How to know whether a Gibbs sampler is irreducible?

I know that the Gibbs sampler in e.g. two variable case constructs a sequence of r.v.s $(X_1^{(i)}, X_2^{(i)})$ by sampling from the related conditional distributions, rather than the joint distribution of these r.v.s

However, how would I infer, whether the Markov chain in this is irreducible or not? I've only seen examples reading that from the transition matrix, but I don't know how to even construct such matrix in this case.

  • 1
    $\begingroup$ If the support of the random variables do not depend on the other variables, you get irreducible for free because you can transition anywhere in one Gibbs sweep. $\endgroup$ – guy Feb 23 '18 at 0:27
  • $\begingroup$ This question has been addressed several times on X validated. $\endgroup$ – Xi'an Feb 23 '18 at 7:36

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