Assume I have data X with normal likelihood and Prior Mean C as Laplace prior as inputs to the model. For easier convergence, I multiply each observation of Data by C and then use Prior Mean 1. And I observe the results are the same exactly.

However, I am not able to visualize the posterior equations and how are they equal in both the cases and bring about the same results.

Case 1 Posterior:

\propto \exp\left(-\frac{(X - \theta)^2}{2\sigma^2} - \frac{|\theta - C|}{b}\right)

Case 2 Posterior:

\propto \exp\left(-\frac{(CX - \theta)^2}{2\sigma^2} - \frac{|\theta - 1|}{b}\right)