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Confidence and central tendency in perceptual
judgment
Yang Xiang,^1 Thomas Graeber,^2 Benjamin Enke,^3 Samuel J. Gershman^4 ,^5
1 Sloan School of Management, MIT
2 Harvard Business School
3 Department of Economics, Harvard University
4 Department of Psychology and Center for Brain Science, Harvard University
5 Center for Brains, Minds and Machines, MIT
∗Corresponding author: yyx@mit.edu
March 11, 2021
Abstract This paper theoretically and empirically investigates the role of noisy cognition in percep- tual 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 stim- ulus 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 exoge- nously 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.
Introduction
One of the most robust empirical regularities in studies of human perception is the central tendency (or regression) effect: across various perceptual domains, estimates of stimulus magnitude are consistently biased towards the center of the magnitude distribution (Hollingworth, 1910, Stevens and Greenbaum, 1966). One prominent explanation for the central tendency effect is that sensory signals are “regularized” to reduce the influence of noise. Intuitively, when the signal-to-noise ratio is very low, then the best guess is the center of the magnitude distribution. When the signal-to- noise ratio is very high, then the best guess will ignore the magnitude distribution and only use the signal. At intermediate levels, the best guess will be somewhere in between the signal and the center of the magnitude distribution. These intuitions can be formalized in a Bayesian framework (Petzschner et al., 2015), where the prior expresses the effect of the magnitude distribution, and
the likelihood expresses the effect of sensory noise. Bayes’ rule prescribes how these two sources of information should be optimally combined. Central to the Bayesian framework is the role of noise in predicting the strength of the central tendency effect. However, typically we cannot directly measure the noise level. This empirical gap is frequently filled by ad hoc assumptions or by fitting free parameters to the data, rendering the Bayesian framework possibly unfalsifiable (Jones and Love, 2011, Marcus and Davis, 2013). The same issue vexes non-Bayesian models (e.g., Ratcliff and McKoon, 2018, 2020) that implicitly or explicitly make assumptions about sensory or cognitive noise. One way to get around this issue is to collect other measures that are hypothesized to capture the magnitude of noise. Even if we cannot measure the noise level directly, we can make predictions about the relationships between indirect measures and magnitude estimates as a test of model predictions. We pursue this strategy here, using subjective confidence and response variability as auxiliary measures to triangulate the effects of noise on perception. To make our predictions precise, we develop a simple Bayesian model of magnitude estimation and derive a number of generic predictions from it (i.e., predictions that don’t depend strongly on the parameter values). However, our goal is not to advocate for the Bayesian model versus alternative models, but rather to formalize some hypothetical regularities which (if true) would need to be satisfied by any model of magnitude estimation. Our model makes several key predictions, elaborated in the next section: (i) confidence should decrease with sensory noise; (ii) the sensitivity of magnitude estimates to actual magnitudes should increase with subjective confidence and decrease with sensory noise; (iii) the central tendency effect should decrease with confidence and increase with sensory noise; (iv) both confidence and sensitivity should decrease with stimulus magnitude, assuming sensory noise that grows with stimulus magni- tude; and (v) response variability should increase with sensory noise and decrease with confidence when prior uncertainty is high relative to sensory noise. We test these model predictions in two ways. First, we implement a new large-scale magnitude estimation experiment. We elicit both magnitude estimates and subjective confidence. A key fea- ture of our experiment is that we exogenously vary the objective stimulus, the mean of the stimulus distribution, and the magnitude of sensory noise. We collect these data because, as explained be- low, existing data sets on magnitude estimation and subjective confidence lack sufficient variation in some of the key elements of Bayesian models. Second, moving beyond our new experiment, we re-analyze data from several earlier studies that measured both continuous reports of stimulus mag- nitude and subjective confidence. Although these earlier studies are limited in several ways (which motivated our new experiment), the results from our re-analysis provide converging evidence for our model’s main predictions.^1 Whether or not one accepts the Bayesian framework, these findings provide constraints on any model of confidence and central tendency in perceptual judgment.
Theoretical Framework
To motivate our empirical predictions, we will first lay out a theoretical framework based on a simple Bayesian estimation problem, which mirrors the experimental tasks given to subjects.
(^1) The data and code are available at https://github.com/yyyxiang/confidence_central_tendency.
