What's the H in H‐likelihood: A Holy Grail or an Achilles' Heel?*
H‐likelihood refers to a likelihood function of both fixed parameters and random “unobservables,” such as missing data and latent variables. The method then typically proceeds by maximizing over the unobservables via an adjusted profile H‐likelihood, and carries out a Fisher‐information‐like calculation for (predictive) variance estimation. The claimed advantage is its avoidance of all “bad” elements of Bayesian prediction, namely the need for prior specification and posterior integration. This talk attempts to provide an in‐depth look into one of the most intriguing mysteries of modern statistics: why have the proponents of the H‐likelihood method (Lee and Nelder, 1996, 2001, 2005, 2009) been so convinced of its merits when almost everyone else considers it invalid as a general method? The findings are somewhat intriguing themselves. On the one hand, H‐likelihood turns out to be Bartlizable under easily verifiable conditions on the marginal distribution of the unobservables, and such conditions point to a transformation of unobservables that makes it possible to interpret one predictive distribution of the unobservables from three perspectives: Bayesian, fiducial and frequentist. On the other hand, the hope for such a Holy Grail in general is diminished by the fact that the log H‐ likelihood surface cannot generally be summarized quadratically due to the lack of accumulation of information for unobservables, which seems to be the Achilles' Heel of the H‐likelihood method.
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