Normalizing constant in Dirichlet distribution According to references (e.g. Wikipedia and elsewhere), the Dirichlet distribution, parametrized
by $\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_K)$, is
$$
D(x_1, \ldots, x_K) = \frac{1}{\mathrm{B}(\boldsymbol\alpha)} \prod_{i=1}^K x_i^{\alpha_i - 1}
$$
where
$$
\mathrm{B}(\boldsymbol\alpha) = \frac{\prod_{i=1}^K \Gamma(\alpha_i)}{\Gamma\left(\sum_{i=1}^K \alpha_i\right)}.
$$
So, if $K = 2$ and $\alpha_1 = \alpha_2 = 1$ then this gives
$
D(x_1, x_2) = 1/\mathrm{B(\boldsymbol\alpha)}
$
where
$$
\mathrm{B}(\boldsymbol\alpha) = \Gamma(1)^2 / \Gamma(2) = 1
$$
so, $D(x_1, x_2) = 1$ for all $x_1, x_2$.  However,
$D(x_1, x_2)$ is defined on the standard $1$-simplex defined in
$R^2$ by $x_i \ge 0$ and $x_1 + x_2 = 1$.  This is the span (or affine hull) of the
two points $(0, 1)$ and $(1, 0)$.  Since this is a line segment of length $\sqrt{2}$,
the integral of the Dirichlet distribution over this simplex is $\sqrt{2}$, not $1$
as expected.  What am I missing here?
The same problem comes in higher dimensions.  For instance, for $K=3$, the simplex is a triangle with side $\sqrt{2}$, but the normalization constant becomes $B(\boldsymbol\alpha) = 1/\Gamma(3) = 1/2$, which is not the area of this triangle.
What is wrong here?
 A: I think there are subtle issues with the way Wikipedia has presented the density. (Those who are more knowledgeable are free to correct me.)
The normalizing constant is for a slightly different form of the density, where the simplex is parameterized as
$\{(x_1, x_2, \ldots, x_{K-1}, 1-(x_1 + \cdots + x_{K-1}) : x_i \in [0,1]\}$
rather than $\{(x_1, \ldots, x_K) : \sum_i x_i = 1, x_i \in [0,1]\}$.
This is completely analogous to how the beta distribution is parameterized, which is a special case of the Dirichlet distribution where $K=2$: note that beta densities are univariate functions on $[0,1]$, rather than a distribution over a line segment in $\mathbb{R}^2$.
Then the density is
$$\frac{1}{B(\alpha)} x_1^{\alpha_1 - 1}\cdots x_{K-1}^{\alpha_{K-1} - 1} (1-(x_1 + \cdots + x_{K-1}))^{\alpha_K-1}$$
and integrals are taken over $(x_1, \ldots, x_{K-1}) \in [0,1]^{K-1}$ rather than the $(K-1)$-dimensional simplex in $K$-dimensional space, and this computation yields the normalizing constant. (For instance in your simple examples, you easily get $1$.) The discrepancy with the original parameterization of the simplex involves a factor due to the change of variables, which would give the extra $\sqrt{2}$ or area measurements that you mention.
I am not sure why Wikipedia maintains the problematic notation involving the other parameterization. In any case, the Dirichlet distribution is most prominently used in a Bayesian setting as a prior distribution, and the normalizing constant often is not important.
