg in the Multinomial model. I deem the confusion to stem from the impression either that uniform means non-informative (what we could dub Laplace’s daemon!) or that it could remain uniform for all realisations of the sampled rv.

objectivity in prior distributions for the multinomial model

Today, Danilo Alvares visiting from the Universitat de Valencià gave a talk at CREST about choosing a prior for the Multinomial distribution. Comparing different Dirichlet priors. In a sense this is an hopeless task, first because there is no reason to pick a particular prior unless one picks a very specific and a-Bayesian criterion to discriminate between priors, second because the multinomial is a weird distribution, hardly a distribution at all in that it results from grouping observations into classes, often based on the observations themselves. A construction that should be included within the choice of the prior maybe? But there lurks a danger of ending up with a data-dependent prior. My other remark about this problem is that, among the token priors, Perk’s prior using 1/k as its hyper-parameter [where k is the number of categories] is rather difficult to justify compared with 1/k² or 1/k³, except for aggregation consistency to some extent. And Laplace’s prior gets highly concentrated as the number of categories grows.