Robots that can effectively understand human intentions from actions are crucial for successful human-robot collaboration. In this work, we address the challenge of a robot navigating towards an unknown goal while also accounting for a human's preference for a particular path in the presence of obstacles. This problem is particularly challenging when both the goal and path preference are unknown a priori. To overcome this challenge, we propose a method for encoding and inferring path preference online using a partitioning of the space into polytopes. Our approach enables joint inference over the goal and path preference using a stochastic observation model for the human. We evaluate our method on an unknown-goal navigation problem with sparse human interventions, and find that it outperforms baseline approaches as the human's inputs become increasingly sparse. We find that the time required to update the robot's belief does not increase with the complexity of the environment, which makes our method suitable for online applications.
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