What is a multidimensional generalization of the Beta distribution, in compliance with the following specification?

I am not looking for the Dirichlet distribution.

I am looking for a generalization where the distribution is defined on the hypercube with each side having length 1, in contrast to the Dirichlet, where all sides add up to 1.

I need a pdf for $d$-dimensional data with $\#p=d^2$ parameters, so that correlations between variables of the data are represented by the distribution. (Actually there are more parameters needed, the number of unique correlations betwixt $d$ variables $ = d\left(\large\frac {d+1}2\right)$ and since in the special case where $d=2$ (the beta distribution) we know we have 2 parameters, we could expect there to be $\#p = d(d+1)$ parameters.)

So basically I am looking for a multivariate normal distribution, but defined on the interval (0,1) for each variable, and allowing only one stationary point (mode/maximum/peak, antimode/minimum/dip, or saddle point, depending on the parameters) within the hypercube.

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    $\begingroup$ Generally, with multivariate distributions, there's generally not a 'the multivariate' case, but simply various possibilities. $\endgroup$ – Glen_b -Reinstate Monica Feb 20 '14 at 22:19
  • $\begingroup$ Of course. A Dirichlet distribution is a generalization of the beta distribution, but not the one I am looking for. But yes, I will change the word the' into a'. $\endgroup$ – Angelorf Feb 20 '14 at 23:49
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    $\begingroup$ You have not supplied enough information to pin down the one you are looking for: even a finite number of parameters cannot describe all the distributions satisfying your criteria. Perhaps we could help you narrow down the possibilities if you shared your reasons for looking for this distribution. $\endgroup$ – whuber Feb 21 '14 at 0:06
  • $\begingroup$ Please supply any distribution satisfying my criteria, or name multiple. I can't imagine there being a lot of plausible possibilities. I am looking for a pdf which can model output data of an unsupervised learning algorithm. This output data consists of data points consisting of several variables with values between 0 and 1. Furthermore, the pdf should be differentiable w.r.t. the output data points. I am going to do something similar to decorrelation-backpropagation. $\endgroup$ – Angelorf Feb 21 '14 at 12:19
  • $\begingroup$ One possibility: arxiv.org/abs/1406.5881 $\endgroup$ – kjetil b halvorsen Nov 30 '17 at 9:06

Have you considered using a copula? Here is a very intuitive explanation: https://twiecki.io/blog/2018/05/03/copulas/

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    $\begingroup$ This is basically a link-only answer, and furthermore, doesn't explain how a copula can address the actual question asked. You should expand it, e.g., by constructing a copula that satisfies the question's requirements. $\endgroup$ – jbowman Oct 16 '19 at 17:07

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