Parametric and nonparametric Bayesian statistical inference in animal science
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Date
2000-11
Authors
Pretorius, Albertus Lodewikus
Journal Title
Journal ISSN
Volume Title
Publisher
University of the Free State
Abstract
Chapter 1 illustrated an extension of the Gibbs sampler to solve problems arising in animal
breeding theory. Formulae were derived and presented to implement the Gibbs sampler where-after
marginal densities, posterior means, modes and credibility intervals were obtained from the Gibbs
sampler.
In the Bayesian Method of Moment chapter we have illustrated how this approach, based on a few
relatively weak assumptions, is used to obtain maximum entropy densities, realized error terms and
future values of the parameters for the mixed linear model. Given the data, it enables researchers to
compute post data densities for parameters and future observations if the form of the likelihood
function is unknown. On introducing and proving simple assumptions relating to the moments of the
realized error terms and the future, as yet unobserved error terms, we derived post-data moments of
parameters and future values of the dependent variable. Using these moments as side conditions,
proper maxent densities for the model parameters were derived and could easily be computed. It was
also shown that in the computed example, where use was made of the Gibbs sampler to compute
finite sample post-data parameter densities, some BMOM maxent densities were very similar to the
traditional Bayesian densities, whilst others were not.
It should be appreciated that the BMOM approach yielded useful inverse inferences without using
assumed likelihood functions, prior densities for their parameters and Bayes' theorem, also it was the
case that the BMOM techniques extended in the present thesis to the mixed linear model provided
valuable and significant solutions in applying traditional likelihood or Bayesian analysis in animal
breeding problems.
The important contribution of Charter 3 and 4 revolved around the nonparametrie modeling of the
random effects. We have applied a general technique for Bayesian nonparametries to this important
class of models, the mixed linear model for animal breeding experiments. Our technique involved
specifying a non parametric prior for the distribution of the random effects and a Dirichlet process
prior on the space of prior distributions for that nonparametric prior. The mixed linear model was
then fitted with a Gibbs sampler, which turned an analytical intractable multidimensional integration
problem into a feasible numerical one, overcoming most of the computational difficulties usually
experience with the Dirichlet process.
This proposed procedure also represented a new application of the mixture of Dirichlet process
model to problems arising from animal breeding experiment. The application to and discussion of
the breeding experiment from Kenya was helpful for understanding the importance and utility of the
Dirichlet process, and inference for all the mixed linear model parameters. However, as mentioned
before, a substantial statistical issue that still remains to be tackled is the great discrepancy between
resulting posterior densities of the random effects as the value of the precision parameter, M changes.
We believe that Bayesian nonparametries have much to offer, and can be applied to a wide range of
statistical procedures. In addition to the Dirichlet Process Prior, we will look in the future at other
non parametric priors like the Pólya tree priors and Bernoulli trips.
Whilst our feeling in the final chapter was that study of performance of non-informative was
certainly to be encouraged, we have found the group reference priors to generally be high
satisfactory, and felt reasonably confident in using them in situations in which further study was
impossible. Results from the different theorems yielded that the group orderings of the mixed model parameters are very important since different orderings will frequently result in different reference
priors. This dependencél of the reference prior on the group chosen and their ordering was
unavoidable. Our motivation and idea for the reference prior was basically to choose the prior, which
in a certain asymptotic sense maximized the information in the posterior that was provided by the
data.
The thesis has surveyed a range of current research in the area of Bayesian parametric and
nonparametrie inference in animal science. The work is ongoing and several problems remain
unresolved. In particular, more work is required in the following areas: a full Bayesian
nonparametrie analysis involving covariate information; multivariate priors based on stochastic
processes; multivariate error models involving Pólya trees; developing exchangeable processes to
cover a larger class of problems and nonparametric sensitivity analysis.
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Keywords
Bayesian statistical decision theory, Animal breeding -- Statistical methods, Thesis (Ph.D. (Mathematical Statistics))--University of the Free State, 2000