 Bernoulli saturated log-likelihood

For individual Bernoulli observations, the statistical saturated model assigns one independent probability to each response. Each likelihood factor is at most one and is exactly one at ${\pi }_i=y_i$. Hence the maximum log-likelihood is zero, with $0\log 0=0$. This boundary maximum is attained when probabilities zero and one are allowed; restricting them to the open interval gives the same supremum.