When Bayesian analyses with noninformative priors are carried out, it
is usual to achieve results similar to those of classical statistics. It
is not clear to me if that could be the case with mixed linear models
(straightforward in animal breeding) when Bayesian point estimators of
variance components are posterior modes. Classical inference obtains
BLUPs of the random variables traditionally from Henderson�s Mixed Model
Equations (MME). Such equations are functions of the variance
components, which are generally unknown. There are iterative methods,
however, which yields estimates of the components and BLUPs as well
(some call this procedure EBLUP), like the EM algorithm.
In Bayesian analysis, if point estimators of the variance components
are the modes of the marginal posterior distributions, shouldn�t they be
equal to the REML estimates (since posterior distributions with
noninformative priors are essentially likelihoods) and consequently to
the BLUPs?
Wright, Stern and Berger, in an excellent recent paper (JABES,
5(2):240-256, 2000) show significant differences when posterior means
are taken as the Bayesian point estimators. But they say (pg.250) that
�Figure 3 shows (...) where it is evident that the posterior modes are
in fact similar to the REML estimates of the variance components�. They
are supposed to be similar or equal? (that is, are the differences from
the values yielded by Gibbs sampler merely random?) If they are not
equal to REML estimates, are they equal to ML estimates?
I would greatly appreciate comments on the task.
Eduardo Bearzoti
Department of Exact Sciences
Federal University of Lavras
Lavras Minas Gerais State
Brazil
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