Uncertainty and prior assumptions, rather than innate logarithmic encoding, explain non-linear number-to-space mapping

Mapping number to space is natural and spontaneous, but often non-veridical, showing a clear compressive nonlinearity, thought to reflect intrinsic logarithmic encoding of number. We asked 78 adult participants to map dot-arrays onto a numberline, each contributing only 9 trials. Combining participant data, we confirmed that on the first trial the numberline was heavily compressed, then linearized over trials. Responses were well described by logarithmic compression, but also by a parameter-free Bayesian model of central tendency, which predicts quantitatively the relationship between non-linearity and number acuity. To test experimentally the Bayesian hypothesis, 90 new participants completed a colourline task, mapping noise-perturbed colour patches to space. With more noise at the high end of the colourline, the mapping was logarithmic; but it became exponential with noise at the low end. We conclude that the non-linearity of both number and colour mapping reflects contextual Bayesian inference processes, rather than intrinsic logarithmic encoding.