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In the middle of page 64 of the third edition of Bayesian Data Analysis, Gelman writes...

We saw in Chapter 2 that a sensibly vague prior for $\mu$ and $\sigma^2$, assuming prior indipendance of scale and location, is uniform on $(\mu, \sigma^2)$, or eqivalently $$ p(\mu, \sigma^2) \propto (\sigma^2)^{-1} $$

I don't think I saw where this was referenced. Gelman writes that a prior or a normal with known mean but unknown variance is an Inverse Chisquared distribution. This distribution has two hyperparameters, $\alpha, \beta$, which if were both set to 0 would yeild $(\sigma^2)^{-1}$.

Is that the logic behind the prior for a normal with two unknown params?

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The flat prior for $\mu$ with support on $(-\infty,\infty)$ is $p(\mu) \propto 1$, i.e. an improper uniform prior over the real line.

The independence assumption gives you $p(\mu,\sigma^2) = p(\mu)p(\sigma^2) \propto (\sigma^2)^{-1}$.

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