9.1 Maximum Likelihood Estimation (MLE)

The likelihood of observing the response vector $Y_i$ for student $i$ given attribute profile ${\alpha }_c$ is $$L({\alpha }_c;Y_i)=\prod _{j=1}^{J}P(Y_{ij}=1|{\alpha }_c{)}^{Y_{ij}}[1-P(Y_{ij}=1|{\alpha }_c){]}^{1-Y_{ij}}$$ The log likelihood is $$\log L({\alpha }_c;Y_i)=\sum _{j=1}^J{Y}_{ij}\log P(Y_{ij}=1|{\alpha }_c)+(1-Y_{ij})\log [1-P(Y_{ij}=1|{\alpha }_c\left)\right]$$ The Maximum Likelihood Estimation (MLE) of attribute profile for student $i$ is ${\alpha }_c$ that maximize $L({\alpha }_c;Y_i)$ or $\log L({\alpha }_c;Y_i)$.