  Despite increasingly precise observations and sophisticated theoretical
models, the discrepancy between measurements of H0 from the cosmic microwave
background or from Baryon Acoustic Oscillations combined with Big-Bang
Nucleosynthesis versus those from local distance ladder probes -- commonly
known as the $H_0$ tension -- continues to perplex the scientific community. To
address this tension, Early Dark Energy (EDE) models have been proposed as
alternatives to $\Lambda$CDM, as they can change the observed sound horizon and
the inferred Hubble constant from measurements based on this. In this paper, we
investigate the use of Bayesian Model Averaging (BMA) to evaluate EDE as a
solution to the H0 tension. BMA consists of assigning a prior to the model and
deriving a posterior as for any other unknown parameter in a Bayesian analysis.
BMA can be computationally challenging in that one must approximate the joint
posterior of both model and parameters. Here we present a computational
strategy for BMA that exploits existing MCMC software and combines
model-specific posteriors post-hoc. In application to a comprehensive analysis
of cosmological datasets, we quantify the impact of EDE on the H0 discrepancy.
We find an EDE model probability of $\sim$90% whenever we include the H0
measurement from Type Ia Supernovae in the analysis, whereas the other data
show a strong preference for the standard cosmological model. We finally
present constraints on common parameters marginalized over both cosmological
models. For reasonable priors on models with and without EDE, the H0 tension is
reduced by at least 20%.
