9.1 Maximum Likelihood Estimation (MLE)

The likelihood of observing the response vector \(\mathbf{Y}_i\) for student \(i\) given attribute profile \(\mathbf{\alpha}_c\) is \[ L(\mathbf{\alpha}_c;\mathbf{Y}_i)=\prod_{j=1}^JP(Y_{ij}=1|\mathbf{\alpha}_c)^{Y_{ij}}[1-P(Y_{ij}=1|\mathbf{\alpha}_c)]^{1-Y_{ij}} \] The log likelihood is \[ \log L(\mathbf{\alpha}_c;\mathbf{Y}_i)=\sum_{j=1}^J{Y_{ij}}\log P(Y_{ij}=1|\mathbf{\alpha}_c)+{(1-Y_{ij})}\log[1-P(Y_{ij}=1|\mathbf{\alpha}_c)] \] The Maximum Likelihood Estimation (MLE) of attribute profile for student \(i\) is \(\mathbf{\alpha}_c\) that maximize \(L(\mathbf{\alpha}_c;\mathbf{Y}_i)\) or \(\log L(\mathbf{\alpha}_c;\mathbf{Y}_i)\).