The CASCADE project has four major research branches centered on low-likelihood high-impact (LLHIs) events. Here, I present work derived from two of those branches; innovation of computational statistical methods for LLHIs and LLHI events in the observational record. Determining the occurrence and severity of LLHI weather events is difficult due to the relatively few recorded events. Instead of making inference on the probability of the weather event directly, we shift our focus to the drivers of the climatology surrounding weather events. We represent the climate system in terms of climatological forcing and internal variability and specify a statistical model for each. The climatological forcing represents changes in the system due to anthropogenic induced climate change which we model using a set of physical drivers. The internal variability represents the variation in the system due to its natural cycle, which we model using our novel Bayesian singular value decomposition methodology. By decomposing the climate system in terms of its drivers, we can determine which combination of drivers result in high-impact weather events and the probability of these events occurring. We apply our statistical framework to two-meter air temperature data from ERA5 over the Pacific Northwest. Our analysis provides additional insight into the 2021 heatwave regarding the contribution of climatological forcing and internal variability to the LLHI event and how the impact of each on the LLHI are changing in time.