Influence of the prior on the crossover point.

<div> The ability to rapidly estimate non-symbolic numerical quantities is a well-conserved sense across species with clear evolutionary advantages. However, despite its importance, this sense is surprisingly imprecise and biased, and a formal explanation for this seemingly irrational behavior remains unclear. We develop a unified normative theory of numerosity estimation that parsimoniously incorporates in a single framework information processing constraints alongside (i) Brownian diffusion noise to capture the effects of time exposure of sensory information, (ii) logarithmic encoding of numerosity representations, and (iii) optimal inference via Bayesian decoding. We show that for a given allowable biological capacity constraint our model naturally endogenizes time perception during noisy efficient encoding to predict the complete posterior distribution of numerosity estimates. This model accurately predicts many features of human numerosity estimation as a function of temporal exposure, indicating that humans can rapidly and efficiently sample numerosity information over time. Additionally, we demonstrate how our model fundamentally differs from a thermodynamically-inspired formalization of bounded rationality, where information processing is modeled as acting to shift away from default states. The mechanism we propose is the likely origin of a variety of numerical cognition patterns observed in humans and other animals. </div>