Calculate log likelihood of Dirichlet distribution using Gamma distribution

Question

Let $X_1,\dots,X_{k+1}$ be mutually independent random variables, each having a gamma distribution with parameters $\alpha_i,i=1,2,\dots,k+1$ show that $Y_i=\frac{X_i}{X_1+\cdots+X_{k+1}},i=1,\dots,k$, have a joint ditribution as $\text{Dirichlet}(\alpha_1,\alpha_2,\dots,\alpha_k;\alpha_{k+1})$

According to the above theorem, we can get Dirichlet random variables from Gamma distribution.
So, I think that it's possible to calculate log likelihood of Dirichlet Distribution using Gamma.

Let $\vec{X} = (X_1,X_2,...,X_n)$ be samples from $Gamma(\alpha_i, 1)$. Then we normalize $\vec{X}$ and get $\vec{Y}$.
Then, Let $A = log\,p(\vec{X}) = \sum_{i=1}^nlog\,p(X_i) = \sum_{i=1}^n [(\alpha_i-1)\,log\,X_i - X_i - log\,\Gamma(\alpha_i)]\,$
$B = log\,p(\vec{Y})=\Sigma_{i=1}^n[(\alpha_i-1)log\,Y_i-log\,\Gamma(\alpha_i)]+log\,\Gamma(\bar{\alpha}) = \Sigma_{i=1}^n[(\alpha_i-1)(logX_i-log\bar{X})-log\,\Gamma(\alpha_i)]-log\,\Gamma(\bar{\alpha})$.

With the calculation, I think the result says that $A \neq B$. I don't know why this happen. I think that dirichlet random variable $\vec{Y}$ comes from Gamma random variable $\vec{X}$. So, I think they have the same log likelihood, but they didn't. Would you expalin about this?

Answer 1

A Gamma sample of $X_i\sim\text{Ga}(\alpha_i,1)$ differs from a Dirichlet sample $(Y_1,\ldots,Y_n)\sim\text{Dir}(\alpha_1,\ldots,\alpha_n)$ and it is thus no surprise that the likelihood functions on $\mathbf \alpha$ differ.

More importantly, the Dirichlet sample contains less information about the $\alpha_i$'s than the Gamma sample since $\sum_i Y_i=1$. Intuitively, it looses one dimension. In other words, when observing the $X_i$'s, turning them into $Y_i$'s by a normalisation step is wasting away information. (The exact information matrix for both the Gamma and the Dirichlet involves digamma functions, hence making the comparison delicate.)

The quoted theorem is about simulation, exhibiting a connection between the distribution of the $X_i$'s and the distribution of the $Y_i$'s. But the $X_i$'s are latent variables in statistical terms, they are not observed.