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I want to learn a decision-tree based on dirichlet distribution (namely a derichlet-tree). Which Python (or possibly other) libraries/packages enables such algorithm?

SSSOF
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I'm not 100% sure if it's exactly that implementation you look for, but I found this paper:

März, A. (2022). Multi-Target XGBoostLSS Regression. arXiv preprint arXiv:2210.06831.

and the author references its GitHub repo:

https://github.com/StatMixedML/XGBoostLSS/tree/master

and there is an implementation of this distribution inside the repo and it appears that this model can use Dirichlet distribution.