Subjective confidence
Magnitude estimation tasks require subjects to report a single point estimate of the magnitude, but Bayesian models hypothesize that subjects are representing an entire distribution over magnitudes. Subjective confidence judgments can potentially provide a probe of this distributional representa- tion. According to the Bayesian confidence hypothesis (Aitchison et al., 2015, Fleming and Daw, 2017, Meyniel et al., 2015, Pouget et al., 2016, Rahnev et al., 2015, Sanders et al., 2016), subjective confidence corresponds to the posterior probability that an action is optimal (in this context, the probability that the subjective estimate equals the objective stimulus magnitude). Note that for continuous magnitudes, the probability that the point estimate ˆx equals the objective magnitude x is 0. However, we can evaluate the posterior probability that the true magnitude falls within an infinitesimally small region around the posterior mean estimate. In the limit, this probability be- comes the density of the normally distributed estimate, evaluated at its mean: (^) σ^1 xˆ
2 /π ≡^ c. Thus, we see that Bayesian confidence (c) for the Gaussian estimation problem is inversely proportional to the posterior standard deviation. This result motivates the use of the standard deviation as a measure of “cognitive uncertainty” (Enke and Graeber, 2020).^2 Most importantly, note that un- der a constant prior variance σ x^2 , Bayesian confidence c decreases in the variance of sensory noise, σ ^2. Because our experimental design controls the prior variance as argued above, we will in the following interpret Bayesian confidence as an approximate measure of sensory noise.
Predictions
Using this framework, we can now lay out a set of theoretical predictions, which we will empirically test in subsequent sections.
Prediction 1 Sensitivity (λ) monotonically increases with confidence (c).
This prediction follows from the expression relating sensitivity to confidence:
λ = 1 −
πc^2 σ^2 x
Although the model quantity λ is not directly observable, we can approximate it empirically as described in the experimental analysis below. The next prediction concerns the relationship between confidence and the central tendency effect, a straightforward corollary of Prediction 1 because ∂ ˆx ∂μx = 1^ −^ λ. As discussed above, we follow previous work in assuming that the prior is determined by the experienced or instructed magnitude distribution, and hence μx is the mean of the magnitude distribution.
Prediction 2 The central tendency effect ( (^) ∂μ∂^ xˆx ) monotonically decreases with confidence.
Here we have defined the central tendency effect formally as the degree to which the perceptual estimate changes with the average stimulus magnitude. We now state the causal analogs of these predictions based on exogenous changes in sensory noise. (^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.
Prediction 3 An exogenous increase in sensory noise (σ ^2 ), which reduces subjective confidence, decreases sensitivity.
Prediction 4 An exogenous increase in sensory noise increases the central tendency effect.
Next, we theoretically explore how the strength of the central tendency effect depends on the stimulus magnitude. A well-known phenomenon is that response variability increases with stimulus magnitude according to Weber’s law. A common interpretation is that the signal-to-noise ratio decreases with stimulus magnitude, due either to a non-linear transformation of magnitude (e.g., Fechner, 1860, Nieder and Miller, 2003, Petzschner and Glasauer, 2011, Roach et al., 2017, Stevens,
- or to magnitude-dependent scaling of sensory noise (e.g., Gibbon, 1977, Gibbon and Church, 1981, Treisman, 1964). For concreteness, we examine the implications of the latter assumption:
∂σ ε^2 ∂x
∂c ∂x
< 0 and ∂λ ∂x
Prediction 5 As the stimulus magnitude increases, confidence and sensitivity decrease. The latter effect implies a stronger central tendency effect for larger magnitudes.
Finally, we turn to response variability. Theoretically, variability is affected both by the effect of sensory noise on estimates (which increases variability) and the counteracting central tendency effect (which decreases variability; see Enke and Graeber, 2020). The expression for response variability is given by:
Var(ˆx|x) = λ^2 σ^2 =
σ x^2 σ^2 x + σ^2
σ^2 . (8)
The relationship between response variability and other quantities depends on the degree of prior uncertainty relative to sensory noise.
Prediction 6 When sensory noise is small relative to prior uncertainty (σ ^2 < σ x^2 ), response vari- ability increases in sensory noise variance and decreases in confidence. When sensory noise is large relative to prior uncertainty (σ ^2 > σ^2 x), response variability decreases in sensory noise variance and increases in confidence.
Intuitively, when sensory noise is zero, the response is exactly equal to the stimulus and there is no residual variability. In the limit of large sensory noise, the response equals the prior mean and again there is no residual variability. Thus there is response variability only for intermediate values of sensory noise.
