In brain-computer interfaces (BCI), most of the approaches based on event-related potential (ERP) focus on the detection of P300, aiming for single trial classification for a speller task. While this is an important objective, existing P300 BCI still require several repetitions to achieve a correct classification accuracy. Signal processing and machine learning advances in P300 BCI mostly revolve around the P300 detection part, leaving the character classification out of the scope. To reduce the number of repetitions while maintaining a good character classification, it is critical to embrace the full classification problem. We introduce an end-to-end pipeline, starting from feature extraction, and is composed of an ERP-level classification using probabilistic Riemannian MDM which feeds a character-level classification using Bayesian accumulation of confidence across trials. Whereas existing approaches only increase the confidence of a character when it is flashed, our new pipeline, called Bayesian accumulation of Riemannian probabilities (ASAP), update the confidence of each character after each flash. We provide the proper derivation and theoretical reformulation of this Bayesian approach for a seamless processing of information from signal to BCI characters. We demonstrate that our approach performs significantly better than standard methods on public P300 datasets.