Gibbs Sampling for Discrete Data

The objective of this project is to develop new and convenient Markov chain Monte Carlo (MCMC) schemes for practical Bayesian estimation of generalized linear models with latent state vectors. The main contribution is to show how Markov chains may be constructed for practical Bayesian estimation that posses a Gibbsian transition kernel. The resulting sampling schemes require only random draws from standard distributions such as multivariate normals, inverse Gamma, exponential and discrete distributions with a few categories. These new MCMC sampling schemes avoid the often inconvenient need of constructing proposal densities within Metropolis-Hastings kernels, as is necessary for earlier published MCMC methods for these type of models. It is shown within this research project, how Gibbsian transition kernels are achieved by introducing two sequences of latent variables through data augmentation. The introduction of the first sequence of latent variables eliminates the non-linearity of the observation equation, whereas the second sequence eliminates the non-normality. Thus we obtain a linear Gaussian model, if we condition on these two additional sequences within a Markov chain Monte Carlo scheme, and straightforward Gibbs sampling becomes feasible. Specific Gibbs sampling schemes are developed for the Bayesian estimation of many useful and complex models for binary data, multinomial data, and count data, such as random effects models, state space models, and models involving spatial structures. Cooperations with researcher in the area of marketing, labor market research, and road safety research are planned, in order to evaluate the practical impact of the prosed methods.

The objective of this project is to develop new and convenient Markov chain Monte Carlo (MCMC) schemes for practical Bayesian estimation of generalized linear models with latent state vectors. The main contribution is to show how Markov chains may be constructed for practical Bayesian estimation that posses a Gibbsian transition kernel. The resulting sampling schemes require only random draws from standard distributions such as multivariate normals, inverse Gamma, exponential and discrete distributions with a few categories. These new MCMC sampling schemes avoid the often inconvenient need of constructing proposal densities within Metropolis-Hastings kernels, as is necessary for earlier published MCMC methods for these type of models. It is shown within this research project, how Gibbsian transition kernels are achieved by introducing two sequences of latent variables through data augmentation. The introduction of the first sequence of latent variables eliminates the non-linearity of the observation equation, whereas the second sequence eliminates the non-normality. Thus we obtain a linear Gaussian model, if we condition on these two additional sequences within a Markov chain Monte Carlo scheme, and straightforward Gibbs sampling becomes feasible. Specific Gibbs sampling schemes are developed for the Bayesian estimation of many useful and complex models for binary data, multinomial data, and count data, such as random effects models, state space models, and models involving spatial structures. Cooperations with researcher in the area of marketing, labor market research, and road safety research are planned, in order to evaluate the practical impact of the prosed methods.