Manipulation check

We predicted that the slow-motion prior would bias participants’ perceptual decisions in the Bias condition, and hence that the Bias condition would require a smaller θ angle in order to lead to the same decision rates as the No-Bias condition. This effect would serve as a basic manipulation check to ensure that the conditions and slow-motion prior had the intended impact. Additionally, it was necessary to quantify this effect in each participant before the experiment such that we could choose the θ values that would be expected to lead to matched perceptual decision rates. To do this, before the main task we ran a staircasing procedure which sampled different θ values in both conditions and then fit psychometric functions to these data, including the bias and sensitivity per condition. The fitted psychometric functions across all pooled participants are shown in Fig 1C, and the bias effect of interest was verified in each individual participant (Fig A in S1 Text). Motion towards the orange region was encoded with negative θ’s and blue region with positive θ’s. We found a clear bias effect, such that the psychometric was shifted leftward when the preferred motion direction from the prior was 35° from the reference in the blue region and rightward when it was 35° from the reference in the orange region, compared to the No-Bias condition (Fig 1C). The point of subjective equality (PSE) when there was a blue bias was -14.57°, meaning that when the line motion direction was 14.57° into the orange region, participants nonetheless chose orange and blue at equal rates. In contrast, the PSE when there was an orange bias was 9.53°, and the PSE in the No-Bias condition was 0.30°. Together, this suggests that the bias manipulation worked as planned.

Motion-direction decisions

After running the staircasing procedure, we chose the values of θ angles that corresponded to a 0.15, 0.35, 0.65 and 0.85 expected probability of choosing blue for each participant. This procedure was followed both for the No-Bias condition and for the Bias condition (for a total of eight different θ values). In the Bias condition, the values corresponding to a 0.15 and 0.35 P(‘Blue’) were taken from cases in which the slow-motion prior induced an orange bias (the lines were oriented to have the orthogonal direction in the orange region), and values corresponding to a 0.65 and 0.85 P(‘Blue’) were taken from cases in which the prior induced a blue bias (the lines were oriented to have the orthogonal direction in the blue region). Overall, this meant that we chose stimulus levels that were expected to lead to matched perceptual decision rates between conditions. Crucially, however, the perceptual decision rates in the Bias condition would be due in part to the slow-motion prior; whereas the perceptual decision rates in the No-Bias condition would be solely driven by the stimulus evidence. These four θ values in each condition were paired in all possible combinations for the two intervals of the confidence forced-choice task, except for pairs repeating the identical θ in the same condition.

To check that the perceptual decision rates were matched overall across the two conditions, we ran a logistic mixed effects model on Perceptual Decision including Expected P(‘Blue’) (as an index of the stimulus difficulty), Condition (Bias vs No-Bias), and their interaction as fixed effects, as well as by-participant random intercepts. If the chosen θ values from the staircasing procedure failed to produce matched perceptual decision rates between conditions, this would generate an interaction between Expected P(‘Blue’) and Condition. Indeed, a significant interaction, χ²(3)=302.97, p < 0.001, BF₁₀ = 1.19 × 10⁶⁰, revealed that the decision rate per stimulus setting additionally depended on the condition, making our analysis more complex. The perceptual decision rates were more extreme in the Bias condition (Fig 2A), with higher odds of choosing blue when the stimulus was blue (Expected P(‘Blue’) of 0.65: odds ratio of the Bias to the No-Bias condition (OR_Bias/No-Bias) = 1.55, 95% confidence interval (CI) [1.32, 1.81], Z = 8.75, p < 0.001; Expected P(‘Blue’) of 0.85: OR_Bias/No-Bias = 1.13, 95% CI [0.92, 1.38], Z = 1.81, p = 0.071), and lower odds when the stimulus was orange (Expected P(‘Blue’) of 0.15: OR_Bias/No-Bias = 0.54, 95% CI [0.42, 0.69], Z = -7.91, p < 0.001; Expected P(‘Blue’) of 0.35: OR_Bias/No-Bias = 0.49, 95% CI [0.41, 0.58], Z = -13.43, p < 0.001), compared to the No-Bias condition, although this contrast was not significant for the strongest blue setting. The more extreme perceptual decision rates in the Bias condition suggest that the bias effect was stronger during the main experimental trials, compared to what was calibrated in the staircasing procedure. To quantify this, we estimated the strength of the bias effect in the experiment by fitting a cumulative normal distribution function with one symmetrical bias level (+/- μ_Bias) for the orange and blue directions relative to the No-Bias condition, which we fit separately as a baseline (Fig 2A). The sensory sensitivity was also allowed to differ between conditions, as it was possible that the influence of the prior impacted the overall sensitivity. This was done for each participant and the resulting mean μ_Bias across participants was 20.81° (Fig 2A) with a SEM of 2.68° (individual fits in Fig B in S1 Text), indeed indicating a stronger bias effect than in the staircasing (Fig 1C). Hence, we needed to account for this residual difference in perceptual decision rates in our analysis when testing for a confidence bias.

