On the posterior distribution of the number of components in a finite mixture

The posterior distribution of the number of components k in a finite mixture satisfies a set of inequality constraints. The result holds irrespective of the parametric form of the mixture components and under assumptions on the prior distribution weaker than those routinely made in the literature on Bayesian analysis of finite mixtures. The inequality constraints can be used to perform an ``internal'' consistency check of MCMC estimates of the posterior distribution of k and to provide improved estimates which are required to satisfy the constraints. Bounds on the posterior probability of k components are derived using the constraints. Implications on prior distribution specification and on the adequacy of the posterior distribution of k as a tool for selecting an adequate number of components in the mixture are also explored.Comment: Published at http://dx.doi.org/10.1214/009053604000000788 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

The Employment of Airships for the Transport of Passengers

It was a conclusion of this detailed study of the practicality of using airships for carrying passengers that, although slow, airships are capable of carrying useful loads over long distances. However, it is noted that there is a certain limit to the advantages of large cubature. Beyond a certain point, the maximum altitude of the airship goes on decreasing, in spite of the fact that the range of action in the horizontal plane and the useful load go on increasing. The possibility of rapid climb is an essential factor of security in aerial navigation in the case of storms, as is velocity. To rise above and run ahead of storms are ways of avoiding them. However, high altitude and high speed are antithetical. This investigation concluded that a maximum velocity of 120 km/h is as far as we ought to go. This figure can only be exceeded by excessive reduction of the altitude of ceiling, range of flight, and useful load. The essential requisites of a public transport service are discussed, as are flight security, regularity of service, competition with other forms of passenger transportation, and the choice between rigid and semi-rigid airships

Sparse approximation of multilinear problems with applications to kernel-based methods in UQ

We provide a framework for the sparse approximation of multilinear problems and show that several problems in uncertainty quantification fit within this framework. In these problems, the value of a multilinear map has to be approximated using approximations of different accuracy and computational work of the arguments of this map. We propose and analyze a generalized version of Smolyak's algorithm, which provides sparse approximation formulas with convergence rates that mitigate the curse of dimension that appears in multilinear approximation problems with a large number of arguments. We apply the general framework to response surface approximation and optimization under uncertainty for parametric partial differential equations using kernel-based approximation. The theoretical results are supplemented by numerical experiments