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Confidence and central tendency in perceptual judgment


This paper theoretically and empirically investigates the role of noisy cognition in perceptual judgment, focusing on the central tendency effect: the well-known empirical regularity that perceptual judgments are biased towards the center of the stimulus distribution. Based on a formal Bayesian framework, we generate predictions about the relationships between subjective confidence, central tendency, and response variability. Specifically, our model clarifies that lower subjective confidence as a measure of posterior uncertainty about a judgment should predict (i) a lower sensitivity of magnitude estimates to objective stimuli; (ii) a higher sensitivity to the mean of the stimulus distribution; (iii) a stronger central tendency effect at higher stimulus magnitudes; and (iv) higher response variability. To test these predictions, we collect a large-scale experimental data set and additionally re-analyze perceptual judgment data from several previous experiments. Across data sets, subjective confidence is strongly predictive of the central tendency effect and response variability, both correlationally and when we exogenously manipulate the magnitude of sensory noise. Our results are consistent with (but not necessarily uniquely explained by) Bayesian models of confidence and the central tendency.

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  1. 1.

    The data and code are available at

  2. 2.

    Note that the measure of Bayesian confidence described here does not directly map onto our experimental instructions, which simply asked subjects to report their subjective confidence using a Likert scale. However, this formulation has the advantage of being directly related to earlier formulations for discrete decisions.

  3. 3.

    Note that since we assume that the noise-corrupted signal to be unbiased, \(\mathbb {E}[s|x]=x\), the average estimate of a given objective stimulus magnitude x converges to \(\mathbb {E}[\hat {x}|x] = \lambda x+ (1-\lambda ) \mu _{x}\) (following Eq. 5). Under the simplifying assumption of s = x for a given signal realization (which holds in expectation), we can empirically approximate an analogue of λ for each individual estimate \(\hat {x}\):

    $$ \hat{\lambda} := \frac{\hat{x}-\mu_{x}}{x - \mu_{x}}. $$

    In regression analyses, we excluded \(\hat {\lambda }\) values of + / − infinity and those where \(\hat {\lambda }\) was undefined due to a denominator of zero. We excluded 3% of the sample this way.

  4. 4.

    We used standard deviation instead of variance because we found it to be slightly better behaved. The predictions are qualitatively the same for both standard deviation and variance.

  5. 5.

    This result came from an additional regression, where we regressed Confidence on Stimulus and Condition, with random effects for the intercept, Stimulus, and Condition grouped by participants.

  6. 6.

    These are trials with estimates 999 or NaN in DB18, trials with estimates 9999 or 999 in DB19, and trials on which participants did not respond.


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We are grateful to Lucy Lai and Henrik Singmann for helpful guidance. This research was supported by the Office of Naval Research (N00014-17-1-2984), the Center for Brains, Minds and Machines (funded by NSF STC award CCF-1231216), and a research fellowship from the Alfred P. Sloan Foundation.

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Correspondence to Yang Xiang.

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All the data and code are available at The experiments were not preregistered.

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Xiang, Y., Graeber, T., Enke, B. et al. Confidence and central tendency in perceptual judgment. Atten Percept Psychophys 83, 3024–3034 (2021).

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  • Visual perception
  • Bayesian modeling