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Fig 2. Motion-direction and confidence decision results.

A. Motion-direction decisions. Choice results for the first-order orange/blue decisions in each condition. The four θ values in each condition were targeting 0.15, 0.35, 0.65 and 0.85 probabilities of choosing blue (P(‘Blue’)). In the Bias condition, this consists of θ values targeting 0.65 and 0.85 P(‘Blue’) when inducing a blue bias (blue points) and targeting 0.15 and 0.35 P(‘Blue’) when inducing an orange bias (orange points). The targeted perceptual decision rates are shown with horizontal dashed gray lines. Although we targeted these matched rates between conditions, the observed perceptual decision rates were less extreme in the No-Bias condition than in the Bias condition. Error bars reflect the SEM across participants. The psychometric functions were fitted to each individual participant to capture the strength of the bias from the slow-motion prior relative to the No-Bias baseline, and to capture possible differences in sensitivity between conditions. The functions shown here reflect the mean fit bias (μ) and noise (σ) values across participants (individual fits in Fig B in S1 Text). The mean μ_Bias was 20.81, mean μ_No-Bias was 0.44, mean σ_Bias was 18.50, and mean σ_No-Bias was 20.46. B. Non-parametric confidence bias results. On the x axis is the difference in the logit transformed rates of choosing the expected color between the Bias and No-Bias condition. Each gray point represents this difference for one pair of stimulus settings assigned to the two intervals of a confidence pair (16 points per participant). On the y axis is the probability of choosing the Bias condition interval as the more confident interval. When there is no difference in perceptual decision response proportions between conditions (x = 0), and the perceptual difficulties are therefore matched for that pair of settings, we expect equal confidence choices between conditions. If this is not the case, this indicates a confidence bias. The red psychometric function captures the fit cumulative normal distribution function to the relationship between this response proportion difference and the confidence choice rates from the pooled data. The PSE is shifted leftward to -1.47 (SEM = 0.24 from performing this analysis in individual participants, see Fig C in S1 Text), indicating a confidence bias favoring the Bias condition. C. CFC-model confidence bias results. We fit the cfc-model [33] across 100 bootstrapped runs and extracted the fit confidence bias parameter β from each. Each red point corresponds to one fit β parameter result. The boxplot shows the median, interquartile range (IQR) with hinges showing the first and third quartiles, and vertical whiskers stretching to most extreme data point within 1.5*IQR from the hinges. The horizontal black line at y = 1 shows the expected β if there was no confidence bias. D. Quality of CFC-model fit. Predicted against observed confidence choice rates across all participants. Each point shows a pair of stimulus settings with one interval in each condition and a given set of two orange/blue responses, and the size of the point reflects the number of trials. The closer the points are to the x = y line, the better the model is at modeling confidence choice rates.

