As far as I know, the multinomial can be defined as:
Given a sequence of n independent trials each having identical probabilities $p = (p_1, \ldots , p_k)$ for $k$ possible outcomes, the vector of the associated counts $X = (X_1, \ldots ,X_k)$ is said to follow a multinomial distribution and it is denoted as $Mu(n, p)$.
In the context of bayesian networks, we are interested in the joint probability distribution, say, for example, $p(X_1=A, X_2=B, \ldots, X_k=Z)$.
Bayesian Networks in R with Applications in Systems Biology, by R. Nagarajan, M. Scutari and S. Lèbre, says this is a Multinomial Distribution, Can someone explain why?
UPDATE: I post bellow the fragment where this is stated, in Bayesian Networks in R with Applications in Systems Biology, by R. Nagarajan, M. Scutari and S. Lèbre, 2013, Springer (US) (page $7$):
This has nothing much to do with Bayesian Networks in particular.
Are you conflating the term multinomial with multivariate? They are not the same word.
A multivariate distribution is any joint probability distribution of multiple random variables. That is all. Bayesian Networks deal with multinomial distributions since they do concern the interrelation of multiple random variables.
A multinomial distribution is a rather specific familily of multinomial distribution; one where the random variables have a particular definition, as described below. If you do not have this set up, then you are not dealing with a multinomial distribution.
Multinomial.
The Binomial Distribution is a special case of the family: where $k=2$ (the two trial-outcomes of "fail" and "success"), then $\vec p=(1-p, p)$ and $X_1$ is the count for failures, and $X_2$ the count for successes in $n$ iid independent trials.
$$\mathsf P\big(\vec X=(n-x, x)\big)~=~ \dbinom {n}{x}(1-p)^{n-x}p^x\quad\mathbf 1_{x\in[0;n]\cap\Bbb Z}$$
Where $\binom n x$ is the count of distinct arrangements for $n$ trials consisting of $x$ successes and $n-x$ failures, and $(1-p)^{n-x}p^x$ is the probability for obtaining those results for each arrangment.
The Multinomial Distribution generalises this to a sequence of trials each resulting in exactly one from $k$ possible outcomes, and $\vec X$ is the multivariate vector of the counts for each of the $k$ outcomes occuring in $n$ such independent and identically distributed trials.
$$\mathsf P(\vec X=\vec x) ~{=~ \mathsf P(X_1=x_1, \ldots, X_k=x_k) \\ =~\binom{n}{x_1,\ldots,x_k} p_1^{x_1}\cdots p_k^{x_k}\quad\mathbf 1_{\vec x\in \Bbb N^k, \sum_{i=1}^n x_i=n}}$$
Where, $\dbinom{n}{x_1,\ldots,x_k}$ is called the multinomial coefficient, and equals $\dfrac{n!}{x_1!\cdots x_k!}$. Hence the name.