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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?

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up vote 8 down vote accepted

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.

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just adding to @JMS answer, a great reference to check out about is Bayesian Inference in Statistical Analysis by Box and Tiao. It presents conceptual ideas behind it too. –  suncoolsu Feb 21 '11 at 6:20
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@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.

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+1 for the additional references. My favorite derivation for the change of variables formula starts with the cdf, like in your other answer. –  JMS Mar 27 '11 at 15:31
    
@JMS - the cdf rule is the only one I don't get confused with, I usually struggle to remember if its $\frac{dy}{dx}$ or $\frac{dx}{dy}$ with the jacobian –  probabilityislogic Mar 27 '11 at 22:40
    
same for me - Pitman gives a nice geometric explanation, which is why I referenced it in my answer, but I can't ever remember it when it counts :) When I TA'd a probability class we used this text and some students found it very helpful. –  JMS Mar 28 '11 at 1:56
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