Abstract:
Different strategies have been proposed to improve mixing and convergence properties of Markov, chain Monte Carlo algorithms. These are mainly concerned with customizing the proposal density in the Metropolis-Hastings algorithm to the specific target density and require a detailed exploratory analysis of the stationary distribution and/or some preliminary experiments to determine an efficient proposal. Various Metropolis-Hastings algorithms have been suggested that make use of previously sampled states in defining an adaptive proposal density, We propose a general class of adaptive Metropolis-Hastings algorithms based on Metropolis-Hastings-within-Gibbs sampling. For the case of a one-dimensional target distribution, we present two novel algorithms using mixtures of triangular and trapezoidal densities. These can also be seen as alternatives of the all-purpose adaptive rejection Metropolis sampling (ARMS) algorithm to sample from non-logconcave univariate densities. Using various different examples, we demonstrate their properties and efficiencies and point out their advantages over ARMS and other adaptive alternatives. We also present an alternative algorithm to ARMS that uses truncated Normal distributions instead of piecewise exponential distributions for sampling from log-concave and non-log-concave distributions. Nonparametric and semiparametric Bayesian methods are now quite popular and well accepted in practice since they offer a more general modeling strategy containing fewer assumptions. Bayesian survival analysis based on semi-parametric mixture models have certain advantages over classical approaches. We describe a generalised mixture of unnormalised B-splines including beta mixtures proposed by Gelfand and Mallick (1995) and triangular mixtures proposed by Perron and Mengersen (2001). In practical clinical trials highly stratified data are frequently encountered. We investigate methods based on the methods by Carlin and Hodges (1999) to construct hierarchical proportional hazards regression models using mixtures of B-splines. We illustrate the methods using two highly stratified real data one of which contains two different types of missing values. The parametric Weibull model, the semiparametric model based on the beta mixture model, the semiparametric model based on the triangular mixture model and the mixture model based on the mixture of B-splines with higher degrees are investigated in terms of the accuracy of estimates, the ease of use, the detailed information provided and robustness.