Abstract: While the impacts of heat waves, droughts, and floods have been increasing along with rising greenhouse gas concentrations, the complex structure of natural variability in the climate system makes it challenging to precisely quantify the extent to which human activities are responsible for observed changes. The statistical methods used by high-profile scientific bodies to address this connection have been observed in recent findings to underestimate the magnitude of variability, resulting in potentially misleading over-confidence. To address this issue, I propose a physically-informed basis function parameterization of the covariance structure within a regularized Bayesian selection method to avoid over-fitting the limited amount of data and to propagate the estimation uncertainty to the final inference. When evaluated on statistically and dynamically simulated data, this method achieves lower RMSE scores and better-calibrated posterior coverage rates than methods that rely on potentially uncertain principal components. Incorporating the physically-informed basis representation into a mixture model allows for the error in the dynamical climate simulations informing the natural variability component to be assessed and accounted for in the inference procedure. Motivated by the need for policymakers and the public at large to understand the extent of human responsibility for climate impacts at specific locations, ongoing work funded aims to leverage the global covariance structure to provide robust quantification of causal connections at fine spatial scales. Longer-term extensions include the use of deep learning techniques to understand more complex distributions and non-linear causal relationships within a Bayesian framework.