https://doi.org/10.1371/journal.pcbi.1013826.g002

Confidence forced-choices

We sought to explore whether, after accounting for any residual difference in orange/blue decision rates, there is still a confidence bias favoring one of the conditions. If so, this could indicate a different influence of the prior on perception and confidence, which could then be further explored using Bayesian confidence modeling. Because the orange/blue decision rates were not perfectly matched across conditions, and following our pre-registered plan for this potential situation, we used a non-parametric approach to assess confidence bias, which accounted for potential mismatches in perceptual decision rates. For this analysis we excluded trials with the same condition in both intervals. We took the difference in logit transformed orange/blue response rates for the preferred stimulus between conditions for every stimulus setting pair for every participant, following previous work [35]. For example, for the stimulus settings expected to lead to a 0.85 probability of choosing blue in each condition, we took the observed rates of choosing blue, took their log-odds to transform them to the (- ∞ ,∞) domain, and took the difference between them. A difference of zero then indicated matched perceptual decision rates, positive differences indicated more extreme decision rates in the Bias condition, and negative differences indicated more extreme decision rates in the No-Bias condition. Then, regardless of whether these values are shifted positively overall, as we expect due to the result of more extreme perceptual decision rates in the Bias condition, we can fit a cumulative normal distribution function to the relationship between these transformed response proportion differences and the probability of choosing the Bias condition as the more confident interval. If the PSE is at 0, this indicates that when participants have matched perceptual decision rates they also have matched confidence choice rates between conditions, suggesting no confidence bias. If, however, the PSE is shifted away from 0, this indicates a confidence bias. We found a clear negative shift of this function, fit to the pooled participants, with a PSE of -1.47 (SEM = 0.24 from performing this analysis in individual participants, see Fig C in S1 Text), suggesting a confidence bias favoring the Bias condition (Fig 2B). When perceptual decision rates are matched (transformed response proportion difference = 0), the fit psychometric function predicts the participants to still choose the Bias condition as the more confident interval at a rate of 0.62. Note that this reflects the degree to which confidence was more biased towards the Bias condition than perceptual decisions, rather than the overall degree to which the slow-motion prior influenced confidence.

Quantifying confidence bias: CFC-model

To further quantify this confidence bias, we fit the cfc-model developed in previous work [33] for use with the confidence forced-choice paradigm. This model includes a confidence bias term, β, which, for a single task or condition, scales the estimated sensory sensitivity of a participant, such that β  = 1 captures correctly estimated sensitivity, β  > 1 captures an overestimation of sensitivity – indicating overconfidence –, and β  < 1 captures an underestimation of sensitivity – indicating underconfidence. With two different conditions, the model fits β as the ratio of biases between them (one can only estimate whether participants are over- or under-confident in one condition relative to the other). In our case, we chose the ‘No-Bias’ condition as baseline, so a fit β larger than 1 would indicate an overconfidence in the ‘Bias’ condition. We fit the model to the pooled normalized data from all participants across 100 bootstrapped runs, which revealed the mean fit β to be 1.44 (SEM = 0.006; Fig 2C). The quality of the model fit is shown in Fig 2D. These results, in agreement with the non-parametric analysis above, indicate a confidence bias favoring the Bias condition. While this model can be used to naively quantify a confidence bias relative to first-order perceptual decision rates, it does not provide an account of the computations occurring in our design, under the influence of the slow-motion prior. To investigate these computations, in exploratory analyses we consider a Bayesian decision and confidence model. We use this model as a normative benchmark, specifically focussing on a Bayesian use of the slow-motion prior. We then explore several alternative generative models that may help capture deviations from this Bayesian baseline, in light of the observed confidence bias.

Bayesian model

In the Bayesian model, we first need to define the prior to represent the preferred motion directions that are orthogonal to the lines’ orientation. Even though we relied on the slow-speed prior to bias one condition, this prior has a direct impact on perceived direction [30], and since we are measuring perceived direction rather than speed, we directly model the prior in terms of the preferred motion directions. This motion direction prior is implemented as the sum of two Gaussians of equal variance, whose means are separated by 180° (Fig 3A). In the No-Bias condition, these means are 90° away from the reference in either direction. In the Bias condition, one mean is 35° away from the reference in the bias direction, and the other is 145° away from the reference in the opposite direction. This latter component is outside the allowed decision region and thus will have a negligible effect in our model (see below). The likelihood is modelled as a single Gaussian centered around the internal signal generated from the stimulus with added internal noise. These combine to form the posterior distribution (Fig 3A). Orange/blue perceptual decisions in this model are based on the area under the posterior on either side of the sensory reference within the allowed decision region (from -90° to +90°). When this area is larger on the negative side, an “orange” decision is made, and when it is larger on the positive side, a “blue” decision is made (Fig 3A–B). Following the literature capturing Bayesian confidence as the estimated posterior probability of being correct [5–10], confidence in the decision is then equal to the proportion of the total area (within the allowed decision region) that falls on the chosen side (thus varying between 0.5 and 1). The confidence forced-choice is based on whichever of these confidence values is larger across two intervals. The free parameters of this Bayesian model are the standard deviation of each of the two Gaussians that form the Gaussian mixture prior, σ_P, and the standard deviation of the likelihood, σ_L, that represents the sensory uncertainty. These were fit individually for each participant to find the parameters that could best explain their orange/blue decisions. We used a simplification to deal with computational complexity and considered the prior in the Bias condition to be a single Gaussian centered around 35° towards the bias direction. Since most of the other Gaussian fell well outside the allowed decision region, this component had a negligible impact. The results from these model fits are shown in Fig 3C by simulating from the full unsimplified model, showing that the fit σ_P and σ_L parameters can still well account for motion-direction decisions.

