In this thesis, we describe three Markov Chain Monte Carlo (MCMC) algorithms, the Metropolis-Hastings (MH) algorithm, the Delayed Rejection Adaptive Metropolis (DRAM) algorithm, and the Differential Evolution Adaptive Metropolis (DREAM) algorithm, as well as simple parallelizations for the first two algorithms.

We explain the root mean square error and mean absolute error as error functions, the Monte Carlo Standard Error, Effective Sample Size and autocorrelation, as well as the $\hat{R}$ diagnostics for convergence analysis, with which we examine the three MCMC algorithms combined with trace plots, histograms, and other diagrams. We determine appropriate parameter values for each algorithm according to these diagnostics first, by evaluating them with the HBV-SASK hydrological model on the Banff river basin. Then, we use each MCMC algorithm with those parameters to obtain parameters for the HBV-SASK model trained on additional training datasets. Afterwards, we run the HBV-SASK model with those parameters on different evaluation sets of the Banff and Oldman river basins.

Overall, all algorithms perform similarly well regarding the error functions, although the parameters we obtained through DREAM perform worse than those of the other two MCMC algorithms if we use them on evaluation sets with different behaviors compared to the respective training set. However, DREAM performs far better for the Effective Sample Size and autocorrelation, as well as the convergence diagnostic, for which MH performs the worst.