Simple question: What is the update equation of a $x_i$ in a Gibbs sampling update, $p(x_i | x_{-i})$, if I have the Model:
$\theta$ $|$ $\alpha \sim Dir(\alpha)$
$X_i$ $|$ $\theta \sim Discrete(\theta), i = 1, 2, 3$
Is it the predictive distribution of a dirichlet multinomial?
$\frac{N_i + \alpha_i}{\sum^k_{j=1}(N_j + a_j)}$
yes.
$$ \begin{align} \\ P(X_i | X_{-i}, \boldsymbol{ \alpha } ) &= \frac{ P(X_i, X_{-i} | \boldsymbol{ \alpha } )}{P(X_{-i} | \boldsymbol{\alpha})} \\ &= \frac{ P(X|\boldsymbol{\alpha})}{P(X_{-i} | \boldsymbol{\alpha})}\\ &\propto P(X|\boldsymbol{\alpha}) \text{ because the denominator doesn't depend on } X_i \\ \end{align} $$
