Parametric and nonparametric Bayesian statistical inference in animal science
Pretorius, Albertus Lodewikus
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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.