Multinomial logistic regression: Difference between revisions

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== 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))= - \sum_{j=1}^K\sum_{y_i=j}\log(P(Y_i=j)).</math>

==Application in natural language processing==