Studies of cognitive processes via electroencephalogram (EEG) recordings often analyze group-level event-related potentials (ERPs) averaged over multiple subjects and trials. This averaging procedure can obscure scientifically relevant variability across subjects and trials, but has been necessary due to the difficulties posed by inference of trial-level ERPs. We introduce the Bayesian Random Phase-Amplitude Gaussian Process (RPAGP) model, for inference of trial-level amplitude, latency, and ERP waveforms. We apply RPAGP to data from a study of ERP responses to emotionally arousing images. The model estimates of trial-specific signals are shown to greatly improve statistical power in detecting significant differences in experimental conditions compared to existing methods. Our results suggest that replacing the observed data with the de-noised RPAGP predictions can potentially improve the sensitivity and accuracy of many of the existing ERP analysis pipelines.