Experiment
Although a considerable number of studies have implemented magnitude estimation tasks while also measuring subjective confidence, none of these studies explicitly vary all of the variables that are of interest in light of our theoretical predictions above. Specifically, we require a study setup that features (i) variation in stimuli; (ii) a meaningful degree of exogenous variation in the mean of the stimulus distribution; (iii) exogenous variation in sensory noise; and (iv) a large sample size to allow for sufficiently powered statistical analyses of the relationship between confidence and the precise mechanics of central tendency. Earlier studies typically had sample sizes of fewer than 50 participants (see Table 2).
Report Confidence
Fixation Cross Report Estimate (500 ms)
Dot Array (100 ms or 2000 ms)
Estimate (10 s or until response)
Confidence (until response)
0-
Figure 1: Illustration of the experiment. Each trial begins with a fixation cross, followed by a dot array. Participants then report their numerosity estimate and confidence.
in a block (AveStim), with random effects for the intercept, Stimulus, and AveStim grouped by participants. In the absence of a central tendency effect, the regression coefficient of the true stimulus should be one, while the coefficient of average stimulus should be zero. The central tendency effect is indicated by a stimulus coefficient of less than one and an average stimulus coefficient of greater than zero. To explore the role of sensory noise and confidence for the central tendency effect, we ran Model
- Here, we added as additional regressors subjective confidence, the interaction between confidence and the stimulus, and the interaction between confidence and average stimulus, and added random effects for subjective confidence, the interaction between confidence and the stimulus, and the interaction between confidence and average stimulus grouped by participants. As derived above, our hypothesis is that higher confidence (as a consequence of lower sensory noise) is associated with a higher responsiveness to the stimulus and a lower responsiveness to the average stimulus, meaning that the first interaction effect should be positive and the second one negative. In addition, Model 2 also accounts for our exogenous variation in stimulus duration, which we conceptualize as exogenous variation in sensory noise that translates into confidence. We include as regressors a binary Condition indicator (0 for the 100 ms condition, 1 for the 2000 ms condition), the interaction between Stimulus and Condition, and the interaction between average stimulus magnitude and Condition, and random effects for Condition, the interaction between Stimulus and Condition, and the interaction between average stimulus magnitude and Condition grouped by participants. To see how the proportional regression to the prior changes in stimulus magnitude, we specified Model 3. Here, we regress an empirical estimate of λ on stimulus magnitude with random effects for the intercept and stimulus magnitude grouped by participants, and hypothesize a negative
coefficient.^3 Turning to the analysis of response variability, in Model 4, we regressed Variability (response standard deviation^4 across multiple repetitions of the same magnitude) on Confidence (averaged across stimulus repetitions), with random effects for the intercept and Confidence grouped by par- ticipants. In Model 5, we regressed Variability on Condition, with random effects for the intercept and Condition grouped by participants. For all five models, we included random effects for the in- tercept and each regressor. For better interpretability, we standardized the Confidence coefficients.
Results
Preliminaries
Our study rests on two prerequisites: (i) The existence of a central tendency effect; and (ii) variation in subjective confidence as a function of stimulus duration. The results of Model 1 (column 1 of Table 1) confirm the existence of a central tendency effect in our data: the coefficient of the true stimulus is substantially smaller than one [F (1, 69599) =
- 20 , p < 0 .0001], and the coefficient of the average stimulus is considerably larger than zero [F (1, 69599) = 117. 34 , p < 0 .0001]. To visualize the central tendency effect, we plotted subjective estimates as a function of stimulus magnitude. As shown in Figure 2A, the slopes of the estimation functions are considerably smaller than 1, and the level of bias in the estimation functions increases in the mean of the stimulus distribution. Our regression analysis revealed that confidence was higher for longer durations [t(71405) =
- 20 , p < 0 .0001]^5 , consistent with the hypothesis that lower sensory noise registers as higher confidence (Figure 2B).
