# Is there a Continuous Multinomial Distribution??

In Multinomial Distribution, we have \begin{align} f(x_1,\ldots,x_k;n,p_1,\ldots,p_k) & {} = \Pr(X_1 = x_1\mbox{ and }\dots\mbox{ and }X_k = x_k) \\ \\ & {} = \begin{cases} { \displaystyle {n! \over x_1!\cdots x_k!}p_1^{x_1}\cdots p_k^{x_k}}, \quad & \mbox{when } \sum_{i=1}^k x_i=n \\ \\ 0 & \mbox{otherwise,} \end{cases} \end{align}

where $x_i$ is an integer. Besides, we should note that x_i have a constant sum, and the sum of p_k equals to 1.0 (another constant sum).

But now, I need a Continuous Multinomial Distribution, where $x_i$ doesn't need to be an integer, and the sum of $x_i$ still equals $n$. I cannot find such a distribution, could any one help me?

p.s. I found a related question in this site. Someone says that Dirichlet Distribution can be helpful. However, the alpha parameters in Dirichlet do not have a constant sum, which is not perfect for my problem.

Thanks very much!

• "the parameters in Dirichlet do not have a constant sum" Sure they do: en.wikipedia.org/wiki/Dirichlet_distribution
– user856
May 16 '15 at 15:13
• Sorry, I mean the alpha parameters in Dirichlet Distribution do not have a constant sum. I've corrected my expression. May 17 '15 at 1:52
• Unfortunately I don't know the answer; I'm also searching for it. But take a look at Johnson, Kotz, & Balakrishnan: Discrete Multivariate Distributions (Wiley 1996) for some limit forms, e.g. for $n\to\infty$: you can then consider the variables $Y_i := X_i/n \in [0,1]$, which have a continuous distribution similar to the original multinomial in this limit. This limit continuous distribution is not a Dirichlet. I'd like to point out that the Dirichlet distribution is not the continuous version of the multinomial distribution, as some comments around claim. Jun 16 '17 at 8:46
• Each variable $X$ of the multinomial has a probability proportional to $p^X/X!$, whereas each variable $X$ of the Dirichlet has a probability proportional to $X^p$. These are very different functional dependences. It's true that the dependence of the multinomial on its parameters is the same as the Dirichlet's on its variables. For this reason the Dirichlet is the conjugate prior of the multinomial. See again Johnson & al. above. Jun 16 '17 at 8:46

Here's a distribution which seems to satisfy your criteria, but which I very much suspect won't be what you're looking for. However, refining your criteria to rule it out might help clarify what it is that you are looking for. Let $$\ A\$$ be a finite subset of $$\ \left\{\alpha\in\mathbb{R}^k\,\vert\ \sum_j\alpha_j=n , \alpha_i\not\in\mathbb{Z}, \alpha_i > 0 \mbox{ for all } i\right\}$$ and $$\ p_i, i=1,2,\dots,k,\$$ be real numbers with $$\ 0, and $$\ G\ \ =\ \sum_{\alpha\in A}\frac{\Gamma(n+1)}{\Gamma(\alpha_1+1)\Gamma(\alpha_2+1)\dots\Gamma(\alpha_k+1)}\,p_1^{\alpha_1} p_2^{\alpha_2}\dots p_k^{ \alpha_k}\ .$$ Then take \begin{align} f(&\alpha_1,\ldots,\alpha_k;n,p_1,\ldots,p_k) {} = \Pr(X_1 = \alpha_1\land\dots\land X_k = \alpha_k) \\ \\ & {} = \begin{cases} G^{-1}\frac{\Gamma(n+1)}{\Gamma(\alpha_1+1)\Gamma(\alpha_2+1)\dots\Gamma(\alpha_k+1)}p_1^{\alpha_1} p_2^{\alpha_2}\dots p_k^{ \alpha_k}, & \mbox{when } \alpha\in A \\ \\ 0 & \mbox{otherwise,} \end{cases} \end{align}
I am not aware of the distribution you were looking for, but there is a "continuous" categorical distribution (special case of multinomial distribution with $$n=1$$) recently popped up in the machine learning community.