Natural disasters may have considerable impact on society as well as on (re)insurance industry. Max-stable processes are ideally suited for the modeling of the spatial extent of such extreme events, but it is often assumed that there is no temporal d
ependence. Only a few papers have introduced spatio-temporal max-stable models, extending the Smith, Schlather and Brown-Resnick spatial processes. These models suffer from two major drawbacks: time plays a similar role as space and the temporal dynamics is not explicit. In order to overcome these defects, we introduce spatio-temporal max-stable models where we partly decouple the influence of time and space in their spectral representations. We introduce both continuous and discrete-tim
We attempt to trace the history and development of Markov chain Monte Carlo (MCMC) from its early inception in the late 1940s through its use today. We see how the earlier stages of Monte Carlo (MC, not MCMC) research have led to the algorithms curre
ntly in use. More importantly, we see how the development of this methodology has not only changed our solutions to problems, but has changed the way we think about problems.
Nested sampling is a simulation method for approximating marginal likelihoods proposed by Skilling (2006). We establish that nested sampling has an approximation error that vanishes at the standard Monte Carlo rate and that this error is asymptotical
ly Gaussian. We show that the asymptotic variance of the nested sampling approximation typically grows linearly with the dimension of the parameter. We discuss the applicability and efficiency of nested sampling in realistic problems, and we compare it with two current methods for computing marginal likelihood. We propose an extension that avoids resorting to Markov chain Monte Carlo to obtain the simulated points.
In Scott (2002) and Congdon (2006), a new method is advanced to compute posterior probabilities of models under consideration. It is based solely on MCMC outputs restricted to single models, i.e., it is bypassing reversible jump and other model explo
ration techniques. While it is indeed possible to approximate posterior probabilities based solely on MCMC outputs from single models, as demonstrated by Gelfand and Dey (1994) and Bartolucci et al. (2006), we show that the proposals of Scott (2002) and Congdon (2006) are biased and advance several arguments towards this thesis, the primary one being the confusion between model-based posteriors and joint pseudo-posteriors. From a practical point of view, the bias in Scotts (2002) approximation appears to be much more severe than the one in Congdons (2006), the later being often of the same magnitude as the posterior probability it approximates, although we also exhibit an example where the divergence from the true posterior probability is extreme.
