AI assistants that assume users to be fully rational generally fail to understand as well as predict users' sub-optimal behaviors, making it difficult to provide adaptive assistance. In many cases, such behaviors are not a result of irrationality, but rather a rational decision made given inherent cognitive bounds and biased beliefs about the world. In this paper, we formally introduce a class of computational-rational (CR) user models for cognitively-bounded agents acting optimally under biased beliefs. The key novelty lies in explicitly modeling how a bounded cognitive process, such as imperfect memory, leads to a dynamically inconsistent and biased belief state, and consequently, sub-optimal sequential decision-making. We address the challenge of identifying the latent user-specific bound and inferring biased belief states from passive observations on the fly. We argue that for our formalized CR model family with an explicit and parameterized cognitive process, this challenge is tractable. To support our claim, we propose an efficient online inference method based on nested particle filtering that simultaneously tracks the user's latent belief state and estimates the unknown cognitive bound from a stream of observed actions. We validate our approach in a representative navigation task using memory capacity as an instance of cognitive bound. With simulations, we show that (1) our CR model generates intuitively plausible behaviours corresponding to different levels of memory capacity, and (2) our inference method accurately and efficiently recovers the ground-truth cognitive bounds from limited observations ($\le 100$ steps). We further demonstrate how this modeling approach provides a principled foundation for developing adaptive AI assistants within an assistive-POMDP, enabling adaptive assistance that takes the user's cognitive bounds into account.