 Solution

Writing ${\eta }_i={\beta }_0+x_i^T\beta$, the log-likelihood isWith 500 observations and 5000 word predictors, the augmented design matrix cannot have full column rank. There are nonzero coefficient directions that leave every ${\eta }_i$ unchanged, so any maximizer belongs to an affine family and is not unique. In addition, the high-dimensional features may completely separate the classes, in which case no finite maximizer exists at all.