Numerical climate models are used to project future climate change due to both anthropogenic and natural causes. Differences between projections from different climate models are a major source of uncertainty about future climate. Emergent relationships shared by multiple climate models have the potential to constrain our uncertainty when combined with historical observations. We combine projections from 13 climate models with observational data to quantify the impact of emergent relationships on projections of future warming in the Arctic at the end of the 21st century. We propose a hierarchical Bayesian framework based on a coexchangeable representation of the relationship between climate models and the Earth system. We show how emergent constraints fit into the coexchangeable representation, and extend it to account for internal variability simulated by the models and natural variability in the Earth system. Our analysis shows that projected warming in some regions of the Arctic may be more than 2 degrees C lower and our uncertainty reduced by up to 30% when constrained by historical observations. A detailed theoretical comparison with existing multi-model projection frameworks is also provided. In particular, we show that projections may be biased if we do not account for internal variability in climate model predictions. for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.