Softmax Regression for Multiclass Classification

In binary logistic regression, a linear score is transformed into a probability. For K mutually exclusive classes, softmax regression assigns one score to each class. Each score is a linear function of the inputs, and softmax exponentiates those scores and divides each exponential by their sum. This produces nonnegative class probabilities that sum to one. The predicted class is commonly the one with the largest probability, but a decision system may use costs or review thresholds instead. Suppose a hypothetical three-class case has scores 2, 1 and 0. Their exponentials are approximately 7.39, 2.72 and 1. The sum is about 11.11, so the probabilities are approximately 0.665, 0.245 and 0.090. The values sum to one because each is normalized by the same denominator. A score difference matters: adding the same constant to every class score leaves the probabilities unchanged. Training typically minimizes multinomial cross-entropy, also called log loss, which penalizes assigning low probability to the observed class. Regularization may be added to control coefficient magnitude. With multiple classes, implementations may parameterize coefficients differently, and some software uses one-versus-rest strategies for particular estimators. Read the selected estimator's documentation before assuming all multiclass logistic models optimize the same objective. Softmax probabilities are not automatically calibrated. A model may rank the most likely class correctly while being systematically overconfident or underconfident. Evaluate classwise recall and precision, confusion matrices, log loss and calibration on data that reflect the deployment population. For imbalanced classes, an overall accuracy score can hide failures on the less common outcomes. Also verify that category order in output arrays matches the library's class labels. The method assumes each case receives one class from the modeled set; multilabel problems, where several labels can occur together, need a different output formulation.