Discussion of "What’s the H in H-likelihood: A Holy Grail or an Achilles’ Heel?" by Meng, X.-L.

Meng raises an interesting issue concerning the relationship among pivotal predictive distribution, posterior predictive distribution and h-distribution. He emphasizes how it is important using the "appropriate" scale both for observations and parameters and presents an example in which the three distributions coincide. We show why this result is not completely surprising and how part of the explanation can be found by extending a result due to Lindley who explains the relationship between a fiducial distribution and a posterior distribution.