Creating a uniform prior on the logarithmic scale

Question

A uniform prior for a scale parameter (like the variance) is uniform on the logarithmic scale.

What functional form does this prior have on the linear scale? And why so?

Answer 1

It's just a standard change of variables; the (monotone & 1-1) transformation is $y = \exp(x)$ with inverse $x=\log(y)$ and Jacobian $\frac{dx}{dy} = \frac{1}{y}$.

With a uniform prior $p_y(y) \propto 1$ on $\mathbb{R}$ we get $p_x(x) = p_y(x(y)) |\frac{dx}{dy}| \propto \frac{1}{y}$ on $(0, \infty)$.

Edit: Wikipedia has a bit on transformations of random variables: http://en.wikipedia.org/wiki/Probability_density_function#Dependent_variables_and_change_of_variables. Similar material will be in any intro probability book. Jim Pitman's "Probability" presents the material in a pretty distinctive way as well IIRC.

Answer 2

@JMS answer is adequate for the nuts and bolts of changing variables. However, This question may help you a bit with why it is uniform on that scale.

My answer to this question goes through a slightly longer derivation of the "jacobian rule" result given in @JMS's answer. It may help with understanding why the rule applies.