# Meaning and importance of 'Gibbs update' in MCMC

I am studying MCMC by "Handbook of Markov Chain Monte Carlo" by Brooks, Gelman This book is nice to explaining many fundamental concepts regarding MCMC. Especially in first chapter, they explain how Metropolis-Hastings algorithm coming from. However, I can't really get what "Gibbs update" (or Gibbs sampling) explain here and why it is important.

In this text "Gibbs Update" is the conditional distribution that always accepted. Also Conditional distribution of X_{n+1} given f(X_n) is same as conditional distribution X_n given f(X_n).

Suppose it is some special case in MCMC (Metropolis-Hasitings) algorithm, the parameter vector X_n is changed to X_{n+1} but both have same probability? It could be understood that this sampling (or Markov chain) should be move of "contour" of same probability? Or how it can be same conditional probability?

Also this contents talking about "block Gibbs" which is several components "omitted". If we think of vectors of parameters, it means just some parameters are not perturbed or not considered? General confusion also arise that Markov Chain Monte Carlo usally don't know probability distribution exactly. Then how "Gibbs update" is possible for exactly same conditional probability?

Thanks. Any general comments regarding the contents would be helpful.

• I searched from another source jwmi.github.io/BMS/chapter6-gibbs-sampling.pdf and got the concept of "Gibbs sampling" which is interesting for some special type of MCMC. Sorry for some basic question but any comments welcome regarding the meaning of gibbs sampling in MCMC. Commented Aug 25, 2023 at 4:40