When reading the paper "A Generative Shape Regularization Model for Robust Face Alignment" by Leon Gu and Takeo Kanade i found an interesting formula $$p(b,\theta,S,h,z) \propto p(S|b,z)p(b)p(z)p(\mathcal{Q}(h)|S,\theta)p(I|h)$$
Useful information is that $p(\theta)$ is non-informative.
Also it is given that($Q_{nk}$ is a 2 dimentional point while I is the whole image):
$p(I|h_{nk})=p(I|Q_{nk})=\pi_{nk}$
$n$-th element of $\mathcal{Q}(h)$ is given as:
$\mathcal{Q}_n(h)=(\sum_{k=1}^{K} h_{nk} Q_{nk}^x,\sum_{k=1}^{K}h_{nk})^T$
$\mathcal{Q}(h)$ is generated from $S$ and $\theta$ adding gaussian noise:
$p(\mathcal{Q}(h)|S,\theta)=\mathcal{N}(T (S;\theta) ;\Sigma)$
Prior is defined as:
$p(S|b)=\sum_{l=1}^{L}\pi_l \mathcal{N}(\Phi_l b_l+ \mu_l;\sigma_l^2 I)$
Also there are formulas for $p(b_l)$ , $p(z_l=1)$ and $p(S|b,z)$ in case they are needed:
$p(b_l)=\mathcal{N}(0;diag(\lambda_{l1},\dots,\lambda_{lM_l}))$
$p(z_l=1)=\pi_l$
$p(S|b,z)=\prod_{l=1}^{L}\mathcal{N}^{z_l}(\Phi_l b_l + \mu_l;\sigma_l^2 I)$
How do i prove the first formula ?
I'm guessing it's just Bayes formula application but i always seem to get lost when trying to prove it.