# What are the definitions of semi-conjugate and conditional conjugate priors?

What are the definitions of semi-conjugate priors and of conditional conjugate priors? I found them in Gelman’s Bayesian Data Analysis, but I couldn’t find their definitions.

Using the definition in Bayesian Data Analysis (3rd ed), if $\mathcal{F}$ is a class of sampling distributions $p(y|\theta)$, and $\mathcal{P}$ is a class of prior distributions for $\theta$, then the class $\mathcal{P}$ is conjugate for $\mathcal{F}$ if
If $\mathcal{F}$ is a class of sampling distributions $p(y|\theta,\phi)$, and $\mathcal{P}$ is a class of prior distributions for $\theta$ conditional on $\phi$, then the class $\mathcal{P}$ is conditional conjugate for $\mathcal{F}$ if