Is there a Continuous Multinomial Distribution??

Question

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!

Answer 1

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.