Logistic regression as a Bernoulli likelihood

The relation p_i=\sigma(w^Tx_i+b),\quad -\ell= -\sum_i[y_i\log p_i+(1-y_i)\log(1-p_i)] summarizes Logistic regression maps a linear score to a Bernoulli probability and estimates coefficients by maximizing likelihood or minimizing negative log-likelihood. Read it as a model for defined quantities, not as an automatic claim about every real situation. In the example For two observations with scores −ln 3 and +ln 3 and labels 0 and 1, correct-class probabilities are 0.75 each. Likelihood is 0.75²=0.5625 and negative log-likelihood is −ln(0.5625)≈0.5754 nats. This is distinct from the zero-one classification error., symbols are connected to concrete values and the result is interpreted in context. Validity limit: For independent binary observations, p_i=σ(wᵀx_i+b) and log-likelihood is Σ[y_i ln p_i+(1−y_i)ln(1−p_i)]. The sigmoid-only transform is not the estimation objective; maximum likelihood chooses parameters from data. Complete separation can make unregularized Source: scikit-learn User Guide, §1.1 Linear Models: Logistic regression and log loss.

First write down the requested quantity and all given values with their symbols. Substitute them into the defining relation and check unit consistency. The example For two observations with scores −ln 3 and +ln 3 and labels 0 and 1, correct-class probabilities are 0.75 each. Likelihood is 0.75²=0.5625 and negative log-likelihood is −ln(0.5625)≈0.5754 nats. This is distinct from the zero-one classification error. provides a worked check: formula and result must have compatible scale and meaning; round only at the end. Validity limit: For independent binary observations, p_i=σ(wᵀx_i+b) and log-likelihood is Σ[y_i ln p_i+(1−y_i)ln(1−p_i)]. The sigmoid-only transform is not the estimation objective; maximum likelihood chooses parameters from data. Complete separation can make unregularized Source: scikit-learn User Guide, §1.1 Linear Models: Logistic regression and log loss.

The formula applies only when quantities are defined as in the source and background. Check boundary cases, measurement conditions and whether an approximation rather than equality is intended. Substitution can be algebraically correct yet technically wrong when assumptions fail or the population or system boundary changes. Validity limit: For independent binary observations, p_i=σ(wᵀx_i+b) and log-likelihood is Σ[y_i ln p_i+(1−y_i)ln(1−p_i)]. The sigmoid-only transform is not the estimation objective; maximum likelihood chooses parameters from data. Complete separation can make unregularized Source: scikit-learn User Guide, §1.1 Linear Models: Logistic regression and log loss.

The stated units are: p and labels are dimensionless; negative log-likelihood is in nats with natural logarithms. Convert values to a consistent unit system before substitution; counts, percentages and dimensionless ratios are not interchangeable. The variable list states the role of each symbol. Dimensional analysis catches many errors but does not replace checking the formula's assumptions. Validity limit: For independent binary observations, p_i=σ(wᵀx_i+b) and log-likelihood is Σ[y_i ln p_i+(1−y_i)ln(1−p_i)]. The sigmoid-only transform is not the estimation objective; maximum likelihood chooses parameters from data. Complete separation can make unregularized Source: scikit-learn User Guide, §1.1 Linear Models: Logistic regression and log loss.

Comparison helps when both quantities address the same question but emphasize different properties. Report the definition, input data, threshold or averaging with the value. The source scikit-learn User Guide, §1.1 Linear Models: Logistic regression and log loss. supports the definition used here; other conventions can differ. Choose the measure to match the learning or analysis goal. Validity limit: For independent binary observations, p_i=σ(wᵀx_i+b) and log-likelihood is Σ[y_i ln p_i+(1−y_i)ln(1−p_i)]. The sigmoid-only transform is not the estimation objective; maximum likelihood chooses parameters from data. Complete separation can make unregularized Source: scikit-learn User Guide, §1.1 Linear Models: Logistic regression and log loss.