Multivariate slice sampling

In Bayesian decision theory, stochastic simulation techniques have been widely used by decision makers. The Markov chain Monte Carlo method (MCMC) plays a key role in stochastic simulation within the framework of Bayesian statistics. The notion of "slice sampling", or employing auxiliary variables in sampling, has been recently suggested in the literature for improving the efficiency of the traditional MCMC methods. In the existing literature, the one dimensional slice sampler has been extensively studied, yet the literature on the multidimensional case is sparse. In this study, we utilize multiple auxiliary variables in our sampling algorithms for multivariate normal distributions, which adapt better to the local properties of the target probability distributions. We show that these methods are flexible enough to allow for truncation to rectangular regions and/or the exclusion of any n dimensional hyper-quadrant. We compare these algorithms for both efficiency and accuracy via simulation experiments. Our results show that our new sampling techniques are accurate and more effective than the traditional single auxiliary variable slice sampler, especially for truncated multivariate distributions. Further, we extend our methods to general multivariate distributions including multivariate student t distributions, multivariate elliptical distributions, multivariate skew normal distributions, multivariate skew t distributions and a general class of multivariate skew elliptical distributions. We also discuss and outline some applications of our algorithms in the real business world, especially in the production and operations management and the finance areas. With regard to production and operations management, we find that the proposed multivariate normal slice sampler can be implemented in a stochastic optimization process for finding the optimal fulfillment rate of an assembly system. This system consists of n components, from which m products are assembled via a periodic review control policy. In finance area, we show that multivariate slice samplers can be used to update model parameters and predict future asset returns in a higher moments Bayesian portfolio optimization model, which is based on a general class of multivariate skew t distributions. The application of a multivariate slice sampler improves the model's ability of handling large data sets and saves computation time for deriving complicated posterior probability distributions.