Biophysical models are increasingly used for medical applications at the organ scale. However, model predictions are rarely associated with a confidence measure although there are important sources of uncertainty in computational physiology methods. For instance, the sparsity and noise of the clinical data used to adjust the model parameters (personalization), and the difficulty in modeling accurately soft tissue physiology. The recent theoretical progresses in stochastic models make their use computationally tractable, but there is still a challenge in estimating patient-specific parameters with such models. In this work we propose an efficient Bayesian inference method for model personalization using polynomial chaos and compressed sensing. This method makes Bayesian inference feasible in real 3D modeling problems. We demonstrate our method on cardiac electrophysiology. We first present validation results on synthetic data, then we apply the proposed method to clinical data. We demonstrate how this can help in quantifying the impact of the data characteristics on the personalization (and thus prediction) results. Described method can be beneficial for the clinical use of personalized models as it explicitly takes into account the uncertainties on the data and the model parameters while still enabling simulations that can be used to optimize treatment. Such uncertainty handling can be pivotal for the proper use of modeling as a clinical tool, because there is a crucial requirement to know the confidence one can have in personalized models.