Bruce Hardie, an authority on statistical marketing wrote this reference, which covers various probabilistic methods for various marketing use cases. Specifically, on slide 23 he details a Beta-Geometric mixture model. However, this looks more like a Heirarchical Bayesian model to me.
He motivates this model with two customer groups $\theta_i$, each with their respective probability of being drawn, $\pi_i$. This to me, does sound like a mixture.
However, he goes on to extend the concept to an infinite number of customer segments, where the Beta distribution stands to represent the relative proportion of every possible customer segment.
To me, this seems quite odd. From the multi-level perspective, the Beta distribution is simply a prior placed over the $\theta$ parameter. I gather that I'm wrong; I just don't know why. Could anyone explain how this model functions as a mixture, not a multi-level model?