Using Genetic Algorithms for Model Selection in Graphical Models

This paper considers an additive model y = f(x) + e when some observations on x are missing at random but corresponding observations on y are available. Especially for this model missing at random is an interesting case because of the fact that the complete case analysis is not expected to be suitable. A simulation study is reported and methods are compared based on superiority measures as the sample mean squared error, sample variance and estimated sample bias. In detail, complete case analysis, zero order regression plus random noise, single imputation and nearest neighbor imputation are discussed.

The incidence of underweight amongst children under five in Western Africa has been increasing over the last decade (UNICEF, 2002). In Asia, where about two thirds of the world's underweight children live, the rate of underweight declined from about 36 per cent to some 29 per cent between 1990 and 2000. In sub-Saharan Africa, the absolute number of underweight children has increased and is now about 36 per cent. Using new data from Demographic and Health Surveys, I estimate the probability of underweight or a sample of West African children, controlling for selective survival.

This paper gives a detailed overview of the problem of missing data in parametric and nonparametric regression. Theoretical basics, properties as well as simulation results may help the reader to get familiar with the common problem of incomplete data sets. Of course, not all occurences can be discussed so this paper could be seen as an introduction to missing data within regression analysis and as an extension to the early paper of Little (1992).

Based on a reversible jump Markov Chain Monte Carlo (RJMCMC) algorithm which was developed by Fronk and Giudici (2000) to deal with model selection for Gaussian dags, we propose a new approach for the pure discrete case. Here, the main idea is to introduce latent variables which then allow to fall back on the already treated continuous case. This makes it also straightforward to tackle the mixed case, i.e. to deal simultaneously with continuous and discrete variables. The performance of the approach is investigated by means of a simulation study for different standard situations. In addition, a real data application is provided.

We investigate the geographical and socioeconomic determinants of childhood undernutrition in Malawi, Tanzania and Zambia, three neighboring countries in Southern Africa using the 1992 Demographic and Health Surveys. We estimate models of undernutrition jointly for the three countries to explore regional patterns of undernutrition that transcend boundaries, while allowing for country-specific interactions. We use semiparametric models to flexibly model the effects of selected so-cioeconomic covariates and spatial effects. Our spatial analysis is based on a flexible geo-additive model using the district as the geographic unit of anal-ysis, which allows to separate smooth structured spatial effects from random effect. Inference is fully Bayesian and uses recent Markov chain Monte Carlo techniques. While the socioeconomic determinants generally confirm what is known in the literature, we find distinct residual spatial patterns that are not explained by the socioeconomic determinants. In particular, there appears to be a belt run-ning from Southern Tanzania to Northeastern Zambia which exhibits much worse undernutrition, even after controlling for socioeconomic effects. These effects do transcend borders between the countries, but to a varying degree. These findings have important implications for targeting policy as well as the search for left-out variables that might account for these residual spatial patterns.

Model selection in graphical models is still not fully investigated. The main difficulty lies in the search space of all possible models which grows more than exponentially with the number of variables involved. Here, genetic algorithms seem to be a reasonable strategy to find good fitting models for a given data set. In this paper, we adapt them to the problem of model search in graphical models and discuss their performance by conducting simulation studies.

Dempster-Shafer theory allows to construct belief functions from (precise) basic probability assignments. The present paper extends this idea substantially. By considering SETS of basic probability assignments, an appealing constructive approach to general interval probability (general imprecise probabilities) is achieved, which allows for a very flexible modelling of uncertain knowledge.

After a short introduction of the model, the missing mechanism and the method of inference some imputation procedures are introduced with special focus on the simulation experiment. Within this experiment, the simple additive model y = f(x) + e is assumed to have missing values in the independent variable according to MCAR. Besides the well-known complete case analysis, mean imputation plus random noise, a single imputation and two ways of nearest neighbor imputation are used. These methods are compared within a simulation experiment based on the average mean square error, variances and biases of \hat{f}(x) at the knots.

Nonignorable nonresponse is a common problem in bivariate or multivariate data. Here a selection model for bivariate normal distributed data (Y1 ; Y2) is proposed. The missingness of Y2 is supposed to depend on its own values. The model for missingness describes the probability of nonresponse in dependency of Y2 itself and it is chosen nonparametrically to allow exible patterns. We try to get a reasonable estimate for the expectation and especially for the variance of Y2 . Estimation is done by data augmentation and computation by common sampling methods.

Do reported reservation wages correspond to the concept of reservation wages that economists have? Using panel data on British unemployed I calculate reservation wages from a search model and compare these with reported reservation wages. It is shown that men's reported reservation wages are greater than what the model predicts, and that for women there is hardly a relation between the two variables.