Double electron-electron resonance (DEER) spectroscopy measures distance distributions between spin labels in proteins, yielding important structural and energetic information about conformational landscapes. Analysis of an experimental DEER signal in terms of a distance distribution is a nontrivial task due to the ill-posed nature of the underlying mathematical inversion problem. This work introduces a Bayesian probabilistic inference approach to analyze DEER data, using a multi-Gauss mixture model for the distance distribution. The method uses Markov chain Monte Carlo (MCMC) sampling to determine a posterior probability distribution over model parameter space. This distribution contains all the information available from the data, including a full quantification of the uncertainty about the parameters. The corresponding uncertainty about the distance distribution is captured via an ensemble of posterior predictive distributions. Several synthetic examples illustrate the method. An experimental example shows the importance of model checking and comparison using residual analysis and Bayes factors. Overall, the Bayesian approach allows for more robust inference about protein conformations from DEER spectroscopy.