    # $p\left(z | x_{1}, \ldots, x_{N}\right) \propto \frac{\prod_{i=1}^{N} p\left(z | x_{i}\right)}{\prod_{i=1}^{N-1} p(z)} \approx \frac{\prod_{i=1}^{N}\left[\tilde{q}\left(z | x_{i}\right) p(z)\right]}{\prod_{i=1}^{N-1} p(z)}=p(z) \prod_{i=1}^{N} \tilde{q}\left(z | x_{i}\right)$  
