Verifying the operational history of nuclear reactors and by proxy the production of plutonium is key to nuclear disarmament efforts. For this, we develop a nuclear archaeology approach through which the history can be verified based on nuclide measurements of spent fuel and reprocessing waste. Our focus is on reconstructing burnup and cooling time values. Pursuing a probabilistic Bayesian inference framework allows us to include various sources of information beyond the measurements such as declarations and records. The goal is an ability to identify inconsistencies and to gain an insight into the actual history, including robust uncertainty estimates. This paper addresses two stages of our research: First, seeking to simultaneously take various isotopic ratios into account, we have identified those that minimize the uncertainties of our reconstruction. Since the number of theoretically possible ratios is very large, we have employed a computational technique for this purpose, Approximate Bayes Computation. Second, we have begun to test our approach initially for simple scenarios based on hypothetical simulation-based studies, but also using actual measurement data. These were obtained from the Spent Fuel Isotopic Composition 2.0 database and from the Joint Research Centre in Karlsruhe, where High Active Raffinate samples were measured.