otherwise. The expression for $\nu_{t,\ell}(\phi_{t, \ell})$ is given in Appendix \[app: nu expression\], which is a continuous function and $\lim_{\phi_{t, \ell} \to \infty} \nu_{t,\ell}(\phi_{t, \ell}) = 0$.

See Appendix \[app: nu expression\].

Lemma \[lem: behavior of integrated likelihood\] shows that both the flat prior $\pi({\boldsymbol \phi}_t) \propto 1$ and the noninformative prior $\pi({\boldsymbol \phi}_t)\propto \prod_{\ell=1}^d \phi_{t,\ell}^{-1}$ with $a_t=1$ in lead to improper posteriors with the condition  violated.

Objective Priors
----------------

The posterior can be improper under certain common choices of priors in . In what follows, several objective priors are derived and they are shown to yield proper posteriors.

Following [@Berger2001], the parameters of interest are chosen to be $({\boldsymbol \sigma}^2, {\boldsymbol \phi})$ and $({\boldsymbol \beta}, {\boldsymbol \gamma})$ are treated as the nuisance parameters. This specification leads to the prior factorization $\pi^R({\boldsymbol \theta})=\pi^R({\boldsymbol \beta}, {\boldsymbol \gamma}\mid {\boldsymbol \sigma}^2, {\boldsymbol \phi})\pi^R({\boldsymbol \sigma}^2, {\boldsymbol \phi})$. The Jeffreys-rule prior $\pi^R({\boldsymbol \beta}, {\boldsymbol \gamma}\mid {\boldsymbol \sigma}^2, {\boldsymbol \phi})\propto 1$ is considered for the location parameters $({\boldsymbol \beta}, {\boldsymbol \gamma})$ when other parameters are assumed known. Then the reference prior $\pi^R({\boldsymbol \sigma}^2, {\boldsymbol \phi})$ is computed based on the integrated likelihood with respect to $\pi^R({\boldsymbol \beta}, {\boldsymbol \gamma}) \propto 1$. Standard calculation yields the integrated 