# What's the role of the data in Gibbs Sampling?

Trying to wrap my mind around Gibbs Sampling. Across many answers in this same forum, I constantly notice that the examples shown do not actually require an observed data set (First example (with R code); The D&D example*), the same for other sources in the web that try to explain.

Whereas in every equation there is always the conditional on y component:

• As a secondary auxiliary question, I see the D&D example introduces an Accept-Reject algorithm, whereas in most other sources I just see that there is a sampling done from the conditional with no extra step (e.g. here "[Gibbs Sampling] accepts all proposals"). Is there something else I'm missing?
• The accept-reject algorithm in Gibbs sampling is normally the uniform distribution, so you see no extra step. You may need to know the MH Algorithm first. – Lerner Zhang Jun 19 at 23:03
• The images attached come from course notes, but an equivalent notation is given in Regression (Fahrmeir, Kneib, Lang & Marx, 2013, p. 675): "In most cases no (simple) methods for directly drawing random numbers from the density p(theta|y) of the entire parameter vector are available. Often, however, random numbers can be directly drawn from the conditional densities p(theta_1 | ·), ... ,p(theta_s | ·), where p(theta_s | ·) denotes the conditional density of theta_s given all other blocks theta1, ... theta_s-1, theta_s+1, ... theta_s and the data y." – Kuku Jun 19 at 23:13

• In any case where you are using the conditional distribution, given the data, that data will show up as part of the function (through the sufficient statistic). So yes, this might entail the conditional distributions being functions of $\bar{y}$ and $n$ in some cases. – Ben Jun 20 at 23:07