A modified and slightly
extended version of the material in Ch. 7 was published in *
Technometrics* in 1975. It invokes very general prior informative
assumptions which incorporate linear models for both the normal
means and the log-variances. After I much later taught this approach
on my Statistics 775 course at the University of Wisconsin-Madison,
a modified form of the methodology was very beneficially used in the
Animal Breeding to smooth the log-variances. See, for example the
article by Jean-Louis Foulley, Daniel Gianola, Magali San
Christobal and Sotan Im in *Computational Statistics and Data
Analysis* (1992).* *
Daniel Gianola’s student
Rob Tempelman did something similar in his 1993 University of
Wisconsin Ph.D. thesis *Poisson Mixed Models for the Analysis of
Counts with an Application to Dairy Cattle Breeding* with the
litter sizes of dairy cattle and the logs of the corresponding
Poisson means.
Gianola is a leading
Bayesian in this general area. Tempelman went on to become a
Professor of Animal Sciences at Michigan State University, where he
has published a number of important Bayesian papers. Jean-Louis
Foulley’s subsequent career at INRA in France has also been very
impressive. He has published numerous applications of Bayesian
inference in Animal Sciences and Genetics.
In the special case
where the normally distributed observations are unreplicated, some
special cases of the prior covariance matrices provide us with some
interesting time series models where the log-variances are taken to
be stochastically related. These formulations may be contrasted with
the various ‘stochastic volatility models’ which have been proposed
in the Economics literature.
My thesis created a
quite general paradigm for the construction of non-conjugate prior
distributions. Firstly, seek a transformation of the parameters such
that the new parameters can be taken to possess a multivariate
normal distribution. If you regard this as the first stage of a
hierarchical prior, then you may, at the second stage, assign
further distributions to the hyperparameters appearing in your first
stage mean vector and covariance matrix. This also provides general
formulation for non-linear random effects models. See also section
6.3 of [15]. |