arXiv subjects
Publications and source records attributed to Tobias Siems.
In my PhD thesis, we elaborate upon Bayesian changepoint analysis, whereby our focus is on three big topics: approximate sampling via MCMC, exact inference and uncertainty quantification. Besides, modeling matters are discussed in an ongoing fashion. Our findings are underpinned through several changepoint examples with a focus on a well-log drilling data.
This work is driven by the ubiquitous dissent over the abilities and contributions of the Metropolis-Hastings and reversible jump algorithm within the context of trans dimensional sampling. We demystify this topic by taking a deeper look into the implementation of Metropolis-Hastings acceptance probabilities with regard to general mixture spaces. Whilst unspectacular from a theoretical point of view, mixture spaces gave rise to challenging demands concerning their effective exploration. An often applied but not extensively studied tool for transitioning between distinct spaces are so-called translation functions. We give an enlightening treatment of this topic that yields a generalization of the reversible jump algorithm and unveils another promising translation technique. Furthermore, by reconsidering the well-known Metropolis within Gibbs paradigm, we come across a dual strategy to develop Metropolis-Hastings samplers. We underpin our findings and compare the performance of our approaches by means of a change point example. Thereafter, in a more theoretical context, we revitalize the somewhat forgotten concept of maximal acceptance probabilities. This allows for an interesting classification of Metropolis-Hastings algorithms and gives further advice on their usage. A review of some errors in reasoning that have led to the aforementioned dissent concludes this paper.
We elaborate the idea behind Markov chain Monte Carlo (MCMC) methods in a mathematically coherent, yet simple and understandable way. To this end, we proof a pivotal convergence theorem for finite Markov chains and a minimal version of the Perron-Frobenius theorem. Subsequently, we briefly discuss two fundamental MCMC methods, the Gibbs and Metropolis-Hastings sampler. Only very basic knowledge about matrices, convergence of real sequences and probability theory is required.
Within a Bayesian retrospective framework, we present a way of examining the distribution of \cps through a novel set estimator. For a given level, $α$, we aim at smallest sets that cover all \cps with a probability of at least $1-α$. These so-called smallest simultaneous credible regions, computed for certain values of $α$, provide parsimonious representations of the possible \cp locations. In addition, combining them for a range of different $α$'s enables very informative yet condensed visualisations. Therewith we allow for the evaluation of model choices and the analysis of \cp data to an unprecedented degree. This approach exhibits superior sensitivity, specificity and interpretability in comparison with highest density regions, marginal inclusion probabilities and confidence intervals inferred by \stepR. Whilst their direct construction is usually intractable, asymptotically correct solutions can be derived from posterior samples. This leads to a novel NP-complete problem. Through reformulations into an Integer Linear Program we show empirically that a fast greedy heuristic computes virtually exact solutions.