Efficient coding of numbers explains decision bias and noise.

Humans differentially weight different stimuli in averaging tasks, which has been interpreted as reflecting encoding bias. We examine the alternative hypothesis that stimuli are encoded with noise and then optimally decoded. Under a model of efficient coding, the amount of noise should vary across stimuli and depend on statistics of the stimuli. We investigate these predictions through a task in which the participants are asked to compare the averages of two series of numbers, each sampled from a prior distribution that varies across blocks of trials. The participants encode numbers with a bias and a noise that both depend on the number. Infrequently occurring numbers are encoded with more noise. We show how an efficient-coding, Bayesian-decoding model accounts for these patterns and best captures the participants' behaviour. Finally, our results suggest that Wei and Stocker's "law of human perception", which relates the bias and variability of sensory estimates, also applies to number cognition.