Saturated statistical model

A saturated model has enough independently varying mean parameters to match every observed response or response cell. In individual Bernoulli distribution data its fitted probabilities are $p_i=y_i$, and its maximum log-likelihood is zero under $0\log 0=0$. It defines the reference in a binomial deviance.

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.