Gaussian process methods for one-dimensional diffusions: optimal rates and adaptation

We study the performance of nonparametric Bayes procedures for one-dimensional diffusions with periodic drift. We improve existing convergence rate results for Gaussian process (GP) priors with fixed hyper parameters. Moreover, we exhibit several possibilities to achieve adaptation to smoothness. We achieve this by considering hierarchical procedures that involve either a prior on a multiplicative scaling parameter, or a prior on the regularity parameter of the GP

Full adaptation to smoothness using randomly truncated series priors with Gaussian coefficients and inverse gamma scaling

We study random series priors for estimating a functional parameter (f\in L^2[0,1]). We show that with a series prior with random truncation, Gaussian coefficients, and inverse gamma multiplicative scaling, it is possible to achieve posterior contraction at optimal rates and adaptation to arbitrary degrees of smoothness. We present general results that can be combined with existing rate of contraction results for various nonparametric estimation problems. We give concrete examples for signal estimation in white noise and drift estimation for a one-dimensional SDE

Recovery, detection and confidence sets of communities in a sparse stochastic block model

Posterior distributions for community assignment in the planted bi-section model are shown to achieve frequentist exact recovery and detection under sharp lower bounds on sparsity. Assuming posterior recovery (or detection), one may interpret credible sets (or enlarged credible sets) as consistent confidence sets. If credible levels grow to one quickly enough, credible sets can be interpreted as frequentist confidence sets without conditions on the parameters. In the regime where within-class and between-class edge-probabilities are very close, credible sets may be enlarged to achieve frequentist asymptotic coverage. The diameters of credible sets are controlled and match rates of posterior convergence.Comment: 22 pp., 2 fi

Breeding for trypanotolerance in African cattle

Trypanosomosis, or sleeping sickness, is one of the most important livestock diseases in Africa. Some West African cattle breeds show a degree of resistance to a trypanosome infection: they are trypanotolerant. At the International Livestock Research Institute (ILRI) in Nairobi, Kenya, an F2 experiment has been established to unravel the genetic background of trypanotolerance. This thesis had two main aims: First to determine the genetic background of trypanotolerance, and second to investigate opportunities to incorporate this information in a breeding scheme to increase performance of cattle in tsetse-infested areas. Based on the results from the F2 experiment, several traits were defined, which reflected features of trypanotolerant cattle. Subsequently, based on preliminary results from an analysis to determine chromosome fractions containing genes (QTL) involved in trypanotolerance performed at ILRI, the mode of expression of these QTL was investigated and one of the QTL was found to be maternally imprinted. These QTL could be utilised in an introgression scheme, but also for within breed selection. Both options were investigated. When introgressing QTL for disease resistance the optimal number of backcross generations from genetic or economic point of view was found to be different. The number of animals required is increasing very rapidly with increasing number of QTL to be introgressed. Within breed selection to increase production under constant infection pressure can be applied with or without aid of QTL for disease resistance. Mass selection on production under infection can be applied if no QTL information is available. A non-linear selection response is achieved in both potential production and disease resistance. Important advantage of QTL information for disease resistance is that animals can be selected outside the infected environment. In implementing a breeding scheme it is important to take into account that social-economic values and environments are very different in large parts of Africa as compared to Western countries. This thesis has demonstrated that there are good opportunities for using selection to improve the results of local farming systems.</p