Comparison of Dirichlet and Multivariate Beta Regression

I am conducting some analysis on my data I found a strange behavior and would greatly appreciate some guidance or suggestions.

I am trying to investigate the effect of a categorical variable ( cl ) to three percentages that sum 1 ( M ). Naturally, I conducted a dirichlet regression on my dataset and a multivariate beta regression , but when compared using loo the beta regression presented a significantly better fit the data than the dirichlet .

πβΌπ·ππππβπππ‘([1,π½πβπ‘π,π½πβπ‘π])MβΌDirichlet([1,Ξ²aβtb,Ξ²bβtb])

or

π1βΌπ΅ππ‘π(1,π½πβπ‘π)M1βΌBeta(1,Ξ²aβtb)

π2βΌπ΅ππ‘π(1,π½πβπ‘π)M2βΌBeta(1,Ξ²bβtb)

π3βΌπ΅ππ‘π(1,π½πβπ‘π)M3βΌBeta(1,Ξ²cβtb)

Strangely, the predicted variables sum varies between 50% to 150% which is nonsense. However, the fitted variables sum varies 95% to 105% that is an acceptable error.

Is it fair to compare the models? or due to the natural constraints of a Dirichlet model it yields worst fit than a multivariate beta regression ?

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Can you share your model?

I canβt tell much, but you realise that the prior of a Dirichlet is tricky for regression? You can reparametrise

~ dirichlet(precision * softmax([rates_of_abundance]))

rates of abundance should have N - 1 degrees of freedom, and can be the result of your linear equation (in that case the thing that have N - 1 degrees of freedom is the N intercept terms).