${\boldsymbol \phi}$, their form is the same as the form in , where inverse range priors are chosen for ${\boldsymbol \phi}$: $\pi({\boldsymbol \phi}) = \prod_{t=1}^s \prod_{\ell=1}^d \phi_{t,\ell}^{-1}$ and $a_t=1$. The parameter ${\boldsymbol \phi}$ is estimated by maximizing the distribution $p({\mathbf{y}}\mid {\boldsymbol \gamma}, {\boldsymbol \sigma}, {\boldsymbol \phi})$. In Gratiet [@Gratiet2013], no prior is assumed for ${\boldsymbol \phi}$, and ${\boldsymbol \phi}$ is estimated based on a concentrated restricted likelihood via a restricted maximum likelihood (REML) approach [@Harville1974]. In Section \[sec: integrated likelihood\], we show that constant flat prior or inverse range prior for ${\boldsymbol \phi}$ yields improper posteriors.

Commonly-used Proper Priors
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As the commonly-used improper priors may lead to improper posteriors, one obvious way to guarantee propriety of the posterior distribution is to assume proper priors, assessed either subjectively or from previous data, however, for Gaussian processes, the correlation parameter ${\boldsymbol \phi}$ can be difficult to interpret. Another way is to assume vague proper priors, however, this can only hide the problem when the posterior concentrates its mass at zero. If the posterior impropriety is occurring because the posterior is not decreasing at infinity, then the empirical Bayes estimates of ${\boldsymbol \phi}$ can be bad. In Appendix \[app: Qian\], we discuss 