# 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.

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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.

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+1. What's the URL for the 3rd edition of Bayesian Data Analysis? –  Patrick Coulombe Mar 22 at 2:11
Thanks! Semi-conjugate appears here (2nd ed) books.google.com/…. By the way, how did you get the ebook for the 3rd ed? –  Tim Mar 22 at 3:15
I'm not sure why it says semiconjugate prior there since the prior is fully conjugate. This statement is removed in the 3rd edition. The ebook can be purchased here: crcpress.com/product/isbn/9781439840955. –  jaradniemi Mar 22 at 22:37
@jaradniemi: In the link I gave, on top of p84, it is pointed out that the semiconjugate prior is not a conjugate prior. –  Tim Mar 22 at 22:52
oops...I looked at the left side of eq (3.10) which showed conditioning on $\sigma^2$ but didn't look at the right side where the distribution for $\mu$ is independent of $\sigma^2$. As used in this context, the definition for semi-conjugate is synonymous with conditionally conjugate (I'll update my answer.) –  jaradniemi Mar 25 at 19:28