I came across the following passage in Gelman's Bayesian Data Analysis (chapter 2): A convenient parameterization [of the inverse-gamma conjugate prior for a normal distribution with known mean and unknown variance] is the scaled inverse-chi-square distribution with scale $\sigma^{2}_0$ and $\nu_0$ degrees of freedom; that is, the prior distribution of $\sigma^{2}$ is taken to be the distribution of $\sigma^{2}_0/X$ , where $X$ is a $\chi^{2}_{\nu_0}$ random variable. We use the convenient but nonstandard notation, $\sigma^2 \sim Inv-\chi^2(\nu_0,\sigma^{2}_0)$ . Gelman goes on show the resulting posterior density for $\sigma^2_0$ based on the above relationship between the inverse-gamma and scaled inverse-chi-square. Besides their shared distribution form, what's the intuitive reason or relationship for parameterizing the inverse-gamma with the scaled inverse-chi-square here? Is it that the scaled inverse-chi-square's parameters are more closely linked to the data sample $y_1,...,y_n$ and the posterior parameter of interest, $\sigma^2$ ?

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