Considering two Bayesian models:
- Poisson Likelihood & Beta Prior:
$p(y|\lambda) \sim \text{Pois}(\lambda)$, $p(\lambda) \sim \text{Be}(a, b)$:
$$ p(\lambda|y) \propto \lambda^{a-1}e^{-b\lambda} \times \lambda^{y}e^{-\theta} $$
$$ = \lambda^{a+y-1}e^{-\lambda(1+b)} $$
$$ \sim \text{Ga}(a+y, b+1) $$
- Normal likelihood & Normal Prior:
$p(x|\theta, \phi) \sim \text{N}(\theta, \phi)$, $p(\theta) \sim \text{N}(\theta_{0}, \phi_{0})$:
$$ p(\theta|x) \propto \text{exp}\left\{-\frac{1}{2}(\theta-\theta_{0})^{2}/\phi_{0}\right\} \times \text{exp}\left\{-\frac{1}{2}(x-\theta)^{2}/\phi\right\} $$
$$ = \text{exp}\left\{-\frac{1}{2}\left(\frac{\theta^{2}-2\theta\theta_{0}+\theta_{0}^{2}}{\phi_{0}}+\frac{x^{2}-2x\theta+\theta^{2}}{\phi}\right)\right\} $$
$$ \propto \text{exp}\left\{-\frac{1}{2}\theta^{2}(\phi_{0}^{-1}+\phi^{-1})+\theta\left(\frac{\theta_{0}}{\phi_{0}}+\frac{x}{\phi}\right)\right\} $$
$$ \sim \text{N}(\theta_{1}, \phi_{1}) $$
where:
$\phi_{1} = \frac{1}{\phi_{0}^{-1}+\phi^{-1}}$, $\theta_{1} = \phi_{1}\left(\frac{\theta_{0}}{\phi_{0}}+\frac{x}{\phi}\right)$.
Why is it, in the first model, the terms ($a$, $b$, $y$, $n$) in the first model are retained, but in the second model, the terms $\left(\frac{\theta_{0}^{2}}{\phi_{0}}, \frac{x^{2}}{\phi}\right)$ are dropped?
Finally, once again in the second model, why is it that the posterior mean is equal to $\theta_{1}$ and the posterior variance is equal to $\phi_{1}$?
EDIT
I'm aware that if some term is a multiple of the parameter of interest (e.g. $c\theta$ or even $\frac{\theta}{c}$), then the scalar ($c$, in this instance) is absorbed by the proportionality sign, leaving just the parameter ($\theta$), but, if possible, could somebody explain the general rules of proportionality with regard to exponents?
A few examples:
$e^{c+\theta}$, $e^{c\theta}$, $\theta^{c+x}$, $\theta^{cx}$
Assuming $\theta$ is the parameter of interest, what terms would a proportionality sign absorb in the above expressions?