Abstract
The aim of this thesis is to study the convergence properties of specific MCMC algorithms for sampling from a posterior distribution. The model considered incorporates the uncertainty in the assignment of a legitimate identity of individuals. In a collected sample, observations wrongly recorded might result in duplicates or missing data which seriously affects the posterior inferences of parameters of interest. For instance, the actual sample size may be overestimated by duplicates or underestimated by the missing data. Thus, the underlying problem is a misidentification problem.
This thesis examines four MCMC algorithms. Two of which exist in the current literature (GENUAD and SMERED), however, their convergence properties had not previously been studied. This is the first contribution to the thesis. The GENUAD algorithm is a Gibbs sampler whereby the relevant full conditional densities are a critical aspect for determining the existence of a unique invariant distribution. SMERED is a Metropolis algorithm, in which convergence problems were detected. To correct these convergence issues, a novel algorithm was developed, named SMERED+. Finally, the DIU algorithm attempts to propose an altogether new technique.
The comparison of the algorithms is performed by simulating the posterior distribution of interest which contains a corruption model including the uncertainty in the data. Three different datasets are considered, a fictional toy example and two collected datasets. The advantage of the toy example is the size of the state space, which allows the behaviour of the chains generated by the relevant algorithms to be observed. The other two datasets require a different treatment.