= Solution
Set $\eta_{ik}=\beta_k^T\mathbf x_i$, so $\eta_{i1}=0$. The <multinomial logistic regression> implies $p_{ik}=e^{\eta_{ik}}p_{i1}$. Normalizing gives
$$
p_{ik}=\frac{e^{\eta_{ik}}}{S_i},\qquad
S_i=\sum_{j=1}^Ke^{\eta_{ij}}
=1+\sum_{j=2}^Ke^{\eta_{ij}}.
$$
Multiply the <multinomial likelihoods> for conditionally independent groups. The multinomial coefficients are constant in $\beta$, leaving
$$
\boxed{L(\beta)\propto\prod_{i=1}^I
\frac{\exp\{\sum_{k=1}^K y_{ik}\beta_k^T\mathbf x_i\}}
{\{\sum_{k=1}^K\exp(\beta_k^T\mathbf x_i)\}^{n_i}}.}
$$
The reference constraint $\beta_1=0$ identifies the coefficients: a common shift of every category coefficient otherwise leaves all probabilities unchanged. The PDF places the full sum inside the numerator exponent; the TeX breaks that expression.
Back to article page
