The problem you encountered in part $($b$)$ is called separation $.$ It occurs when the can be perfectly recovered by a linear classifier, i$.$e$.,$ when there is a such that

In order to avoid this behavior, one option is to use a prior on $.$ Let us investigate what happens if we assume that is drawn from a distribution, i$.$e$.,$

What is the joint log likelihood of this Bayesian model? Again, for simplicity, let's plug in and $.($Try to work out the general formula on your own. It will also be provided in the solution.$)$