Multinomial logistic regression: Difference between revisions

Revision as of 18:27, 12 August 2023

Revision as of 18:33, 12 August 2023

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Using the fact that all ''K'' of the probabilities must sum to one, we find:

:<math>\Pr(Y_i=K) \,=\, 1- \sum_{kj=1}^{K-1} \Pr (Y_i = kj) \,=\, 1 - \sum_{kj=1}^{K-1}{\Pr(Y_i=K)}\;e^{\boldsymbol\beta_kbeta_j \cdot \mathbf{X}_i} \;\;\Rightarrow\;\; \Pr(Y_i=K) \,=\, \frac{1}{1 + \sum_{kj=1}^{K-1} e^{\boldsymbol\beta_kbeta_j \cdot \mathbf{X}_i}}</math>.

We can use this to find the other probabilities:

:<math>

\Pr(Y_i=k) = \frac{e^{\boldsymbol\beta_k \cdot \mathbf{X}_i}}{1 + \sum_{kj=1}^{K-1} e^{\boldsymbol\beta_kbeta_j \cdot \mathbf{X}_i}} \;\;\;\;,\;\;k < K

</math>.