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==Estimation of intercept==
When using multinomial logistic regression, one category of the dependent variable is chosen as the reference category. Separate [[odds ratio]]s are determined for all independent variables for each category of the dependent variable with the exception of the reference category, which is omitted from the analysis. The exponential beta coefficient represents the change in the odds of the dependent variable being in a particular category vis-a-vis the reference category, associated with a one unit change of the corresponding independent variable.
== Likelihood function ==
The observed values <math>y_i \in {0,1,\dots K}</math> for <math>i=1,\dots,n</math> of the explained variables are considered as realizations of stochastically independent, [[Categorical distribution|categorically distributed]] random variables <math>Y_1,\dots, Y_n</math>.
The likelihood function for this model is defined by:
:<math>L = \prod_{i=1}^n P(Y_i=y_i) = \prod_{i=1}^n \left( \prod_{j=1}^K P(Y_i=j)^{\delta_{j,y_i}} \right) ,</math> where the index <math>i</math> denotes the observations 1 to n and the index <math>j</math> denotes the classes 1 to K. <math>\delta_{j,y_i}=\begin{cases}1 \text{ for } j=y_i \ 0 \text{ otherwise}\end{cases}</math> is the Kronecker delta.
The negative log-likelihood function is therefore the well-known cross-entropy: :<math>-\log L = - \sum_{i=1}^n \sum_{j=1}^K \delta_{j,y_i} \log(P(Y_i=j)).</math>
==Application in natural language processing==
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