The paper considers how to create a flexible method for modelling Hawkes-like processes with flexibility in the triggering kernel, using Gaussian processes working on a different input than is usually attempted. Section 1-3 is well written, very clear and gives a good motivation and description of how to get a GP regressive point process. The idea in the preliminary work that the likelihood of a Hawkes process factorises across different types is surprising as the triggering kernel in the CIF appears to connect \lambda_{u_{i}}(t) and \lambda_{u_{j}}(t), since u_{i} and u_{j} appear to be part of a single triggering kernel, and hence are connected. Can the authors clarify, is there some conditional independence I am not spotting here? It doesn't appear to be of significance however as the rest of the paper only focusses on a single event type in the end.