Scientists and engineers build quantitative models of our natural and constructed world. Using the process of statistical inference, the models are tested against experimental data in order to constrain their underlying parameters, which leads to an improved understanding of the models and more accurate predictions for future data. Markov Chain Monte Carlo (MCMC) is the most popular algorithm for statistical inference because of its power and simplicity. The goal of this project is to develop a new MCMC inference method, Shrek, which exploits the fact that many models in science and engineering can be calculated using different time-accuracy trade-offs. By performing inference with a combination of low accuracy but computationally cheap models, with high accuracy but computationally expensive models, Shrek can achieve better performance than traditional MCMC algorithms. The project team aims to apply Shrek to the field of cosmology to answer questions about dark matter, galaxy formation, and the expansion rate of the Universe. Furthermore, the team is developing an open-source software package so that scientists across many fields may use the Shrek algorithm. This project exploits the multi-fidelity nature of many scientific and engineering models to help guide MCMC. Many simulations, including the Boltzmann codes ubiquitously used to model the cosmic microwave background, have tunable fidelities. This can be, for instance, a spatial or temporal resolution of the model, or