Parametric and nonparametric Bayesian statistical inference in animal science

[摘要] Chapter 1 illustrated an extension of the Gibbs sampler to solve problems arising in animalbreeding theory. Formulae were derived and presented to implement the Gibbs sampler where-aftermarginal densities, posterior means, modes and credibility intervals were obtained from the Gibbssampler.In the Bayesian Method of Moment chapter we have illustrated how this approach, based on a fewrelatively weak assumptions, is used to obtain maximum entropy densities, realized error terms andfuture values of the parameters for the mixed linear model. Given the data, it enables researchers tocompute post data densities for parameters and future observations if the form of the likelihoodfunction is unknown. On introducing and proving simple assumptions relating to the moments of therealized error terms and the future, as yet unobserved error terms, we derived post-data moments ofparameters 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 wasalso shown that in the computed example, where use was made of the Gibbs sampler to computefinite sample post-data parameter densities, some BMOM maxent densities were very similar to thetraditional Bayesian densities, whilst others were not.It should be appreciated that the BMOM approach yielded useful inverse inferences without usingassumed likelihood functions, prior densities for their parameters and Bayes' theorem, also it was thecase that the BMOM techniques extended in the present thesis to the mixed linear model providedvaluable and significant solutions in applying traditional likelihood or Bayesian analysis in animalbreeding problems.The important contribution of Charter 3 and 4 revolved around the nonparametrie modeling of therandom effects. We have applied a general technique for Bayesian nonparametries to this importantclass of models, the mixed linear model for animal breeding experiments. Our technique involvedspecifying a non parametric prior for the distribution of the random effects and a Dirichlet processprior on the space of prior distributions for that nonparametric prior. The mixed linear model wasthen fitted with a Gibbs sampler, which turned an analytical intractable multidimensional integrationproblem into a feasible numerical one, overcoming most of the computational difficulties usuallyexperience with the Dirichlet process.This proposed procedure also represented a new application of the mixture of Dirichlet processmodel to problems arising from animal breeding experiment. The application to and discussion ofthe breeding experiment from Kenya was helpful for understanding the importance and utility of theDirichlet process, and inference for all the mixed linear model parameters. However, as mentionedbefore, a substantial statistical issue that still remains to be tackled is the great discrepancy betweenresulting 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 ofstatistical procedures. In addition to the Dirichlet Process Prior, we will look in the future at othernon 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 wascertainly to be encouraged, we have found the group reference priors to generally be highsatisfactory, and felt reasonably confident in using them in situations in which further study wasimpossible. 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 referencepriors. This dependencél of the reference prior on the group chosen and their ordering wasunavoidable. Our motivation and idea for the reference prior was basically to choose the prior, whichin a certain asymptotic sense maximized the information in the posterior that was provided by thedata.The thesis has surveyed a range of current research in the area of Bayesian parametric andnonparametrie inference in animal science. The work is ongoing and several problems remainunresolved. In particular, more work is required in the following areas: a full Bayesiannonparametrie analysis involving covariate information; multivariate priors based on stochasticprocesses; multivariate error models involving Pólya trees; developing exchangeable processes tocover a larger class of problems and nonparametric sensitivity analysis.

[发布机构] University of the Free State