Statistics - (Multiclass Logistic|multinomial) Regression
About
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 ?
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Model
The symmetric form:
<MATH>
\begin{array}{rrrl}
Pr(Y = k|X) & = & \frac{\displaystyle e^{\displaystyle B_{0k} + B_{1k} . X_1 + \dots + B_{ik} . X_i }}{\displaystyle \sum^K_{l=1} e^{\displaystyle B_{0l} + B_{1l} . X_1 + \dots + B_{il} . X_i }} \\
\end{array}
</MATH>
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.
R
package glmnet