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Jeffreys' prior on variance (var.), although uninformative, is not flat, but it is equivalent to assuming that the logarithm of the variance is uniformly distributed on the real line. So:

A) how I can see (plot) the bold-faced statement above (e.g., in R)?

B) Regarding Jeffreys' prior on variance itself, do we need to normalize p(Sigma^2) propor. to 1/sigma^2 e.g., by Var. values/sum(Var. values), if we want to call the Y axis of the plot of Jeffreys' prior on variance "Density"?

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  • $\begingroup$ Also see stats.stackexchange.com/questions/156199/… and stats.stackexchange.com/questions/56365/… $\endgroup$ – Carl Nov 15 '16 at 1:58
  • $\begingroup$ Edited to Jeffreys'. Personal opinion: the apostrophe is dispensible but if included belongs in only one possible place. Harold Jeffreys (personal communication 1976, believe it or not) preferred the form Jeffreys's because Jeffreys' is so likely to morph into Jeffrey's -- as happened once in the first version of this. More interestingly, perhaps, Harold Jeffreys and the Bayesian astronomer William H. Jefferys are completely different people. $\endgroup$ – Nick Cox Sep 29 '17 at 14:18
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If $X\sim \frac{1}{\theta}f(x/\theta)$, with $\theta$ a scale parameter, like the standard deviation, then $$Y=\log\{X\}\sim f\circ\exp\{y-\log\{\theta\}\}\,\exp\{y-\log\{\theta\}$$enjoys a location distribution with location $\xi=\log\{\theta\}$. Since Fisher's information on $\xi$ is constant, Jeffreys's prior on $\xi$ is uniform, $$\pi^J(\xi)=c$$ and $$\pi^J(\theta)=\frac{c}{\theta}$$by a change of variable.

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A) how I can see (plot) the bold-faced statement above (e.g., in R)?

Draw values of $\sigma$ on a logarithmic scale:

---- 0.01 ------------ 0.1 ----------- 1 ------------ 10 ---------- 100 ----

Draw the uniform prior: constant pdf. This plot respects information theoretic distances. Colourize in red the area under curve (weight) with $\sigma\in[1;10]$ (for example). You can colourize in blue the area $\sigma\in[0.1;1]$ to show it's the same weight.

Now draw $\sigma$ on a linear scale:

0 -------------- 1 -------------- 2 ------------- 3 ---------------

Draw the prior: pdf $1/{\sigma}$. This plot distorts information theoretic distances. Colourize in red the area under curve (weight) with $\sigma\in[1;10]$. You can colourize in blue the area $\sigma\in[0.1;1]$ to show it's the same weight.

This shows how the weight moves when changing the scale: the weight that was between 1 and 10 on the first chart spreads between 1 and 10 on the second chart. The weight is the same but gathered in a region with a different width. Same for the blue region.

The priors are improper so the notion of weight is... improper. But ok for intuition.

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