Factoring a joint posterior

Question

Suppose the joint posterior density for parameters $(\theta_1, \theta_2)$ can be expressed as \begin{align} \Pr(\theta_1, \theta_2 \mid y) &\propto Pr(y \mid \theta_1, \theta_2) *Pr(\theta_1, \theta_2)\\ &=Pr(y \mid \theta_1, \theta_2) *Pr(\theta_1 \mid \theta_2)*Pr(\theta_2) \end{align} These 2 forms are equivalent, right? Then, can we factor and express the joint posterior as: \begin{align} \Pr(\theta_1, \theta_2 \mid y) &= (\Pr(\theta_1 \mid \theta_2, y)*Pr(\theta_2 \mid y)) \\ &\propto Pr(y \mid \theta_1, \theta_2) *Pr(\theta_1 \mid \theta_2) * Pr(\theta_2 \mid y) \\ &\propto Pr(y \mid \theta_1, \theta_2) *Pr(\theta_1 \mid \theta_2)*Pr(y \mid \theta_2 )* Pr(\theta_2) \end{align}

I think I followed the rules of factorization but because of $Pr(y \mid \theta_2)$ I can't equate the first derivation with the second up to a proportionality constant. $$ Pr(y \mid \theta_1, \theta_2) *Pr(\theta_1 \mid \theta_2)*Pr(\theta_2) \propto Pr(y \mid \theta_1, \theta_2) *Pr(\theta_1 \mid \theta_2)*Pr(y \mid \theta_2 )* Pr(\theta_2) $$

Can someone point me where I made a mistake?

Answer 1

As Vimal points out my mistake was writing, $$Pr(\theta_1 \mid \theta_2, y)*Pr(\theta_2 \mid y) \propto Pr(y \mid \theta_1, \theta_2) *Pr(\theta_1 \mid \theta_2) * Pr(\theta_2 \mid y)$$ When factoring, \begin{align} (\Pr(\theta_1 \mid \theta_2, y)*Pr(\theta_2 \mid y)) &= \Pr(\theta_1 \mid \theta_2, y)*Pr(\theta_2 \mid y) \\ &= \dfrac{Pr(y \mid \theta_1, \theta_2) *Pr(\theta_1 \mid \theta_2)}{Pr(y \mid \theta_2)} * Pr(\theta_2 \mid y) \\ &= \dfrac{Pr(y \mid \theta_1, \theta_2) *Pr(\theta_1 \mid \theta_2)}{Pr(y \mid \theta_2)} * \dfrac{Pr(y \mid \theta_2) Pr(\theta_2)}{Pr(y)} \\ &\propto Pr(y \mid \theta_1, \theta_2) *Pr(\theta_1 \mid \theta_2) * Pr(\theta_2) \end{align} Thus, both approaches factoring the prior and factoring the posterior are equivalent.