Practical importance sampling methods for finite mixture models and multiple imputation

The first past of my thesis studies calculating Bayes factors for mixture models. The first contribution of my research is to exploit the possibility of augmenting the data and sampling over the space of the component labels, z. For the second contribution, I propose using an adaptive defensive mixture importance sampling function (Hesterberg 1995) in order to reduce the variance observed when trying to estimate I via I = sump(y|z) p(z). I suggest a defensive mixture importance sampling approach, which entails using a mixture importance sampling function gz= k=1K-1dkgk z+dKpz where the gk(z) are themselves a mixture of multinomial and Dirichlet-Multinomial sampling functions on the component labels.;The second part of my thesis work addresses questions surrounding the specification of prior distributions for the parameters in mixture models in the absence of specific prior knowledge. I suggest specifying a sensible, approximate unit information prior (Kass and Wasserman 1995) that places mass around maximum likelihood estimates of the parameters. I examine the ramifications of using data-based priors for mixtures and suggest a further modification in order to correct some odd behavior that results when overfitting the number of components.;The last part of my thesis re-analyzes some of the behavior of certain standard imputation generation and parameter estimation techniques for small numbers of copies. I compare results for Rubin's methods to results for the methods of Wei and Tanner (1990) for small numbers of copies. I note that the variance of Rubin's imputation estimators overwhelms the bias for the Wei and Tanner estimators for small numbers of copies and simple models. The initial part of my work thus explores the bias-variance trade-off with respect to these two imputation methods in more complex models.