Inferences in Earth system science rely to a large degree on the numerical output of multiple Earth System Models. It has been shown that for many variables of interest, the multi-model ensemble average often compares better with observations than the output from any one individual model. However, a simple arithmetic average does not reward or penalize models according to their ability to predict available observations, and for this reason, a weighted averaging approach would be preferred for those cases in which there is information on model performance. We propose an approach based on concepts from information theory with the aim to approximate the Kullback-Leibler distance between model output and unknown reality, and to assign weights to different models according to their relative likelihood of being the best-performing model in a given grid cell. This article presents the theory and describes the steps necessary for obtaining model weights in a general form, and presents an example for obtaining multi-model averages of carbon fluxes from models participating in the sixth phase of the Coupled Model Intercomparison Project CMIP6. Using this approach, we propose a multi-model ensemble of land-atmosphere carbon exchange that could be used for inferring long-term carbon balances with much reduced uncertainties in comparison to the multi-model arithmetic average.