Likelihood function: $$L(\zeta) = \zeta^{\alpha} \exp\left[\zeta\sum_{i=1}^{\alpha}(x_i - x)\right]$$
Prior function: $$p(\zeta) = \frac{1}{(\sqrt{2π}σ_\zeta)} \exp\left[-\frac{(\zeta-\zeta_0)^2}{(2σ_\zeta^2)}\right]$$
I am trying to find the log posterior of $\zeta$
Bayes Theorem: $$\log(posterior) \stackrel{\log}{\propto} \log(L(\zeta))+\log(p(\zeta)$$
Solving this I get:
$$\log(posterior) \stackrel{\log}{\propto}\log[\zeta^{\alpha}] +\zeta\sum_{i=1}^{\alpha}(x_i - x)-\frac{(\zeta-\zeta_0)^2}{2\sigma_{\zeta}}-\log(\sigma_{\zeta})$$
but I don't not know how to to proceed.
Edit:
Here
- $x_{i} ≤ x$ where $x$ is a limiting constant
- We have ${x_1,…x_{\alpha}} $independently and identically distributed random sample.