Uniform posterior on bounded space [duplicate]

Question

In a particular Bayesian problem, I have encountered a choice of parameters that leads to a uniform posterior distribution. Given prior

\begin{equation} p(\boldsymbol{\pi}) =Dirichlet(\boldsymbol{\alpha}), \quad \text{with}\ \boldsymbol{\alpha} = [0,\dots,0]. \end{equation}

we consider a multinomial distribution whose support is $\{x_1, x_2, \dots, x_n\}$ and $\text{Pr}(X=x_i)=\pi_i(\sum_i\pi_i=1)$

\begin{equation} \label{eq:likelihood} p(X|\boldsymbol{\pi}) = \mathcal{M}ulti_{k}(n,\boldsymbol{\pi}), \end{equation}

we find the posterior \begin{align*} p(\boldsymbol{\pi}|X) &\propto p(X|\boldsymbol{\pi})p(\boldsymbol{\pi})\\ & \propto \prod_{i}\pi_i\prod_{i}\pi_{i}^{\alpha_i-1}\\ & \propto \prod_i\pi_i^{\alpha_i}. \end{align*}

to be a uniform distribution

\begin{equation} p(\boldsymbol{\pi}|X) =Dirichlet(\boldsymbol{\alpha}), \quad \text{with}\ \boldsymbol{\alpha} = [1,\dots,1]. \end{equation}

I was wondering if it is uniform, why is this called a posterior then? How can we have a posterior distribution that is a uniform distribution?

Answer 1

NOTE: My answer was written based on an earlier formulation of the question, namely on how having a uniform Dirichlet $\mathcal D(1,\ldots,1)$ posterior was at all possible. I proposed this setting as an illustration of a possible case. The question later got edited and, behold!, coincided with this case.

Consider the special following case of

1. an improper prior on $\mathbf p=(p_1,\ldots,p_k)\in\mathfrak S_k$, the $k$-th dimensional simplex, proportional to$$\pi(\mathbf p)\propto \prod_{i=1}^k p_i^{-1}\tag{1}$$
2. an observation $\mathbf x=(x_1,\ldots,x_k)$ of $\mathbf X\sim\mathcal M_k(k;\mathbf p)$ equal to $(1,\ldots,1)$
3. a posterior equal to the Dirichlet $\mathcal D(1,\ldots,1)$ distribution

To comment on this special case,

1. the posterior obviously depends on the data. Were the data $\mathbf x=(0,\ldots,0,k)$ the posterior would be another Dirichlet distribution. Any other observation than $(1,\ldots,1)$ does not produce a Uniform posterior (with this prior (1)).
2. A Uniform posterior is a distribution and as such contains information about the probability parameter $\mathbf p$. It is limited because the data is limited but increasing the number of observations will see the posterior concentrate. There is nothing paradoxical with having a uniform prior with a single observation, at least in a sampling distribution over a finite set. Unless the likelihood is constant in the parameter, this is impossible in a continuous setting.