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.

Answer

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

p(\theta|y)\in \mathcal{P} \mbox{ for all }p(\cdot|\theta)\in \mathcal{F} \mbox{ and }p(\cdot)\in \mathcal{P}.

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

p(\theta|y,\phi)\in \mathcal{P} \mbox{ for all }p(\cdot|\theta,\phi)\in \mathcal{F} \mbox{ and }p(\cdot|\phi)\in \mathcal{P}.

Conditionally conjugate priors are convenient in constructing a Gibbs sampler since the full conditional will be a known family.

I searched an electronic version of Bayesian Data Analysis (3rd ed.) and could not find a reference to semi-conjugate prior. I’m guessing it is synonymous with conditionally conjugate, but if you provide a reference to its use in the book, I should be able to provide a definition.

Attribution
Source : Link , Question Author : Tim , Answer Author : jaradniemi

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