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I have a dirichlet distribution with three dependent variables.What is the range in which the dirichlet parameters (alphas) should lie? I read in a reference only a condition near the pdf of dirichlet distribution that alphas must be greater than 0. Is it correct?
I have alpha1=1.266, alpha 2 = 0.386, and alpha 3 = 0.4935. alpha 1 + alpha 2 + alpha 3 = 2.1455. The mode of first variable is negative (since k =3 and 2.1455 is less than 3) and so the mode of other two variables becomes positive. But when they are added it sums to 1. Is it valid to get a negative mode for the first variable?

Anagha Raveendran
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    Where do you get this information? Obviously the mode cannot be negative, since the variable values must all lie between $0$ and $1$. – whuber Jan 08 '18 at 17:33
  • Since some $\alpha_i$'s are less than $1$, the mode is achieved when either the second or the third variable go to zero, with a value of $+\infty$. – Xi'an Jan 08 '18 at 17:56
  • Value of each variable lies between 0 and 1. I have three observations for each variable. For var1- (0.2, 0.75 and 0.82). For var 2 - (0.2, 0.22 and 0.12). For var3 - (0.6, 0.03 and 0.06). Each set of observations sums to 1. For var 1, I estimated the alpha as 1.266 by the method of moments given by "A. Narayanan,1992, A note on parameter estimation in the multivariate beta distribution".Then I found dirichlet mode for var 1. But I get negative mode as the denominator(2.1455 - 3)is negative. I am confused where I go wrong. I believe the equation given in the reference to estimate alphas. – Anagha Raveendran Jan 08 '18 at 18:18
  • @AnaghaRaveendran It would be helpful if you can add more details in your question. You mention that var1, var2 and var3 are drawn from Dirichlet distribution with different hyper-parameters i.e. alpha. But then you estimate alpha using MOM? – kedarps Jan 10 '18 at 17:05
  • The three variables and their three set of observations are dependent to each other and it constitute my prior data to do the bayesian analysis. As suggested my project co-guide, I fit dirichlet distribution to this data. Then I combined it with multinomial likelihood data (x1,x2 and x3) and derived posterior distribution of Dirichlet form. In order to find posterior parameters (alphas +x's) I used MOM to estimate alphas from prior distribution. Then I need to find 'posterior mode'. I got the 'negative mode' for the prior distribution when I simply tried to find 'prior mode'!. – Anagha Raveendran Jan 11 '18 at 04:05

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