Lemma (Dirichlet Mixture Distribution)

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 $n(i|x^{n})=0$.

Lemma (Krichevsky and Trofimov Coding Distribution)

For all $1\le t\le n$ define $$q(i|x^{t-1})= \frac{n(i|x^{t-1})+\frac{1}{2}}{t-1+\frac{m}{2}}$$Then

1. $q(i|x^{t-1})$ is a conditional pmf on $\mathcal{X}$ given $x^{t-1}$
2. $$\prod_{t=1}^{n}q(x_{t}|x^{t-1})= \frac{\prod_{i=1}^{m}\prod_{j=1}^{n(i|x^{n})}\left( n(i|x^{n})+ \frac{1}{2} -j \right)}{\prod_{j=1}^{n}\left( n+\frac{m}{2}- j \right)}=q(x^{n})$$