Gfrktyl

As far as I understand it, it is (at least, that is how Wikipedia defines it). But I've found this statement by Efron* (emphasis added):

Markov chain Monte Carlo (MCMC) is the great success story of modern-day Bayesian statistics. MCMC, and its sister method “Gibbs sampling,” permit the numerical calculation of posterior distributions in situations far too complicated for analytic expression.

and now I'm confused. Is this just a minor difference in terminology, or is Gibbs sampling something other than MCMC?

[*]: Efron 2011, "The Bootstrap and Markov-Chain Monte Carlo"

The algorithm that is now called Gibbs sampling forms a Markov-chain and uses Monte-Carlo simulation for its inputs, so it does indeed fall within the proper scope of MCMC (Markov-Chain Monte-Carlo) methods. Historically, the method can be traced back at least to the mid-twentieth century, but it was not well-known and was only later popularised by the seminal paper of Geman and Geman (1984) which examined statistical physics in relation to the use of the Gibbs distribution (for some historical references, see Casella and George 1992, p. 167).

For some reason, thoughout his paper, Efron refers to the Gibbs sampler as if it were outside the scope of MCMC. He does this in the quote you have given, and also in some other parts of the paper. Since his opening reference to the technique refers to the "Gibbs sampler" (given in quotes) it is possible that he is alluding to the historical fact that the original method was developed through the Gibbs distribution in statistical physics, and was not incorporated into the general statistical theory of MCMC until much later. This is my best guess as to why he refers to it this way.

Update: Since Prof Efron is still alive I took the liberty of writing to him to ask why he describes the Gibbs sampler in this way. Here is his response (reproduced with his permission):

It was for mainly historical reasons... On the other hand, the Gibbs algorithm looks quite different from the MCMC recipe, and it takes some work to show that it is in some sense the same.
(Efron 2018, personal correspondence, ellipsis in original)

Go with Wikipedia. Better yet, go with these MCMC researchers:

• 
  Tierney (1994), "Markov Chains for Exploring Posterior Distributions";


• 
  Geyer (2011), "Introduction to Markov Chain Monte Carlo";


• 
  Robert and Casella (2011), "A Short History of Markov Chain Monte Carlo".

The Gibbs sampler is an example of a Markov chain Monte Carlo algorithm. Indeed, it is a special case of the Metropolis-Hastings algorithm. Any algorithm that generates random draws from a distribution $pi(theta)$ by simulating a Markov chain that has $pi(theta)$ as its stationary distribution is a Markov chain Monte Carlo algorithm, and that's exactly what the Gibbs sampler does.