In this thesis we study the use of Piecewise Deterministic Markov Processes (PDMPs), such as the Zig-Zag process and the Bouncy Particle Sampler, in rare event probability estimation. We introduce both processes and illustrate how the methods work. To estimate the rare event probabilities we use the splitting method. We also analyze this method and show how the method can be applied by working out a simple example. To make the connection between PDMPs and the splitting method we introduce a setting which suites both well. We consider an example of rare event probabilities outside a d-dimensional sphere for a Gaussian random variable. We compare the results in time complexity, probability estimation, Effective Sample Size and distribution of thresholds between the use of a standard Metropolis-Hastings algorithm and the Bouncy Particle sampler. Furthermore results are obtained to express the eigenvalues of the stochastic process as a function of the boundary on a restricted domain, in particular for the case of the one-dimensional Zig-Zag process with a zero refreshment intensity.

This report examines the possibility of modeling the performance of professional cyclists of Team Sunweb using Bayesian Networks. This research has an objective to see how these structures work and how they fit in the complex world of cycling. We want to compare different cyclists of Team Sunweb in the Grand Tours (Giro d'Italia, Tour de France and Vuelta a Espana) of the year 2016 and build a model for the leader - who is supported by a group of helpers - during different stages in a given race and see if we can predict the pedal power in the crucial part of the race, i.e. the sprint or a last difficult climb. We can conclude that the Bayesian network we created with the combination of help from an expert and the data captures the most common relationships between all variables, but that the model doesn’t reveal surprising relationships or good predictions.