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I know that the joint prior distribution is $p\left( {{a^2},{\beta ^2},\gamma ,\delta ,{\varepsilon ^2}} \right) \propto {\alpha ^{ - 2}}{\varepsilon ^{ - 1}}{\beta ^{ - 2}}$. However, I am confused, as I cannot understand what this distribution is (i.e. it doesn't look like a Gamma, Normal, etc). Is it a uniform? I can't understand what its pdf is.

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  • $\begingroup$ You don't give the interval the prior is defined on. Is it $R^+\times R^+\times R\times R\times R^+$?. If so you have a product of priors (suggesting independence), but each is improper. You can write each as a limit in several ways (e.g. $\varepsilon^{-1}$ can be thought of as the limit of an inverse gamma or a Pareto or a lognormal, among others). $\endgroup$
    – Glen_b
    Jul 16 '15 at 5:18
  • $\begingroup$ Yes this is the interval that it is defined on. Could you please provide me with literature regarding the derivation of these limits? $\endgroup$
    – Noob
    Jul 16 '15 at 7:15
  • $\begingroup$ P.S. am I right to say that ${\alpha ^{ - 2}}$ is inverse-chi-squared with 0 degrees of freedom, while ${\varepsilon ^{ - 1}}$ is inverse-chi-squared with -1 degrees of freedom? $\endgroup$
    – Noob
    Jul 16 '15 at 7:21
  • $\begingroup$ No.... 1. Note that the parameter is actually $\alpha^2$, not $\alpha$, so all three inverse terms are the parameter to the power $-1$.$\:$ 2. Note that the inverse chi-square already has a -1 term in the power, so the df would (if anything) be 0.$\:$ 3. But in any case it can't be any inverse chi square because that also has an exponential term that doesn't go away. 4. However, if you make it inverse Gamma instead, there are two parameters, and by using both (in the limit) you can get the right form (shape parameter $\to 0$ and scale parameter $\to 0$) $\endgroup$
    – Glen_b
    Jul 16 '15 at 10:19
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It's good that you don't recognize this distribution because it isn't one (it has a divergent integral). In Bayesian statistics you interpret this type of thing as more of a "weight function" than an actual density.

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