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What happens when one of the concentration parameters of a Dirichlet distribution tends to zero?

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$begingroup$

If I have a dirichlet distribution with parameter $alpha in mathbbR^k = [alpha_1, alpha_2, cdots, alpha_k]$, 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 $textDir([alpha_1, alpha_2, cdots, alpha_k-1, epsilon])$, will they follow approximately the same distribution as samples from $textDir([alpha_1, alpha_2, cdots, alpha_k-1])$

---

My progress so far:

As a sanity check I started with simulations using Numpy. Sampling from $textDir([alpha_1, alpha_2, cdots, alpha_k-1, 10^-10])$ and $textDir([alpha_1, alpha_2, cdots, alpha_k-1])$ 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 $frac1x_kcdotGamma(epsilon)$. As $epsilon$ decreases the whole expression tends to zero. Not really sure how to proceed from here.

probability probability-distributions

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asked Apr 2 at 9:19

JLaganaJLagana

258113

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0

$begingroup$

If I have a dirichlet distribution with parameter $alpha in mathbbR^k = [alpha_1, alpha_2, cdots, alpha_k]$, 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 $textDir([alpha_1, alpha_2, cdots, alpha_k-1, epsilon])$, will they follow approximately the same distribution as samples from $textDir([alpha_1, alpha_2, cdots, alpha_k-1])$

---

My progress so far:

As a sanity check I started with simulations using Numpy. Sampling from $textDir([alpha_1, alpha_2, cdots, alpha_k-1, 10^-10])$ and $textDir([alpha_1, alpha_2, cdots, alpha_k-1])$ 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 $frac1x_kcdotGamma(epsilon)$. As $epsilon$ decreases the whole expression tends to zero. Not really sure how to proceed from here.

probability probability-distributions

share|cite|improve this question

asked Apr 2 at 9:19

JLaganaJLagana

258113

$endgroup$

add a comment | 

$begingroup$

If I have a dirichlet distribution with parameter $alpha in mathbbR^k = [alpha_1, alpha_2, cdots, alpha_k]$, 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 $textDir([alpha_1, alpha_2, cdots, alpha_k-1, epsilon])$, will they follow approximately the same distribution as samples from $textDir([alpha_1, alpha_2, cdots, alpha_k-1])$

---

My progress so far:

As a sanity check I started with simulations using Numpy. Sampling from $textDir([alpha_1, alpha_2, cdots, alpha_k-1, 10^-10])$ and $textDir([alpha_1, alpha_2, cdots, alpha_k-1])$ 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 $frac1x_kcdotGamma(epsilon)$. As $epsilon$ decreases the whole expression tends to zero. Not really sure how to proceed from here.

probability probability-distributions

share|cite|improve this question

asked Apr 2 at 9:19

JLaganaJLagana

258113

$endgroup$

share|cite|improve this question

asked Apr 2 at 9:19

JLaganaJLagana

258113

share|cite|improve this question

asked Apr 2 at 9:19

JLaganaJLagana

258113

asked Apr 2 at 9:19

JLaganaJLagana

258113

asked Apr 2 at 9:19

JLaganaJLagana

258113