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Fig 3. Bayesian model.

A. Sketch of Bayesian model. The prior on perceived motion direction is a sum of two Gaussians of equal standard deviation (σ_P), with means at the orthogonal motion directions (+/-90° from the reference in the No-Bias condition and either 35°/-145° or -35°/145° from the reference in the Bias condition). The likelihood comes from the internal signal, r, sampled from the stimulus with added internal noise (σ_L). The prior and likelihood are combined to form the posterior distribution. The shaded orange area under the posterior leads to the first-order decision and confidence. B. Posteriors per condition from Bayesian model. Posterior distributions from the Bayesian model across 50 samples of an internal signal from the stimulus to form the likelihood, only including samples leading to an orange decision. These reflect posteriors across different trials of the same stimulus intensity and condition, which led to the same perceptual decision. The confidence in these trials is captured by the ratio of area under the orange part of the posterior (chosen side) to the area under the gray part of the posterior (non-chosen side), between the red vertical lines (allowed decision region). The prior tends to pull the maximum of the posterior in the No-Bias condition away from the reference towards -90°, so the ratio of area under the orange versus gray parts of those posteriors tends to be higher, suggesting higher average confidence in the No-Bias condition despite matched choice rates. C. Modeled motion-direction decision results. Results from the fit Bayesian decision model, simulated across 1001 stimulus intensities ranging continuously (in steps of 0.1) from -50 to 50 using the best-fitting σ_P and σ_L parameters for each participant, against the data. D. Modeled confidence choice results. In two consecutive intervals, a Bias and a No-Bias stimulus were presented (or in the reverse order). Because confidence refers to an estimate that a perceptual decision is correct, we consider each possible combination of difficulty and perceptual decision accuracy. ‘Strongly correct’ refers to trials with either an extreme blue stimulus (Expected P(‘Blue’) of 0.85) and a ‘Blue’ perceptual decision, or an extreme orange stimulus (Expected P(‘Blue’) of 0.15) and an ‘Orange’ perceptual decision. ‘Strongly incorrect’ refers to trials with those extreme stimulus settings but opposing perceptual decisions. The ‘Weakly correct’ and ‘Weakly incorrect’ categories follow the same patterns but with the more difficult stimuli: Expected P(‘Blue’) of 0.35 or 0.65. Results split further by orange versus blue stimulus settings and perceptual decisions are shown in Fig D in S1 Text. The x-axis captures this combination of difficulty and perceptual decision accuracy in the Bias condition interval —also indicated by thin, colored, dashed vertical lines—. The combination of difficulty and perceptual decision accuracy in the No-Bias condition interval is captured by color. The colors correspond to the vertical lines of the Bias condition, such that where a colored point intersects a dashed vertical line of the same color, there is the same degree of difficulty and perceptual decision accuracy across the two conditions, and an unbiased confidence observer would choose both conditions at equal rates for their confidence. The y-axis captures the confidence choice rates favoring the Bias condition. Simulated confidence choice rates from the Bayesian model are shown by the solid lines and the observed confidence data are shown by circles whose size reflects the number of trials. The overall shift upward of the confidence data points compared to the model predictions indicates that observed confidence choices favored the Bias condition more often than would be expected of the Bayesian model, which actually predicts a bias in the opposite direction.

