 Solution

For independent Bernoulli random variables, the log-likelihood isEvery likelihood factor is at most one, so $\ell \le 0$. In the statistical saturated model, choose ${\hat{\pi }}_i=y_i$ independently for each observation. The probability of each realised response is then exactly one. With the continuous convention $0\log 0=0$, the Bernoulli saturated log-likelihood is thereforeThe inclusion of probabilities zero and one is important: if the parameter space required $0<{\pi }_i<1$, zero would be the supremum approached at the boundary, rather than an attained maximum.