Suppose that you have a stocastic variable:
(1) $$ Y\mid\Theta\sim Laplace\left(\mu,\sigma^2\right) $$
(2) $$ \Theta\sim N\left(\mu_0,\sigma^2_0\right) $$ and
(3) $$ Z\sim Laplace\left(\mu,\sigma^2\right) $$ where $$ f_z\left(\mu,\sigma^2\right)=\frac{1}{2\sigma^2}\exp\left(-\frac{1}{\sigma^2}\mid z-\mu\mid\right) $$
How would you find a posterior distribution for theta given this information?
My guess would be to do this numerically but I currently don't have the knowledge for that, so I am interested how you do it algebraically. I guess the first step would be to multiply (1) with (2), however I don't know a distribution that has this value.
My expression is this:
$$ \exp\left(-\frac{1}{\sigma^2}\sum_{i=1}^n\mid z-\mu\mid-\frac{1}{2}\frac{\left(\Theta-\mu_0\right)^2}{\sigma_0^2_{}}\right) $$
For the posterior distribution. However I don't recognize this.
Any thoughts on how to start?