Poisson and Bernoulli losses

data	loss f(x, m)	∂f/∂m
real, constant spread	(x − m)²	−2(x − m)
counts (Poisson)	m − x log m	1 − x/m
binary (Bernoulli, odds link)	log(m + 1) − x log m	1/(m + 1) − x/m

Poisson keeps every prediction positive, and nothing enforces it. The log m term is undefined at zero and enormous just above it, so predicting nearly zero where something was observed costs an unbounded amount. The bound at EPS is only a numerical guard. At x = 0 there is no barrier: a genuinely empty cell may be predicted empty, for free.

Bernoulli models the odds. The model value m is the odds, so m / (1 + m) is a probability by construction.