Exponential family form of multinomial distribution I feel like this must be a duplicate, but I don't know the magic words to find the appropriate post...
The multinomial distribution is a member of the exponential family. I am used to seeing the "standard" formulation of the multinomial distribution as:
$$p(x|\pi) = \frac{n!}{\prod_{i=1}^m x_i!}\prod_{i=1}^m \pi_i^{x_i}$$
I have seen multiple sources express the multinomial, in the form of an exponential family member as:
$$p(x|\pi) = \exp \{\sum_{i=1}^m x_i \ln \pi_i \}$$
My question is, what happened to the multinomial coefficient $\frac{n!}{\prod_{i=1}^m x_i!}$ ??? If I try to go from the exponential family formulation back to the standard form, I get:
$$p(x|\pi) = \prod_{i=1}^m \pi_i^{x_i}$$
From Kevin Murphy's book, it appears that this exponential formulation only holds for a single trial, in which case the multinomial coefficient is 1.
 A: Exponential families are characterised by their densities, which are such that the interaction between the outcome $x$ of the random variable and the parameter $\theta$ occurs in an exponential scalar product, 
$$\exp\{T(\theta)^\text{T} S(x)\}$$
The other terms in the density are the normalising constant $C(\theta)=\exp\{-\psi(\theta)\}$ and a function of $x$, $h(x)$, which comes to complement the dominating measure $\text{d}\nu(x)$. But all that matters is the product $h(x)\text{d}\nu(x)$ which can be seen as a new measure $\text{d}\nu´(x)$.
So in the multinomial example, the term
$${n \choose x_1 \cdots x_m}=\frac{n!}{\prod_{i=1}^m x_i!}$$
can either be seen as $h(x_1,\ldots,x_m)$ completing the counting measure [which qualifies as "standard" in your terms] on $$\left\{(x_1,\ldots,x_m)\in\mathbb{N}^m;\sum_{i=1}^m x_i=n\right\}$$ or as part of a new measure on that set.
A: I was trying to build the softmax regression from scratch, and got stuck here also: why all the posts just silently omit the coefficient $\frac{n!}{x_1! x_2! \dots x_k!}$ totally, after some research and thinking i share my understanding here:
In reality, we always use a special form of the multinomial distribution with $k>2$ and $n=1$, that is categorical distribution, and since $n=1$, then the coefficient $\frac{n!}{x_1! x_2! \dots x_k!}$ of the multinomial distribution's PMF will be always all 1, that is why we can omit it, more details:
$$
\begin{cases}
  \text{class 1 is chosen: } \frac{1!}{1! \cdot 0! \cdot 0! \dots 0!} = 1 \\
  \text{class 2 is chosen: } \frac{1!}{0! \cdot 1! \cdot 0! \dots 0!} = 1 \\
  \text{class 3 is chosen: } \frac{1!}{0! \cdot 0! \cdot 1! \dots 0!} = 1 \\
  \vdots
\end{cases}
$$
For getting more details and a better full pic of understanding u can refer to my post GLM and exponential family distributions -> Why the PMF has no coefficient
Especially pay attention to understand what does the $x_i$ stand for in multinomial distribution from this section
