The coding distribution corresponding to the Dirichlet $(\alpha_{1},\dots,\alpha_{m})$ pdf is given by $$q(x^{n})=\int\limits _{\Theta}p_{\theta}(x^{n})f_{\alpha_{1},\dots,\alpha_{m}}(\theta) \, d\theta = \frac{\prod_{i=1}^{m}\prod_{j=1}^{n(i|x^{n})}(n(i|x^n)+\alpha_{i}-j)}{\prod_{j=1}^{n}\left( n+\sum_{i=1}^{m}\alpha_{i}-j \right)}$$where $\prod_{j=1}^{n(i|x^{n})}(n(i|x^{n})+\alpha_{i}-j)=1$ if $n(i|x^{n})=0$.