Constructing realistic and real time human-robot interaction models is a core challenge in crowd navigation. In this paper we derive a robot-agent interaction density from first principles of probability theory; we call our approach “first order interacting Gaussian processes” (foIGP). Furthermore, we compute locally optimal solutions—with respect to multi-faceted agent “intent” and “flexibility”—in near real time on a laptop CPU. We test on challenging scenarios from the ETH crowd dataset and show that the safety and efficiency statistics of foIGP is competitive with human safety and efficiency statistics. Further, we compute the safety and efficiency statistics of dynamic window avoidance, a physics based model variant of foIGP, a Monte Carlo inference based approach, and the best performing deep reinforcement learning algorithm; foIGP outperforms all of them.