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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!

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  • $\begingroup$ "the parameters in Dirichlet do not have a constant sum" Sure they do: en.wikipedia.org/wiki/Dirichlet_distribution $\endgroup$ – Rahul May 16 '15 at 15:13
  • $\begingroup$ Sorry, I mean the alpha parameters in Dirichlet Distribution do not have a constant sum. I've corrected my expression. $\endgroup$ – LittleYUYU May 17 '15 at 1:52
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    $\begingroup$ 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. $\endgroup$ – pglpm Jun 16 '17 at 8:46
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    $\begingroup$ 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. $\endgroup$ – pglpm Jun 16 '17 at 8:46
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PDF of the multinomial distribution can be evaluated outside its support, so we can define a distribution taking PDF as its analytic continuation. In fact, this new PDF integrate to 1 on the corresponding stretched simplex for a binomial distribution. integrates to approximately 1 due to rectangular rule.

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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<p_i\le 1, \sum_i p_i=1\ $, 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}

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