Problem: Show that the following distributions are conjugate priors for the corresponding densities..
The multinomial distribution with $k$ categories and
$$ p_{X|\theta_1 , \dots, \theta_k} (x_1, \dots, x_k) = \frac{n!}{x_1! \cdot \dots \cdot x_k!} \prod_{i=1}^{k} \theta_i^{x_i}$$
where $x_i \geq 0 $ are integers and $\sum_{i=1}^{k} x_i = n$ has the Dirichlet distribution,
$$ P_{\Theta_1, \dots, \Theta_k} (\theta_1, \dots, \theta_k) = \frac{\prod_{i=1}^{k} \Gamma(\alpha_i)}{\Gamma(\sum_{i=1}^{k} \alpha_i)} \prod_{i=1}^{k} \theta_i^{\alpha_i-1}$$
Attempted solution:
Need to calculate the posterior $p_{\Theta | X} = \frac{p_{X|\Theta} p_{\Theta}}{p_X}$ and see that it is a Dirichlet distributions for some hyperparameters $\beta_i$.
The numerator is
$$ p_{X|\Theta} p_{\Theta} = \frac{n!}{x_1! \cdot \dots \cdot x_k!} \frac{\prod_{i=1}^{k} \Gamma(\alpha_i)}{\Gamma(\sum_{i=1}^{k} \alpha_i)} \prod_{i=1}^{k} \theta_i^{x_i + \alpha_i-1} := \xi \prod_{i=1}^{k} \theta_i^{x_i + \alpha_i-1}$$
And the denominator \begin{equation} \begin{split} p_X &= \int_{\theta} p_{X|\Theta} p_{\Theta} d\Theta = \xi \prod_{i=1}^{k} \int_{0}^{1} \theta_i^{x_i + \alpha_i-1} d\theta_i \\ &= \xi \prod_{i=1}^{k} [\frac{\theta_i^{x_i + \alpha_i}}{x_i + \alpha_i} ]_{\theta = 0}^{1} \end{split} \end{equation}
Any hints...?
Best regards