Confidence and the central tendency effect
We begin our main analysis by testing Predictions 1 and 2. The results of Model 2 (column 2 of Table 1) show that the central tendency effect is strongly moderated by subjective confidence. The positive interaction effect of confidence and stimulus implies that, for every standard devia- tion of confidence, the responsiveness of subjective estimates to the stimulus value increases by 5 percentage points [t(71399) = 6. 71 , p < 0 .0001]. This result confirms Prediction 1. Similarly, the negative interaction effect of confidence and average stimulus shows that confident subjects place substantially lower weight on the mean of the stimulus distribution [t(71399) = − 3. 18 , p < 0 .01], supporting Prediction 2 that subjective estimates are pulled towards the prior mean to a greater
(^3) Note that since we assume that the noise-corrupted signal to be unbiased, E[s|x] = x, the average estimate of a given objective stimulus magnitude x converges to E[ˆx|x] = λx + (1 − λ)μx (following equation (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 ˆx:
ˆλ := xˆ^ −^ μx x − μx^.^ (9)
In regression analyses, we excluded λˆ values of +/− infinity and those where ˆλ was undefined due to a denominator of zero. We excluded 3% of the sample this way. (^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) 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.
estimates to the average stimulus; (iii) these results hold both in correlational analyses and when we exogenously manipulate confidence; (iv) central tendency increases with the stimulus magnitude; and (v) response variability decreases with confidence and increases under shorter stimulus duration.
Table 1: Regression coefficients and standard errors for stimulus estimates in the new data set (Models 1 and 2). Model 1 Model 2 (Intercept)
Stimulus 0.637**** (0.010)
AveStim 0.110**** (0.013)
Confidence
Stimulus:Confidence
AveStim:Confidence
Condition
Stimulus:Condition
AveStim:Condition
Observations 71408 71408 Note: Condition refers to the 2000 ms stimulus duration condition. The Confidence coefficients are standardized.
- p<0.05, ** p<0.01, *** p<0.001, **** p<0.
Re-analysis of Earlier Studies
While our experiment has the advantage of being specifically tailored to investigate the predictions associated with the core elements of a Bayesian model, we sought to test the validity of our hypoth- esis more generally. Owing to the recent publication of the Confidence Database (Rahnev et al., 2020), we were able to additionally address the relationship between confidence and the central tendency effect by re-analyzing data from several earlier studies.
Materials and Methods
Data sets
Out of a total of 145 studies contained in the Confidence Database (Rahnev et al., 2020), we identified six studies that elicited continuous reports of both stimulus magnitude and subjective confidence (Table 2). Two of these studies use circular stimuli (motion direction and grating
20 40 60 Stimulus
20
40
60
Estimate
AveStim 30- AveStim 37- AveStim 44-
A
20 40 60 Stimulus
0
2
4
6
8
10
Confidence
100 ms 2000 ms
B
20 40 60 Stimulus
20
40
60
Estimate
Low Confidence High Confidence
C
0 2 4 6 8 10 Confidence
2
4
6
8
Response Variability (SD)
E
Stimulus Duration
2
4
6
8
Response Variability (SD)
(^) 100 ms 2000 ms
20 40 60 Stimulus
20
40
60
Estimate
100 ms 2000 ms
D
F
Figure 2: (A) Subjective estimates as a function of objective stimulus magnitude, shown separately for three within-block average stimulus ranges. (B) Confidence as a function of objective stimulus magnitude, shown separately for different stimulus duration conditions. (C) Subjective estimates as a function of objective stimulus magnitude, shown separately for low confidence (confidence levels 0-3) and high confidence (confidence levels 7-10). (D) Subjective estimates as a function of objective stimulus magnitude, shown separately for different stimulus durations. (E) Response standard deviation as a function of confidence. (F) Response standard deviation under short and long stimulus duration. Error bars indicate 95% confidence intervals.
due to insufficient variation in the average stimulus magnitude in prior experiments. In our new study reported above, we remedied this limitation by exploring a wider range of average stimulus magnitudes to be sufficiently statistically powered. The low variation in stimulus magnitude in the earlier studies also makes our analysis of Pre- diction 5 (Model 3) difficult. We hypothesized that the magnitude of central tendency increases in stimulus magnitude, which should produce a negatie coefficient for stimulus magnitude. The results of Model 3 are mixed, with a negative coefficient for only two studies, and neither of them is significant [AB17, t(12454) = − 0. 77 , p = 0.44; DB19, t(8564) = − 0. 31 , p = 0.75]. Again, this null result is to be expected given insufficient variation in (average) stimulus magnitude. Our new experiment deliberately remedies this shortcoming.
