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A prior that express lack of detailed information or lack of any information at all.

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Why is uniform prior on log(x) equal to 1/x prior on x?

When transforming a uniform distribution on $\log(\sigma)$ to a distribution on $\sigma$ you need to take into account the Jacobian of the transformation. This corresponds, as you correctly intuited, …
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4 votes

Haldane's prior Beta(0,0) - Part 1

The second expression is correct as this is an improper distribution, i.e. it doesn't integrate to $1$. Thus it doesn't have a density and you can only specify it up to proportionality.
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