Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 34 / 3 / b / 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 givesMultiply the multinomial likelihoods for conditionally independent groups. The multinomial coefficients are constant in $\beta$, leavingThe 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.