I'm trying to understand Jeffreys prior. One application is for 'scale' variables like the standard deviation $\sigma$ (or its square, the variance $\sigma^2$) of Gaussian distributions. It is often said that using a uniform prior over $\sigma$ is not really non-informative and instead one should either:
Use instead $\ln \sigma$ as the free parameter, with a uniform prior (this is often called a log-uniform prior)
Or keep using $\sigma$ as the free parameter but use $1/\sigma$ as the prior (which is not uniform).
Why are the above two methods/priors equivalent? I feel it has something to do with the fact that the derivative of ln $\sigma$ is $1/\sigma$ but I can't take the next step.
Also, why does this even matter, in simple language with minimal jargon? I see all these complicated explanations online involving the Fisher information matrix but in the end all I see is that the above log-uniform or $1/\sigma$ priors preferentially weight lower values of $\sigma$ more highly. Why? If possible, a simple analytic example or python snippet would be very helpful.