Efficient coding of numbers explains decision bias and noise

Human subjects differentially weight different stimuli in averaging tasks. This has been interpreted as reflecting biased stimulus encoding, but an alternative hypothesis is that stimuli are encoded with noise, then optimally decoded. Moreover, with efficient coding, the amount of noise should vary across stimulus space, and depend on the statistics of stimuli. We investigate these predictions through a task in which participants are asked to compare the averages of two series of numbers, each sampled from a prior distribution that differs across blocks of trials. We show that subjects encode numbers with both a bias and a noise that depend on the number. Infrequently occurring numbers are encoded with more noise. A maximum-likelihood decoding model captures subjects9 behaviour and indicates efficient coding. Finally, our model predicts a relation between the bias and variability of estimates, thus providing a statistically-founded, parsimonious derivation of Wei and Stocker9s "law of human perception".