Understanding the influence of the prior distribution on the original parametrization

Be $y_1,y_2,..y_2$ a simple random sampling from $p(y|\theta)$. Be $\theta$ a parameter with a given prior distribution $p(\theta)$. A way to understand how much informative is $p(\theta)$ is to plot it and see how it is distributed on the paramter space $\Theta$.

Now be $y_1,y_2,..y_2$ a simple random sampling from $p(y|\theta)$. Be $\psi=\psi(\theta)$ a reparametrization. Be $\psi$ is a parameter with a given a prior distribution $p(\psi)$. Choosing a specific prior distribution for $\psi$ imply a precise choice for the prior distribution of $\theta$. Suppose that I choose a specific prior distribution for $\psi$. I'd like to understand the influence of the prior distribution $p(\psi)$ on the original parametrization $\theta$. How can I understand it? How can I understand how much informative is $p(\psi)$ on $\theta$? As far as I know a uninformative prior $p(\psi)$ for $\psi$ may be informative for $\theta$.

Should I generate some values from $p(\psi)$, apply the inverse transformation $\theta=\theta(\psi)$ and look to its histogram?

Let's consider an example.
Be $y$ an observation from $Y|\theta \sim Bin(n,\theta)$. Be $\psi=\psi(\theta)=log(\theta/(1-\theta))$ a reparametrization. Let's choose for $\psi$ a normal prior distribution: $\psi\sim N(0,\sigma^2)$. If $\sigma^2$ is large, $p(\psi)$ is a weak informative prior for $\psi$. I'd like to understand how much informative is $p(\psi)$ for $\theta$.

Thank you.