https://doi.org/10.1371/journal.pcbi.1013826.g003

We then used this fit model and investigated the confidence forced-choice patterns that would be expected of a Bayesian observer given the first-order perceptual decision rates that we found (Fig 3D, solid lines), and compared them to the observed confidence patterns (Fig 3D, data points). For the analyses based on the Bayesian model, we removed all trials with the same condition in both intervals. Because of the confidence forced-choice task structure and the fact that confidence refers to an estimate that a perceptual decision is correct, we should distinguish confidence choices depending on the stimulus difficulty and perceptual decision (blue and orange) of each of two intervals on a given trial. For example, if the Bias condition interval is easy and the No-Bias condition interval is difficult, there is a high probability of a confidence choice favoring the Bias condition. Likewise, if the Bias condition perceptual decision is correct and the No-Bias condition perceptual decision is incorrect, there would be a tendency for the confidence choice to favor the Bias condition with all else equal. Hence, the confidence choice rates in Fig 3D (y-axis) can be viewed as a function of the combination of stimulus difficulty and perceptual decision accuracy in each condition, with the Bias condition on the x-axis and the No-Bias condition in color. ‘Strongly correct’ trials refer to those with extreme θ values targeting a 0.85 rate of either ‘Blue’ or ‘Orange’ perceptual decisions, and on which the participant made the corresponding, expected ‘Blue’ or ‘Orange’ perceptual decision. ‘Strongly incorrect’ trials refer to those with the extreme θ values but on which the participant made the opposing perceptual decision. ‘Weakly correct’/’Weakly incorrect’ trials follow the same pattern but refer to trials with the more difficult θ values targeting a 0.65 rate of either ‘Blue’ or ‘Orange’ perceptual decisions. When the combination of difficulty and perceptual decision accuracy is matched across conditions (when the color of the No-Bias condition intersects the vertical dashed line of the same color of the Bias condition), an unbiased confidence observer would have a confidence choice rate of 0.5. Instead, observed confidence choice rates are shifted towards the Bias condition overall, suggesting the same confidence bias seen in Fig 2B. However, it is important to note that the stimulus difficulties shown are grouped by targeted perceptual decision probability, and we were not perfectly successful at achieving these decision rates. So, the shift of the data points towards the Bias condition cannot be interpreted in isolation, as it may reflect the failure to match perceptual decision rates. Instead, the data must be compared to the model predictions in these Figs. The raw confidence choices are also in striking contrast to the predicted confidence choice rates of the Bayesian model, shown in the solid lines of 3D, which actually show a confidence bias in the opposite direction.

The expected rates of confidence choices favoring a condition depend not only on stimulus and perceptual decision accuracy, but on the relative strengths of the posteriors in each condition, and therefore on the use of the prior. In the Bayesian model, because of the shape of the Gaussian mixture priors, the distribution of posteriors across different samples of the stimulus (or in other words across trials) is quite different between the conditions (Fig 3B). In the No-Bias condition, the maxima of these posteriors are pulled towards the peaks of the prior at either + or -90°, so a large proportion of the area is distributed relatively far from the sensory reference on the chosen side (Fig 3B). Having a higher proportion of the area on the chosen side like this leads to higher confidence, and therefore the Bayesian observer will on average have higher confidence in the No-Bias condition despite matched perceptual decision rates. This pattern is shown clearly in the model simulations of the Bayesian confidence observer given the first-order decision results (Fig 3D). This suggests that, if our participants used the information from the slow-motion prior in a Bayesian optimal manner for their confidence, there would have been a confidence bias favoring the No-Bias condition, which is the opposite of what is seen in the data. So, we next explore some alternative confidence models that might better explain the patterns found.

Distance-to-Criterion confidence model

In the Bayesian confidence model, confidence is based on the area under the full Bayesian posterior. However, another possibility for computing confidence is to base it off of a single sample from the posterior as a heuristic, in which case the brain does not need to perform the full integration (see, e.g., [36]). Such posterior sampling is still debated in the literature, but there is some evidence in support of it [37]. In this model, confidence is proportional to the distance between the posterior sample and the sensory reference, or ‘criterion’ (Fig 4A). Samples further from the criterion very clearly support that perceptual decision and lead to higher confidence, whereas samples close to the criterion indicate uncertain decisions and lead to lower confidence. We also assume that confidence would never oppose the perceptual decision, so we only considered samples for confidence that fell on the same side of the criterion as the perceptual decision. This confidence model has no free parameters beyond the first-order sensitivity parameters fit to perceptual decisions and used in all models. We simulated confidence forced-choices from it to investigate whether it could better account for the data. This revealed even more extreme predictions of a confidence bias favoring the No-Bias condition than in the Bayesian model, suggesting this model could only poorly capture the confidence data (Fig 4B). Beyond posterior sampling, other possible heuristics include basing confidence on the mean of the posterior distribution, but this is more computationally costly, and does not substantially change the results (Fig E in S1 Text).

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Fig 4. Distance-to-Criterion confidence model.

