Abstract:
The posterior in general is sensitive to the inputs of a Bayesian analysis, consequently study of posterior sensitivity to the prior and likelihood are the focus of much research. Different approaches to sensitivity analysis, including local and global methods have been developed by such authors as Berger (2000) and Gustafson (2000). The sensitivity of the posterior to the prior due to its often subjective origin has attracted most attention. In this thesis, a particular form of parametric local sensitivity analysis is discussed and developed, and the results found in Geweke (1999), Millar (2004), Millar and Stewart (2005) and Millar and Stewart (2007) are extended, and new results provided. Many new results are immediate due to the functional form of Bayes formula so that results for local sensitivity to the prior transfer easily into formulae for local sensitivity to the likelihood. Of particular note is the sensitivity of a Bayes estimator to a geometrically defined case weight, шi. Information theoretic results are derived for the curvature of the Kullback-Leibler ‘distance’ between two posteriors relative to the geometric case weight шi. The local sensitivity of ø-divergence is discussed and a new general expression for the curvature with respect to a prior hyper-parameter is derived including the calibration scheme of McCulloch (1989) for the Kullback-Leibler distance (ø= ղlogղ). Sensitivity expressions for the Bayes factor are reviewed and a new local sensitivity expression is derived. This expression is of a very simple form and easily determined. The result is made available for both sensitivity to the prior and likelihood. Sensitivity expressions of Bayes estimators and Bayes factors to e-contamination and directed geometric priors are discussed and developed. A special application of the theory to sensitivity of Bayes estimators to intervals used in the construction of priors (as would be made from a meta-analysis) is derived. The sensitivity of a posterior highest density region to a prior hyper-parameter is approximated. Expressions for the sensitivity of a Bayes estimator to hyper-parameters of a prior from an exponential distribution are presented along with some specific local sensitivities with respect to conjugate priors. The theory developed is implemented by way of four examples, each revealing the ease with which the local sensitivity can be found.