I'm seeking the right distribution for my 4D data, where the sum of values in each sample equals one. Currently, I've chosen to employ the Dirichlet distribution. However, upon applying this distribution, I've noticed that the marginal Dirichlet distribution doesn't align well with the first dimension. It appears that the influence of the other dimensions is causing the distribution to shift to the left. Surprisingly, when I exclude the fourth dimension or fourth and third dimensions, the fit improves significantly. This discrepancy seems to be due to interdimensional interactions.
My question is whether it's advisable to explore alternative distributions to the Dirichlet, or if there's a means to introduce a parameter that accommodates this variance, if this is due to the variance.
If you're aware of any articles or methods in this field, your guidance would be greatly appreciated.
Please note that I'm working with a limited sample size, approximately 200 samples, so a distribution with a large number of parameters may not be suitable.
Edit:
thanks to @whuber, I have removed histogram plots to make the question more clear.
Update (another example of a 4D data modeling with dirichlet distribution with 4D vs 3D vs 2D )
4D
3D