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. $$\prod_{t=1}^{n}q(x_{t}|x^{t-1})= \frac{\prod_{i=1}^{m}\prod_{j=1}^{n(i|x^{n})}\left( n(i|x^{n})+ \frac{1}{2} -j \right)}{\prod_{j=1}^{n}\left( n+\frac{m}{2}- j \right)}=q(x^{n})$$