Response variability decreases with confidence
Finally, we again find strong evidence for confidence-dependent variability. Five studies show a significantly negative coefficient: AB17 [t(150) = − 2. 07 , p < 0 .05], DB18 [t(86) = − 3. 96 , p < 0 .001], DB19 [t(466) = − 5. 62 , p < 0 .0001], RZ14 [t(2538) = − 10. 58 , p < 0 .0001], and SP17 [t(4169) = − 4. 60 , p < 0 .0001]. These results are very similar to the ones observed in our own experimental data.
Table 3: Regression coefficients and standard errors for the re-analysis of stimulus estimates (Model 2). Dataset AB17 DB18 DB19 DB20 RZ14 SP (Intercept)
Stimulus
AveStim
Confidence
Stimulus:Confidence
AveStim:Confidence
Observations 15260 14438 8569 9657 8980 14065 Note: The Confidence coefficients are standardized. p<0.05, ** p<0.01, *** p<0.001, **** p<0.
General Discussion
Using data from earlier studies and a new data set, we have established a relationship between confidence and central tendency that conforms with (but is not necessarily unique to) the generic predictions of Bayesian models. First, we showed that the central tendency effect is lower on high confidence trials. Second, we showed that when sensory noise was exogenously increased via a
stimulus duration manipulation, the central tendency effect increased and confidence decreased, demonstrating the causal role of sensory noise. Third, we showed a stronger central tendency effect at higher magnitudes, which is in line with subjective confidence decreasing in stimulus magnitude. Fourth, we showed that across-trial variability in responses decreased in subjective confidence and increased in sensory noise whenever prior uncertainty is relatively large. Our findings bridge several disparate lines of research on confidence and central tendency effects. Some theories assert that confidence judgments in perceptual decision making tasks reflect the posterior probability of being correct—the Bayesian confidence hypothesis (Fleming and Daw, 2017, Meyniel et al., 2015, Pouget et al., 2016, Rahnev et al., 2015, Sanders et al., 2016). While past experimental work on the Bayesian confidence hypothesis has focused on discrete choice tasks (Aitchison et al., 2015), here we analyzed continuous report tasks, which allowed us to relate confidence judgments to the central tendency effect. Our finding that this relationship held across several different stimulus domains (time, visual numerosity, auditory numerosity, line length, motion direction, and grating orientation) lends support to the generality of our conclusions. While our findings are specifically consistent with the Bayesian confidence hypothesis, they might also be compatible with alternative models. For example, Adler and Ma (2018) developed sev- eral models that map probability representations of uncertainty onto confidence in a non-Bayesian way, and presented experimental evidence that some of these models outperformed the Bayesian model in predicting confidence judgments. Li and Ma (2020) developed a different non-Bayesian model, which determined confidence based on the difference in probability between the top two hy- potheses. For our purposes, all of these non-Bayesian models share the key property that confidence is lower when uncertainty (due to sensory noise) is greater. Some authors have argued that previous data supporting Bayesian models can be explained by simpler heuristic models. For example, Huttenlocher et al. (2000) tested a Bayesian model of perceptual judgment similar to the one analyzed here, but their conclusions were questioned by later work showing that the same patterns of behavior could be fit by a model that simply reports an average of recent stimulus magnitudes (Duffy and Smith, 2020). This alternative model has limited explanatory scope for the data we discuss here, because it is silent about the role of confidence in generating judgment and the effects of stimulus duration. Similarly, their account does not explain the patterns of a stronger central tendency effect at higher stimulus magnitudes and of predictable heterogeneity in response variability. A second line of research bridged by our theory is on the central tendency effect. Recent research has shown that cognitive load (which ostensibly increases sensory noise) strengthens the central tendency effect, broadly consistent with Bayesian models of perception. For example, Allred et al. (2016) found that asking participants to memorize six-digit numbers (high load condition) increased the central tendency effect in the estimation of line length, compared to a low load condition in which participants memorized two-digit numbers. Similarly, Olkkonen et al. (2014) found that increasing chromatic noise or the delay between stimulus presentation and estimation increased the central tendency effect in a color estimation task. Relatedly, there is evidence that the central tendency effect is stronger when sensory information is less reliable or when the magnitude distribution is more concentrated around the center (Allred et al., 2016, Ashourian and Loewenstein, 2011, Huttenlocher et al., 2000, Olkkonen et al., 2014), again consistent with Bayesian models. Our study took this line of research one step further, showing that confidence judgments respond to an exogenous manipulation of sensory noise, while explaining significant additional response variance not explained by noise alone.
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