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, $\mathbf{\lambda}=[\pi_1,\ldots,\pi_{2^K}]^\top$, 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.