= Observed and latent mixture labels
In a <finite mixture model> with hidden membership, the component label $Z_i$ is a <latent variable> and $x_i$ is observed. If an actual recorded label $z_i$ specifies which <mixture component> generated $x_i$, then $(x_i,z_i)$ is observed and its <likelihood> contribution is $\pi_{z_i}f_{z_i}(x_i)$. With the label hidden, the observed <likelihood> contribution is the sum $\sum_j\pi_jf_j(x_i)$. A hard assignment chosen from <mixture responsibilities> is an inference under the <statistical model>, not an additional recorded label.
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