A. Sketch of Distance-to-Criterion model. In the Distance-to-Criterion model, the prior, likelihood, and resulting posteriors are the same as in the Bayesian model. Motion-direction decisions also work identically. However, at the confidence level, instead of computing the full Bayesian posterior probability of being correct, this model bases confidence on the absolute distance, dist, between a single sample from the posterior, r_c, and the reference, or criterion (c). Larger distances lead to higher confidence and smaller distances lead to lower confidence. B. Confidence predictions from Distance-to-Criterion model against data. Confidence forced-choice predictions simulated from the Distance-to-Criterion model (solid lines) against the confidence data (points) in the same format as in Fig 3D. Like in the Bayesian model, the solid lines well below the data points show that this model predicts more confidence choices favoring the No-Bias condition, unlike what is seen in the data.

https://doi.org/10.1371/journal.pcbi.1013826.g004

Decoupled Prior confidence model

The Bayesian and Distance-to-Criterion models both base confidence on the same posterior as the first-order decisions, and fail to capture the confidence bias seen in the data favoring the Bias condition. It is possible that the confidence data can be better explained by using a weaker likelihood relative to the prior for computing confidence than for perceptual decisions, leading to a stronger influence of the prior in confidence, similar to the pattern found in previous work [18]. In other words, the strength of the prior that is used in confidence may decouple from that used in the perceptual decision. In the Decoupled Prior (DP) model, the variance of the likelihood is scaled by a multiplicative factor w during the confidence computation to capture a relative over- or underestimation of the likelihood precision and hence an over- or underweighting of the likelihood information relative to the prior (Fig 5A). When w = 1, the likelihood variance is correctly estimated, the relative precision-weighting is optimal, and this model is identical to the Bayesian confidence model. When w > 1, the likelihood variance is overestimated and therefore it has a weaker effect on the computation of confidence, and the prior is relatively overused in confidence. When w < 1, the likelihood variance is underestimated and therefore it has a stronger effect and the prior is relatively underused in confidence. In essence, this model is a naive implementation of the confidence bias parameter in the Confidence-Forced-Choice model [33] where large values of w correspond to small values of β. We fit this model with its free parameter w to the group data using a maximum likelihood estimation approach. This revealed a best-fitting parameter of w = 1.54, indicating that the prior is overused relative to the likelihood in confidence. This is in agreement with the results of previous work which also found an overweighting of prior information at the confidence level using a similar model implementation [18]. It may also align with work finding that people base inference on an underestimation of their own uncertainty [38], though that work focussed only on likelihood variance, assuming uninformative priors.

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Fig 5. Decoupled Prior confidence model.

A. Sketch of Decoupled Prior model. In the Decoupled Prior model, confidence is computed the same way as in the Bayesian model, by taking the area under the posterior distribution on the chosen side. However, in forming this posterior, participants over- or underestimate the precision of the likelihood relative to the prior, leading to the prior being relatively over- or underused in the confidence computation. The over- or underestimation of the likelihood precision is captured by a scaling term w that acts as a multiplicative factor on the variance of the likelihood. When this value is 1, the variance is correctly estimated and this model is the same as the Bayesian model. When w < 1, the prior is relatively underused and when w > 1 the prior is relatively overused in confidence. B. Confidence predictions from fit Decoupled Prior model against data. Confidence forced-choice predictions simulated from the fit Decoupled Prior model (solid lines) with the best-fitting w = 1.54, against the confidence data (points). Although the model fits the data better than the Bayesian model, it still underpredicts the confidence bias favoring the Bias condition compared to the data.

https://doi.org/10.1371/journal.pcbi.1013826.g005

Simulations from this fit model are shown in Fig 5B. These demonstrate the ability of this model to better capture the confidence bias favoring the Bias condition than the Bayesian and Distance-to-Criterion models. However, this model predicts confidence to be particularly high in the Bias condition when the perceptual decision is in agreement with the prior, and to be particularly low when the perceptual decision goes against the prior. This leads to the prediction that confidence choices will favor the Bias condition when the Bias condition perceptual decision agrees with the bias direction. But, the model also predicts that confidence choices will favor the No-Bias condition when the Bias condition perceptual decision goes against the bias direction, which cannot explain the data for these cases (Fig 5B, the red and pink vertical dashed lines).

