3.12 The saturated model for joint attribute distribution

The saturated model assumes the joint attribute distribution is a multinomial (categorical) distribution.

It involves $2^K$ parameters, $\lambda =[{\pi }_1,\dots ,{\pi }_{2^K}{]}^{\top }$, with the constraint of ${\sum }_{c=1}^{2^K}{\pi }_c=1$. Note that ${\pi }_c$ is sometimes referred to as mixing proportion parameter, indicating the proportion of individuals in latent class $c$.

The saturated model is considered the most general parameterization of the joint attribute distribution, but tends to involve too many parameters when $K$ is large.