3.13 The independent model for joint attribute distribution

The independent model assumes that all attributes are independent statistically and can be written by \[\begin{equation} \pi_c=\prod_{k}p(\alpha_{k}=1)^{\alpha_{ck}}[1-p(\alpha_{k}=1)]^{1-\alpha_{ck}}. \end{equation}\]

The independent model involves \(K\) parameters, \(\mathbf{\lambda}=[p(\alpha_{1}=1),\ldots,p(\alpha_{K}=1)]^\top\), where \(p(\alpha_{k}=1)\) is the probability of mastering attribute \(k\).

An example

What are the joint probabilities of two attributes in the following table under an independent model?