This thesis comprises the work conducted during three years as a PhD student at the Department of Mathematics at the University of Oslo (UiO). I will be looking back at these years as an enjoyable phase of my life, in which I learned a lot. The period has also been tough, though – possibly even tougher than I expected it to be. With the finished thesis in my hand, there is however no doubt it was worth it. Still, I have to agree with the author Joseph Epstein who noted that it is a lot better to have written, than to actually be writing. The final product constituting my PhD thesis is inarguably positively correlated with the original project description, although the rho is far from one. Through these years I have been working with methodology within a broad range of fields across the science of statistics. Among them are approximate Bayesian inference, asymptotic theory, Bayesian statistics, copulae, density estimation, frequentist statistics, functional differentiation, Gaussian distribution theory, geostatistics, inverse problems, Markov chain Monte Carlo (MCMC), model averaging, model selection, spatial statistics, stochastic process theory, survival analysis, time series modelling – and I even had to learn a little bit of geophysics, petrophysics and rock physics. I do by no means claim to master all these subjects, but I have learned a fair amount about all of them, and for that I feel incredibly lucky. Upon completing this thesis, I am deeply indebted to my two supervisors Nils Lid Hjort and Odd Kolbjørnsen. I am truly grateful for how you inspired me, and the eagerness you showed while working with the various projects. I will sincerely like to thank you both for that. You also supported me and made it possible for me to spend the autumn of 2014 at Stanford University, visiting Paul Switzer at the Department of Statistics. Paul was an outstanding host during some incredible months over there – our delightful academic and non-academic discussions will not be easily forgotten. Being founded by Statistics for Innovation (SFI2), a centre for research-based innovation, I was lucky enough to be awarded with two offices in Oslo; one at the campus at Blindern and one at the Norwegian Computing Center (NR) at Forskningsparken. Without even being employed at NR, I was very well taken care of and included in the SAND group. I am thankful for that additional dimension and opportunity to learn, and for being exposed to the weekly dose of Thursday-buns – that will be missed! I would also like to thank all my colleagues, both at the statistics group at UiO and the SAND group at NR. Special thanks go to my ‘roommates’ Marie at NR and Reinaldo at UiO for all our inviting discussions and fascinating conversations, and to Gudmund Horn Hermansen for co-authoring one of the papers in the thesis. Finally, I would like to thank my friends, family, and ‘family-in-law’ for filling my life with joy – especially my wonderful Elin for putting up with me, supporting and understanding me, even though I know you really wished I was rather spending those late evenings and weekends with you.