Oh, I see now. You are right with the unidendifieability; I got confused with another model, where I sample from \mathbb R^{n-1} and embed it in \mathbb R^n before the mixing. And I think we stumbled on something interesting here. It almost seems like there is a bifurcation in the model, a set of hyperparameters at which the posterior distribution switches from mono- to multimodality. I will investigate this. If someone shares the interest, please contact me.
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