# Statistics - (Multiclass Logistic|multinomial) Regression

Multiclass logistic regression is also referred to as multinomial regression.

Multinomial Naive Bayes is designed for text classification. It's a lot faster than plain Naive Bayes.

also known as maximum entropy classifiers ?

## Model

The symmetric form:

$$\begin{array}{rrrl} Pr(Y = k|X) & = & \frace^B_{0k} + B_{1k} . X_1 + \dots + B_{ik} . X_i }}\sum^K_{l=1} e^B_{0l} + B_{1l} . X_1 + \dots + B_{il} . X_i }} \\ \end{array}$$

• k is the index of a outcome class
• K is the number of outcome classes (ie bigger than 2)
• in the numerator we've got an exponential to the linear model. This is for the probability that Y is k given X, a small k.
• In the denominator, we've just got the sum of those exponentials for all the classes. In this case, each class gets its own linear model.
• And then we just weigh them against each other with this exponential function, sometimes called the softmax function.
• some cancellation is possible,
• only K - 1 linear functions are needed as in 2-class logistic regression.

package glmnet