Outputs from complex Earth system models (ESMs) participating in the Coupled Model Intercomparison Project (CMIP) are a major source of information for policy-relevant assessments of climate change. The way we view and interpret the CMIP archive shapes our understanding of the real world, yet many assessments present analysis choices only implicitly. We describe a fully Bayesian approach and software package for analyzing CMIP data and present a series of interpretive models with gradually increasing complexity to illustrate the methodology. We also show how to update CMIP-derived posteriors using additional evidence, including observations, emergent constraints, and information about processes in the Earth system models themselves. Using these methods, we show that even apparently strong relationships between observable processes and equilibrium climate sensitivity (ECS) in ESMs do not necessarily tightly constrain ECS. We also show that estimates of β, the terrestrial carbon dioxide fertilization effect, are revised downward when considering the individual processes included in ESMs. Our results illustrate how these and other estimates may be updated as new information arrives.