Exactly. A reasonable uninformative prior for scale parameters is Jeffrey&#x27;s prior 1&#x2F;x equivalent to log(x) being flat. These patterns arise from this fundamental prior.<p>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.<p>For the first case of flat x, it means that adding or subtracting a constant to the variable doesn&#x27;t affect the probability.<p>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.<p>For &quot;scale&quot; parameters that can&#x27;t go into the negatives (you can&#x27;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 &quot;location&quot; parameters that go from -∞ to ∞, a flat x &quot;Uniform distribution&quot; is more natural. Few people question the flat x prior. It seems somehow much more intuitive.<p>Note that these are &quot;improper&quot; distributions that don&#x27;t integrate to 1 but to infinity.