Binary data: Bernoulli stays in [0, 1]

Ask did this kind of crime ever happen here, at this hour, on this day? and the answer is a tensor of ones and zeros, B = (T > 0).

Squared error on it predicts values it has no meaning for: about a tenth of the predictions fall outside [0, 1]. The Bernoulli fit cannot leave the interval, and nothing bounds it — m is the odds and m / (1 + m) maps any positive number into (0, 1). The constraint is in the parameterisation, the same trick logistic regression uses.

Binarising changes what the tensor is about. Counting weights a busy hour heavily; asking only whether gives a quiet hour the same vote. Bernoulli leans towards what is widespread, Poisson towards what is frequent — both correct, about different questions.