Binary logistic regression can give poor results when the two classes are perfectly separated by a linear decision boundary. One way to address this problem is to use the Lasso applied to logistic regression.

1. (a) Write the likelihood function for the logistic regression problem in terms of x, y, 0 and 1. Assume for simplicity that we have n observations and only one variable (i.e. xi is a real number, for i = 1,...,n).
2. (b) Show that the likelihood function L(0,1) is always strictly less than 1.
3. (c) Recall that the logistic regression coefficients 0, 1 are obtained by maximizing L(0, 1). Suppose that all of the xi corresponding to yi = 0 are negative, all other xi are positive. In this case, note that we can get L(0, 1) arbitrarily close to 1. Explain why this means that 0 and 1 are undefined.
4. (d) For computational convenience, it is common to find 0 and 1 by minimizing the neg- ative log-likelihood log L(0, 1), rather than by maximizing the likelihood itself. Ex- plain why both these problems yield the same 0,1.

(e) Inspired by the Lasso, suggest a way to modify the negative log-likelihood function log L(0, 1) so that 0 and 1 become defined even in the separable case above. Justify your answer.