If I have a dirichlet distribution with parameter $\alpha \in \mathbb{R}^k = [\alpha_1, \alpha_2, \cdots, \alpha_k]$, and then I set the component $\alpha_k$ to $\epsilon$. As I decrease $\epsilon$, will this pdf approximate the pdf of a dirichlet distribution with parameters $[\alpha_1, \alpha_2, \cdots, \alpha_{k-1}]$? That is, ignoring the last dimension of the samples from $\text{Dir}([\alpha_1, \alpha_2, \cdots, \alpha_{k-1}, \epsilon])$, will they follow approximately the same distribution as samples from $\text{Dir}([\alpha_1, \alpha_2, \cdots, \alpha_{k-1}])$

My progress so far:

As a sanity check I started with simulations using Numpy. Sampling from $\text{Dir}([\alpha_1, \alpha_2, \cdots, \alpha_{k-1}, 10^{-10}])$ and $\text{Dir}([\alpha_1, \alpha_2, \cdots, \alpha_{k-1}])$ seems very similar indeed.

Then I've tried substituting the new $([\alpha_1, \alpha_2, \cdots, \alpha_{k-1}, \epsilon])$ vector into the expression for the PDF of the Dirichlet distribution. The resulting expression seems very similar to the PDF of Dir$([\alpha_1, \alpha_2, \cdots, \alpha_{k-1}])$, but it's being multiplied by $\frac{1}{x_k\cdot\Gamma(\epsilon)}$. As $\epsilon$ decreases the whole expression tends to zero. Not really sure how to proceed from here.


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