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.