I'd like to know a procedure to recognize whether a given probability distribution over outcomes $\{0, \dots, n\}$ can be expressed as a mixture of $n$-trial binomial distributions with different success probabilities $p$.
Interpreting distributions as vectors over $\mathbb{R}^{n+1}$ (or the simplex $\Delta^n$), the binomial distributions corresponds to vectors $v_p$ whose coordinates are given by $(v_p)_i=\binom{n}{i} p^i (1-p)^{n-i}.$ Then, the question is whether a given vector $w$ is in the convex hull of the set $\{v_p \mid 0\leq p\leq1\}$.
Note that Carathéodory's Theorem guarantees that a point $w$ in the convex hull is a convex combination of at most $n+1$ binomial distributions. Determining such a mixture explicitly would also be helpful.
This question is motivated by Bayesian uncertainty. Suppose one will perform $n$ independent identical trials of some experiment whose success probability $p$ is unknown, but for which one has a prior distribution $\alpha$. Then, one's subjective distribution for the number of successes is the infinite mixture $\int _{0}^1 v_p \alpha(p) \thinspace \mathrm{d}p$ (this generalizes to formal distributions $\alpha$). The convexity argument shows that finite mixtures suffice. I'd like to understand what class of distributions can be modeled this way.