This paper proposes a data-driven technique to estimate the wave excitation force (WEF) which is an essential signal for wave forecasting and implementing power efficiency maximization control of Wave Energy Converters (WECs). A Bayesian probabilistic model of a WEC hydrodynamic system is described to generate robust WEF estimates. Specifically, the WEF uncertainty can be estimated based on observations through Gaussian Process (GP) modeling. It is shown that this modern way of incorporating the first principle modelling into a probabilistic framework has stronger robustness properties than the alternative of calculating estimates of a parametric function representation. Unlike the sample-based non-linear Kalman Filter, the means and covariances of the joint probabilities can be directly computed based on analytic moment matching that allow for reliable state-dependent uncertainty propagation. The results presented demonstrate the accuracy and robustness of the proposed data-driven wave excitation force estimator.