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I have a question regarding the probabilistic setting of clustering and classification. More specifically regarding Gaussian Mixture Models and probabilistic generative models for classification.

In GMM, we are trying to model the evidence $p(x)$ by using an auxiliary and hidden variable $z$ that represents the clusters, then, we model the prior and likelihood using some variables $\pi_{k}$ (prior) and a Gaussian distribution $\mathcal{N}(x|\mu_k, \Sigma_k)$ for the likelihood. Then, we apply EM for solving over $\pi_k, \mu_k$ and $\Sigma_k$.

In case, of classification or regression, we model in the same way, the posterior using and the likelihood (which models the target given some parameters and input $x$) the prior (for the parameters) using Gaussians $\mathcal{N}(t|wx, \Sigma_k)$ (for regression) for the likelihood and $\mathcal{N}(0| w, \alpha)$.

Are we model in case of GMM the prior in this way, since we do not have to data? If we had it, and use Gaussian priors is it similar to probabilistic generative models then?

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