$p(x|\pi)$ follows the categorical distribution (the multinomial with one observation), where $\sum\pi_i=1$ and $x$ is a one-hot vector, and $p(\pi|\alpha)$ follows the Dirichlet distribution.
$p(x|\pi)=\prod\pi_i^{x_i}$
$p(\pi|\alpha)=\frac{1}{B(\alpha)}\prod\pi_i^{\alpha_i-1}$
$p(x,\pi|\alpha)=\frac{1}{B(\alpha)}\prod\pi_i^{\alpha_i-1+x_i}$
$p(x|\alpha)=\int p(x,\pi|\alpha)d\pi=\frac{1}{B(\alpha)}\int\prod\pi_i^{\alpha_i-1+x_i}d\pi=\frac{B(\alpha+x)}{B(\alpha)}$
$\sum p(x^{(i)}|\alpha)=\sum\frac{B(\alpha+x^{(i)})}{B(\alpha)}=\sum\frac{\alpha_i+1}{1+\sum\alpha_j}\neq1$
where $x^{(i)}$ is the one-hot vector with the i-th bit equal to 1.
please correct me where I'm wrong, thanks in advance.