How to write a multinomial logit model as a generalized linear model?

Question

Logistic, linear and probit regression can be written in terms of Generalized Linear Model (GLM). Can the multinomial logit regression be written in terms of GLM as well?

Answer 1

If, by multinomial logit regression, you mean a model such that $$ P(y=k|x) = \dfrac{e^{\alpha_k^\text{T}x}}{\sum_j e^{\alpha_j^\text{T}x}} $$ the answer is yes if we follow the definition of GLMs given by MacCullagh and Nelder:

1. the distribution conditional on $x$ is from an exponential family
2. the parameters of the exponential family depend on $x$ through a linear combination $$ (\alpha_1,\ldots,\alpha_k)^\text{T} x $$