What is the difference between Multinomial Logistic Regression and Softmax Regression?

Multinomial logistic regression and softmax regression (also known as maximum entropy classifier) are both machine learning algorithms used for multiclass classification problems, where the goal is to predict a categorical dependent variable with more than two categories. However, there are some differences between the two:

  • Formulation: Multinomial logistic regression models the relationship between the dependent variable and independent variables through multiple binary logistic regression models, one for each class. Softmax regression models the relationship through a single multiclass logistic regression model that calculates the probablities for all classes.

  • Output: The output of multinomial logistic regression is a set of probabilities for each class, whereas the output of softmax regression is a set of scores that can be transformed into probabilities.

  • Optimization: The optimization problem solved in multinomial logistic regression is a maximum likelihood estimation, while the optimization problem solved in softmax regression is a maximum entropy optimization.

In summary, both algorithms can be used for the same problem, but they differ in the way they model the relationship between the dependent variable and independent variables, the form of their output, and the optimization problem they solve.