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Efficient coding of numbers explains decision bias and noise

Abstract

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.

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Fig. 1: Illustration of the task and participants’ choice behaviour.
Fig. 2: Models with both nonlinear transformation and varying noise best capture participants’ behaviour.
Fig. 3: Best-fitting noise and bias: participants encode less frequent numbers with greater noise.
Fig. 4: Efficient-coding, Bayesian-decoding model: the Fisher information, fitted to the participants’ data, is adapted to the prior.
Fig. 5: The efficient-coding, Bayesian-decoding model reproduces the participants’ behaviour, and the Fisher information functions fitted to the participants’ data improve the performance ratio in the Upward and Downward conditions.

Data availability

The data relating to this study are available at https://doi.org/10.7916/tn94-qn62 (ref. 51).

Code availability

Scripts for analysing the data, implementing the models, fitting their parameters and producing the results and figures are available at https://doi.org/10.7916/tn94-qn62 (ref. 51).

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Acknowledgements

We thank B. Ho for his outstanding help as a research assistant, the National Science Foundation for research support (grant no. SES DRMS 1949418, M.W.) and the Italian Academy for Advanced Studies in America at Columbia University for research support (Spring 2021 Fellowship, A.P.-C.). The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript.

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M.W. conceptualized the study. M.W. and A.P.-C. designed the experiment. A.P.-C. implemented the task and collected the data. M.W. and A.P.-C. analysed the data and wrote the computational models. A.P.-C. implemented the models. M.W. and A.P.-C. interpreted the results and wrote the manuscript.

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Correspondence to Arthur Prat-Carrabin.

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Prat-Carrabin, A., Woodford, M. Efficient coding of numbers explains decision bias and noise. Nat Hum Behav 6, 1142–1152 (2022). https://doi.org/10.1038/s41562-022-01352-4

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