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For logistic regression, set ${\eta }_i=x_i^{\mathsf{T}}\beta$ and $p_i=(1+e^{-{\eta }_i}{)}^{-1}$. Its Bernoulli log-likelihood, score function and Fisher information arewhere $W=\mathrm{diag}\{p_i(1-p_i)\}$. Solve $U(\beta )=0$ by Fisher scoring or iteratively reweighted least squares. One scoring step isEquivalently regress the working response $z_i={\eta }_i+(y_i-p_i)/[p_i(1-p_i)]$ on $X$ using weights $p_i(1-p_i)$. Iterate until the likelihood and coefficients stabilize, with rank and convergence checks. Separation in logistic regression can prevent a finite maximum: if a predictor direction strictly separates the zeros and ones, moving indefinitely along it improves the likelihood rather than producing a finite root.

Under the unrestricted model, each individual probability is estimated independently as ${\hat{p}}_i=y_i$. Every observed binary outcome then has probability one, so, using boundary limits,The resulting residual deviance is $D=-2\ell (\hat{\beta })$. It is a legitimate likelihood contrast and can be used to compare fitted models. However, individual Bernoulli deviance need not have a chi-squared calibration: the saturated model has one boundary-valued parameter for each single observation, and its dimension grows with $n$. The fixed-dimensional regularity behind Wilks theorem does not justify a ${\chi }_{n-p}^2$ reference here.

For a concrete demonstration, let the true model contain only an intercept and have success probability $1/2$. Its fitted probability is $\bar{y}$, andwhereas a ${\chi }_{n-1}^2$ variable divided by $n$ converges to one. Thus treating the raw Bernoulli residual deviance as an ordinary absolute goodness-of-fit statistic can reject a correct model systematically. In contrast, deviance differences between fixed-dimensional nested logistic models have their usual asymptotic chi-squared reference under the relevant regularity assumptions. For absolute fit use meaningful replicated covariate groups and grouped-binomial logistic regression, or assess residual patterns and prediction on held-out data; a model-specific parametric bootstrap can calibrate a chosen statistic.