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If you have a binomial likelihood, $y|n,p\sim\textsf{Bin}(n,p)$, the Jeffreys' prior for the proportion $p$ is $\textsf{Beta}(1/2,1/2)$. If we instead reparameterize the proportion as the log odds $\phi=\log(p/1-p)$, through a change of variables we arrive at the pdf for $\phi$: $p(\phi)=\frac{1}{\pi}\sqrt{\frac{e^{\phi}}{(1+e^{\phi})^2}}$ (assuming I didn't do the math wrong). My question is does this distribution have a name? It looks related to the logistic distribution but I'm not sure how.

It's weird because most introductory sources on Jeffreys' priors use a binomial likelihood with uniform priors on $p$ and $\phi$ to motivate the desire for invariance under 1-1 transformations, find the Jefrreys' prior for $p$, but then don't actually look at what this corresponds to in the log odds parametrization.

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If $p \sim \text{Beta}(a,b)$, then $T = p/(1-p)$ has a Pearson Type VI distribution, also known as a beta-prime, inverted beta, or beta distribution of the second kind, so I suppose you could call the distribution of $\log T$ a log beta-prime distribution.

N. L. Johnson in "Systems of frequency curves generated by methods of translation", Biometrika, 36, in 1949 (online access required, but registration is free), looked at the distribution of $\log p/(1-p)$ briefly (p. 166) in the context of transforming Pearson Type I distributions, which are a generalization of the Beta distribution, and derived several properties which, I am sorry to say, are mainly of historical interest. In the context of his work, it would be referred to as an application of the $S_B$ transform to a Pearson Type I variate with (original parameterization) $\xi = 0, \lambda = 1$, so "log beta-prime distribution" seems better.

I don't know whether $a = b = 1/2$ results in a special distribution with its own name, though!

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