Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

A Bayesian observer model constrained by efficient coding can explain 'anti-Bayesian' percepts

Nature Neuroscience volume 18, pages 1509–1517 (2015)Cite this article

Subjects

Abstract

Bayesian observer models provide a principled account of the fact that our perception of the world rarely matches physical reality. The standard explanation is that our percepts are biased toward our prior beliefs. However, reported psychophysical data suggest that this view may be simplistic. We propose a new model formulation based on efficient coding that is fully specified for any given natural stimulus distribution. The model makes two new and seemingly anti-Bayesian predictions. First, it predicts that perception is often biased away from an observer's prior beliefs. Second, it predicts that stimulus uncertainty differentially affects perceptual bias depending on whether the uncertainty is induced by internal or external noise. We found that both model predictions match reported perceptual biases in perceived visual orientation and spatial frequency, and were able to explain data that have not been explained before. The model is general and should prove applicable to other perceptual variables and tasks.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Get just this article for as long as you need it

$39.95

Learn more

Prices may be subject to local taxes which are calculated during checkout

Figure 1: Bayesian observer model constrained by efficient coding.
Figure 2: Prediction 1: Bayesian perception can be biased away from the prior peak.
Figure 3: Prediction 2: stimulus (external) and sensory (internal) noise differentially affect perceptual bias.
Figure 4: Biases in perceived orientation.
Figure 5: Relative biases in perceived orientation.
Figure 6: Biases in perceived spatial frequency.
Figure 7: Predicted biases for different loss functions.
Figure 8: Equivalent efficient neural representations for the same stimulus distribution.

References

  1. Attneave, F. Some informational aspects of visual perception. Psychol. Rev. 61, 183 (1954).

    Article  CAS  PubMed  Google Scholar 

  2. Barlow, H.B. Possible principles underlying the transformation of sensory messages. in Sensory Communication (ed. Rosenblith, W.A.) 217–234 (MIT Press, 1961).

  3. Olshausen, B.A. & Field, D.J. Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature 381, 607–609 (1996).

    Article  CAS  PubMed  Google Scholar 

  4. Dan, Y., Atick, J.J. & Reid, R.C. Efficient coding of natural scenes in the lateral geniculate nucleus: Experimental test of a computational theory. J. Neurosci. 16, 3351–3362 (1996).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  5. Lewicki, M.S. Efficient coding of natural sounds. Nat. Neurosci. 5, 356–363 (2002).

    Article  CAS  PubMed  Google Scholar 

  6. Helmholtz, H. Treatise on Physiological Optics (transl.) (Thoemmes Press, Bristol, UK, 2000).

  7. Curry, R.E. A Bayesian model for visual space perception. in Seventh Annual Conference on Manual Control NASA SP-281, 187ff (NASA, 1972).

  8. Knill, D.C. & Richards, W. Perception as Bayesian Inference (Cambridge University Press, 1996).

  9. Körding, K.P. & Wolpert, D. Bayesian integration in sensorimotor learning. Nature 427, 244–247 (2004).

    Article  PubMed  Google Scholar 

  10. Stocker, A.A. & Simoncelli, E.P. Noise characteristics and prior expectations in human visual speed perception. Nat. Neurosci. 9, 578–585 (2006).

    Article  CAS  PubMed  Google Scholar 

  11. van den Berg, R., Vogel, M., Josic, K. & Ma, W.J. Optimal inference of sameness. Proc. Natl. Acad. Sci. USA 109, 3178–3183 (2012).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  12. Jazayeri, M. & Shadlen, M.N. Temporal context calibrates interval timing. Nat. Neurosci. 13, 1020–1026 (2010).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  13. Jones, M. & Love, B.C. Bayesian fundamentalism or enlightenment? On the explanatory status and theoretical contributions of Bayesian models of cognition. Behav. Brain Sci. 34, 169–188 (2011).

    Article  PubMed  Google Scholar 

  14. Bowers, J.S. & Davis, C.J. Bayesian just-so stories in psychology and neuroscience. Psychol. Bull. 138, 389 (2012).

    Article  PubMed  Google Scholar 

  15. Tomassini, A., Morgan, M.J. & Solomon, J.A. Orientation uncertainty reduces perceived obliquity. Vision Res. 50, 541–547 (2010).

