Exactly. A reasonable uninformative prior for scale parameters is Jeffrey's prior 1/x equivalent to log(x) being flat. These patterns arise from this fundamenta

Exactly. A reasonable uninformative prior for scale parameters is Jeffrey's prior 1/x equivalent to log(x) being flat. These patterns arise from this fundamental prior. The same way the lack of knowledge of some phenomenons can be described as a flat probability distribution making every point as likely as the other, some other phenomenons are better described as log(x) being flat. For the first case of flat x, it means that adding or subtracting a constant to the variable doesn't affect the probability. For the flat log(x), it means multiplying or dividing by a constant always has the same effect. For example, something twice as big is always twice as unlikely, half as big twice as likely no matter your starting point. For "scale" parameters that can't go into the negatives (you can't have a negative size) the flat log(x) probability distribution is very natural to represent lack of knowledge. I think it is the maximum entropy distribution under reasonable assumptions. For "location" parameters that go from -∞ to ∞, a flat x "Uniform distribution" is more natural. Few people question the flat x prior. It seems somehow much more intuitive. Note that these are "improper" distributions that don't integrate to 1 but to infinity.