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5.2 Latent discrete parameterization

One way to parameterize a mixture model is with a latent categorical variable indicating which mixture component was responsible for the outcome. For example, consider K normal distributions with locations μkR and scales σk(0,). Now consider mixing them in proportion λ, where λk0 and Kk=1λk=1 (i.e., λ lies in the unit K-simplex). For each outcome yn there is a latent variable zn in {1,,K} with a categorical distribution parameterized by λ, zncategorical(λ).

The variable yn is distributed according to the parameters of the mixture component zn, ynnormal(μz[n],σz[n]).

This model is not directly supported by Stan because it involves discrete parameters zn, but Stan can sample μ and σ by summing out the z parameter as described in the next section.