Escobar and West Sampler for Dirichlet Process Parameters

Question

I am reading Escobar&West paper and in particular am interested in their Gibbs sampler for the concentration parameter of Dirichlet Process. The issue I have is at the end of their section 6, where all the interesting stuff is done I guess :). So the authors mention that from the following,

$$p(\alpha|k)\propto p(\alpha)\alpha^{k-1}(\alpha+n)\int^1_0x^{\alpha}(1-x)^{n-1}dx$$

it can be implied that $p(\alpha|k)$ is a marginal distribution from a joint for $\alpha$ and a continuous quantity $\eta$ such that $$p(\alpha,\eta|k)\propto p(\alpha)\alpha^{k-1}(\alpha+n)\eta^\alpha(1-\eta)^{n-1}$$ I don't understand how this is implied from the first equation.

Answer 1

The (conditional) marginal distribution can be computed from the (conditional) joint distribution: $$ p(\alpha|k) = \int p(\alpha,\eta|k)\,d\eta. $$ Therefore, if $$ p(\alpha,\eta|k) = c\,p(\alpha)\,\alpha^{k-1}\,(\alpha+n)\,\eta^{\alpha}\,(1-\eta)^{n-1}, $$ where $c$ is an appropriate constant, then $$ p(\alpha|k) = c\,p(\alpha)\,\alpha^{k-1}\,(\alpha+n)\,\int\eta^{\alpha}\,(1-\eta)^{n-1}\,d\eta. $$ Since $\eta$ is a dummy variable, you can rename it $x$.