    Article  PubMed  Google Scholar 

  16. Girshick, A.R., Landy, M.S. & Simoncelli, E.P. Cardinal rules: visual orientation perception reflects knowledge of environmental statistics. Nat. Neurosci. 14, 926–932 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  17. Geisler, W.S., Najemnik, J. & Ing, A.D. Optimal stimulus encoders for natural tasks. J. Vis. 9, 17.1–17.16 (2009).

    Article  Google Scholar 

  18. Burge, J. & Geisler, W.S. Optimal defocus estimation in individual natural images. Proc. Natl. Acad. Sci. USA 108, 16849–16854 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  19. Brayanov, J.B. & Smith, M.A. Bayesian and “Anti-Bayesian” biases in sensory integration for action and perception in the size-weight illusion. J. Neurophysiol. 103, 1518–1531 (2010).

    Article  PubMed  PubMed Central  Google Scholar 

  20. Wei, X.-X. & Stocker, A.A. Efficient coding provides a direct link between prior and likelihood in perceptual Bayesian inference. Adv. Neural Inf. Process. Syst. 25, 1313–1321 (2012).

    Google Scholar 

  21. Linsker, R. Self-organization in a perceptual network. Computer 21, 105–117 (1988).

    Article  Google Scholar 

  22. Brunel, N. & Nadal, J.-P. Mutual information, Fisher information, and population coding. Neural Comput. 10, 1731–1757 (1998).

    Article  CAS  PubMed  Google Scholar 

  23. McDonnell, M.D. & Stocks, N.G. Maximally informative stimuli and tuning curves for sigmoidal rate-coding neurons and populations. Phys. Rev. Lett. 101, 058103 (2008).

    Article  PubMed  Google Scholar 

  24. Ganguli, D. & Simoncelli, E.P. Implicit encoding of prior probabilities in optimal neural populations. Adv. Neural Inf. Process. Syst. 23, 658–666 (2010).

    Google Scholar 

  25. Fechner, G.T. Elemente der Psychophysik (Breitkopf und Haertel, Leipzig, 1860).

  26. Stocker, A.A. & Simoncelli, E.P. Sensory adaptation within a Bayesian framework for perception. Adv. Neural Inf. Process. Syst. 18, 1289 (2006).

    Google Scholar 

  27. Webb, B.S., Ledgeway, T. & McGraw, P.V. Relating spatial and temporal orientation pooling to population decoding solutions in human vision. Vision Res. 50, 2274–2283 (2010).

    Article  PubMed  PubMed Central  Google Scholar 

  28. Putzeys, T., Bethge, M., Wichmann, F., Wagemans, J. & Goris, R. A new perceptual bias reveals supoptimal population decoding of sensory responses. PLoS Comput. Biol. 8, e1002453 (2012).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  29. Switkes, E., Mayer, M.J. & Sloan, J.A. Spatial frequency analysis of the visual environment: anisotropy and the carpentered environment hypothesis. Vision Res. 18, 1393–1399 (1978).

    Article  CAS  PubMed  Google Scholar 

  30. Coppola, D.M., Purves, H.R., McCoy, A.N. & Purves, D. The distribution of oriented contours in the real world. Proc. Natl. Acad. Sci. USA 95, 4002–4006 (1998).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  31. Jastrow, J. Studies from the University of Wisconsin: on the judgment of angles and positions of lines. Am. J. Psychol. 5, 214–248 (1892).

    Article  Google Scholar 

  32. de Gardelle, V., Kouider, S. & Sackur, J. An oblique illusion modulated by visibility: non-monotonic sensory integration in orientation processing. J. Vis. 10, 6 (2010).

    Article  PubMed  Google Scholar 

  33. Ruderman, D.L. The statistics of natural images. Network 5, 517–548 (1994).

    Article  Google Scholar 

  34. Dong, D.W. & Atick, J.J. Statistics of natural time-varying images. Network 6, 345–358 (1995).

    Article  Google Scholar 

  35. Campbell, F.W. & Robson, J.G. Application of Fourier analysis to the visibility of gratings. J. Physiol. (Lond.) 197, 551 (1968).

    Article  CAS  Google Scholar 

  36. Georgeson, M.A. & Ruddock, K.H. Spatial frequency analysis in early visual processing. Phil. Trans. R. Soc. Lond. B [and discussion] 290, 11–22 (1980).

    Article  CAS  Google Scholar 

  37. Körding, K.P. & Wolpert, D. The loss function of sensorimotor learning. Proc. Natl. Acad. Sci. USA 101, 9839–9842 (2004).

