Separation in logistic regression ($x_i^{\mathsf{T}}v>0(y_i=1),x_i^{\mathsf{T}}v<0(y_i=0)$)

Complete separation means that a predictor direction $v$ strictly separates the binary outcomes as displayed. In logistic regression, replacing $\beta$ by $\beta +tv$ increases each observed-outcome probability as $t$ increases, so the log-likelihood has no finite maximizer. Weak inequalities with some equalities can similarly produce quasi-complete separation. Finite coefficient estimates and ordinary information-matrix inference therefore require an existence check rather than relying only on the formal score equations.