How to obtain the multinomial logistic regression via the framework of GLMs?

Question

From wikipedia, the assumption we are making in multinomial logistic regression is that we take one class as "reference", say class $K$, and we assume that $\ln\left(\frac{\mathbb{P}(X=k_0)}{\mathbb{P}(X=K)}\right) = \beta_{k_0}^TX_i$. This very much feels like a GLM, though I do not know what is the distribution used in the GLM.

An educated guess would be the multinomial distribution, but then we have to specify $y$ as a vector, $y = (n_1, \dots, n_K)^T$, and I'm slightly worried here as I have not seen exponential parameterization of multivariate distributions, are they even a thing?

EDIT: On second thought $y$ is not a vector, rather just which class. Since the goal is multi-class classification, we don't use the general definition of multinomial distribution, but rather $\mathbb{P}(\text{Class}=k) = \binom{1}{1_{k=k_1}, \dots, 1_{k=k_K}}\prod_{i=1}^K p_1^{1_{k=k_1}}, \dots, p_K^{1_{k=k_K}} $

Answer 1

Multivariate generalized linear models are quite possible; see for example Fahrmeir and Tutz, "Multivariate Statistical Modeling Based on Generalized Linear Models," second edition, Springer, 2001. The multinomial distribution is appropriate for a multi-category outcome with no ordering among the levels. The link function is then a vector, representing the log-odds of each non-reference category against the reference category, based on a category-specific linear predictor.