    Article  PubMed  PubMed Central  Google Scholar 

  38. Wang, Z., Stocker, A.A. & Lee, D.D. Optimal neural tuning curves for arbitrary stimulus distributions: Discrimax, Infomax and minimum Lp loss. Adv. Neural Inf. Process. Syst. 25, 2177–2185 (2012).

    Google Scholar 

  39. Salinas, E. How behavioral constraints may determine optimal sensory representations. PLoS Biol. 4, e387 (2006).

    Article  PubMed  PubMed Central  Google Scholar 

  40. Ganguli, D. & Simoncelli, E.P. Efficient sensory encoding and Bayesian inference with heterogeneous neural populations. Neural Comput. 26, 2103–2134 (2014).

    Article  PubMed  PubMed Central  Google Scholar 

  41. Simoncelli, E.P. & Olshausen, B.A. Natural image statistics and neural representation. Annu. Rev. Neurosci. 24, 1193–1216 (2001).

    Article  CAS  PubMed  Google Scholar 

  42. Laughlin, S.B. A simple coding procedure enhances a neuron's information capacity. Z. Naturforsch. C. 36, 910–912 (1981).

    Article  CAS  PubMed  Google Scholar 

  43. Stilp, C.E. & Kluender, R.K. Efficient coding and statistically optimal weighting of covarinace among acoustic attributes in novel sounds. PLoS ONE 7, e30845 (2012).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  44. Chalk, M., Seitz, A.R. & Series, P. Rapidly learned stimulus expectations alter perception of motion. J. Vis. 10, 2 (2010).

    Article  PubMed  Google Scholar 

  45. Crane, B.T. Direction specific biases in human visual and vestibular heading perception. PLoS ONE 7, e51383 (2012).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  46. Cuturi, L.F. & MacNeilage, P.R. Systematic biases in human heading estimation. PLoS ONE 8, e56862 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  47. Rose, D. & Blakemore, C. An analysis of orientation selectivity in the cat's visual cortex. Exp. Brain Res. 20, 1–17 (1974).

    Article  CAS  PubMed  Google Scholar 

  48. Gu, Y., Fetsch, C.R., Adeyemo, B., DeAngelis, G.C. & Angelaki, D.E. Decoding of MSTd population activity accounts for variations in the precision of heading perception. Neuron 66, 596–609 (2010).

    CAS  PubMed  PubMed Central  Google Scholar 

  49. Fischer, B.J. Bayesian estimates from heterogeneous population codes. Proc. Int. Jt. Conf. Neural Netw., 1–7 (IEEE, 2010).

  50. Wei, X.-X. & Stocker, A.A. Bayesian inference with efficient neural population codes. in Artificial Neural Networks and Machine Learning–ICANN 2012 (eds. Villa, A., Duch, W., Erdi, P., Masulli, F. & Palm, G.) 523–530 (Springer, 2012).

  51. Kullback, S. & Leibler, R.A. On information and sufficiency. Ann. Math. Stat. 22, 79–86 (1951).

    Article  Google Scholar 

  52. Mr. Bayes & Mr. Price. An essay towards solving a problem in the doctrine of chances, by the late Rev. Mr. Bayes, F. R. S. communicated by Mr. Price, in a letter to John Canton, A. M. F. R. S. Philos. Trans. 53, 370–418 (1763).

    Article  Google Scholar 

Download references

Acknowledgements

We thank V. DeGardelle, S. Kouider and J. Sackur for providing their data. We also would like to express our gratitude to E. Salinas, J. Gold and D. Swingley for providing valuable feedback on previous versions of the manuscript. The work was supported by the Office of Naval Research (grant N000141110744) and the University of Pennsylvania (including a Benjamin Franklin fellowship to X.-X.W.).

Author information

Authors and Affiliations

  1. Department of Psychology, University of Pennsylvania, Philadelphia, Pennsylvania, USA

    Xue-Xin Wei & Alan A Stocker

  2. Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, Pennsylvania, USA

    Alan A Stocker

Authors
  1. Xue-Xin Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alan A Stocker
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Both authors jointly designed and performed the research, and wrote the paper.

Corresponding author

Correspondence to Alan A Stocker.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Methods Checklist

Supplementary Information (PDF 352 kb)

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wei, XX., Stocker, A. A Bayesian observer model constrained by efficient coding can explain 'anti-Bayesian' percepts. Nat Neurosci 18, 1509–1517 (2015). https://doi.org/10.1038/nn.4105

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nn.4105

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing