Multinomial logistic regression: Difference between revisions

Revision as of 21:22, 16 July 2022

Revision as of 21:28, 16 July 2022

Line 93:

:<math>

\begin{align}

\Pr(Y_i=1) &= \frac{e^{\boldsymbol\beta_1 \cdot \mathbf{X}_i}}{1 + \sum_{j=1, j\neq 1}^{K-1} e^{\boldsymbol\beta_j \cdot \mathbf{X}_i}} \\

\\

\Pr(Y_i=2) &= \frac{e^{\boldsymbol\beta_2 \cdot \mathbf{X}_i}}{1 + \sum_{j=1,j\neq 2}^{K-1} e^{\boldsymbol\beta_j \cdot \mathbf{X}_i}} \\

\cdots & \cdots \\

\Pr(Y_i=K-1) &= \frac{e^{\boldsymbol\beta_{K-1} \cdot \mathbf{X}_i}}{1 + \sum_{j=1, j\neq K-1}^{K-1} e^{\boldsymbol\beta_j \cdot \mathbf{X}_i}} \\

\end{align}

</math>

where the summation runs from <math>1</math> to <math>K-1</math>, but excluding the term with the index of the probability being computed, or generally:

<math>

\begin{align}

\Pr(Y_i=k) = \frac{e^{\boldsymbol\beta_{K-1k} \cdot \mathbf{X}_i}}{1 + \sum_{j=1, j\neq k}^{K-1} e^{\boldsymbol\beta_j \cdot \mathbf{X}_i}}

\end{align}

</math>

where <math>

The fact that we run multiple regressions reveals why the model relies on the assumption of [[independence of irrelevant alternatives]] described above.

\beta_K

</math> is defined to be zero. The fact that we run multiple regressions reveals why the model relies on the assumption of [[independence of irrelevant alternatives]] described above.

===Estimating the coefficients===