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With two variables, you are defining a line segment in R2, as you pointed out. However, due to the simplex constraint, one of these two variables is redundant in terms of specifying the density, since there is a one-to-one relationship between x1 and x2. Therefore, the density is specified over K1 free variables (i.e., in R)

This is actually pointed out in the first line of this section of the Wikipedia article, albeit very subtly.

Therefore, your density function becomes:

Dir1,1(x1,1x1)=Γ(2)Γ(1)2(x1)0(1x1)0=1

Therefore,

10Dir1,1(x1,1x1)dx1=1


Response to OP Comment

Due to the simplex constraints, the two-variable Dirichlet density is actually degenerate in R2, as shown by my construction above (it only requires one variable). While it is true it has a density of 1, it does not have a density of 1 on the line segment connecting (1,0) with (0,1). What the above construction shows is that the marginal density has a value of 1. Your confusion comes from thinking of x2 as a free variable, in which case the support of the Dirichlet on R2 would have a non-zero area. This intuition is fine in cases like the the bivariate gaussian, where the two variables are not perfectly correlated, but not in this case.

We can formally derive this as follows:

Let L be some number in [0,2] specifying the distance from (1,0) to (0,1) along the connecting line segment. Thus, each value of L identifies a unique (x1,x2) pair. Using this notation, your assumption that the density is 1 along this line boils down to:

P(L[a,b])=ba

However, we can show this is not the case through a formal treatment of the joint density of x1,x2:

P(x1,x2)=P(x1)P(x2|x1)

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