Prior-Congruent Evidence (PCE) confidence model

From the previous models, it seems that confidence forced-choices deviate from the Bayesian model and are biased towards the condition that is more driven by the prior, but not simply by overweighting the prior (or downweighting the likelihood) in the integration to form the posterior. This suggests that confidence may instead be biased towards the prior by an additional source of evidence, and a plausible candidate for this additional cue is the prior-congruent evidence (PCE) available in the line-motion stimulus itself. The prior-congruent evidence refers to the component of the stimulus motion speed that is projected on the axis orthogonal to the lines’ orientation. It is not tied to the orange/blue decision made, and is separate from the posterior evidence on which that decision is based. In an extreme case, confidence could be based solely on this PCE, ignoring the posterior evidence entirely and dissociating from the perceptual decision. This is the case illustrated by the PCE confidence model. Confidence here was directly proportional to the length of the prior-congruent vector component of the sensory evidence (a noisy measurement of the stimulus), as shown in Fig 6A. There were no additional free parameters for confidence in this model. Because there tended to be more of this prior-congruent evidence in the Bias condition (when line motion directions were closer to the preferred orthogonal direction), this model predicts a confidence bias favoring the Bias condition (Fig 6B). However, the predicted confidence bias is stronger than what we see in the data. This is not surprising, as this model serves primarily to illustrate the influence of the PCE cue. But, it is included for completeness as it is still a possible confidence strategy, albeit an extreme one.

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Fig 6. Prior-Congruent Evidence models.

A. Sketch of prior-congruent evidence. In the line-motion stimuli, the evidence that is congruent with the prior is the component of the motion that is in the preferred direction, orthogonal to the lines’ orientation, shown as the yellow vector component. The sensory evidence is shown in red and reflects the mean of the likelihood distribution, and the prior-incongruent component, parallel to the lines’ orientation, is shown in black. Stimuli with high PCE have a motion direction that is closer to the preferred one, and therefore a longer vector component along that dimension. In the PCE confidence model, confidence was directly proportional to this PCE, and was not based on the posterior evidence or on the orange/blue decision made. B. Confidence predictions from Prior-Congruent Evidence (PCE) model against data. Confidence forced-choice predictions simulated from the PCE model (solid lines) against the confidence data (points). This model predicts more confidence choices favoring the Bias condition, in line with, though more extreme than, what is seen in the data. C. Confidence predictions from Weighted Posterior and Prior-Congruent Evidence (WPPCE) model against data. Confidence forced-choice predictions simulated from the WPPCE model with best-fitting ɑ = 0.34 (solid lines) against the confidence data (points). In the WPPCE model, confidence is based on a weighted combination of two sources of evidence, the prior-congruent evidence shown in (A), as well as the posterior evidence (as in the Bayesian model).

https://doi.org/10.1371/journal.pcbi.1013826.g006

Weighted Posterior and Prior-Congruent Evidence (WPPCE) confidence model

In the WPPCE model, the PCE acts as an additional confidence cue, but not the sole information on which confidence choices are based. Confidence is based on a weighted combination of the Bayesian posterior evidence (BPE), which drives the first-order decisions, as well as the degree of prior-congruent evidence (PCE). The influence of each of these two evidence sources is determined by the weighting parameter, ɑ, such that confidence is proportional to the sum of (1-ɑ)*BPE and ɑ*PCE. Higher ɑ values then lead to more contribution from the PCE. When ɑ = 1, the PCE is the only cue to confidence and this is identical to the PCE model, and when ɑ = 0, the BPE is the only cue to confidence and this is identical to the Bayesian model. We fit this free weighting parameter to the group data using a maximum likelihood estimation approach, and found that a value of ɑ = 0.34 could best explain the data. Because the BPE and PCE use different units, it is difficult to interpret the raw ɑ value. However, simulations from this fit model are shown in Fig 6C, and can capture the data well qualitatively, showing a similar confidence bias for the Bias condition. In an exploratory analysis, we also examined whether the strength of deviation away from Bayesian confidence, as captured by the weight of the PCE (ɑ), correlates to the strength of the prior at the perceptual level. While it did not reach significance, we did find a trend towards a positive relationship between these variables (Fig F in S1 Text), which could be explored in future research.

Efficient Coding model

We also consider the possibility, in line with previous work [39–41], that there is efficient encoding of sensory evidence such that signals that are more likely to occur are encoded more precisely. This efficient encoding means that the likelihood precision itself is constrained by the prior, before the computation of the Bayesian posterior. So, this is another way in which the prior may effectively have a stronger impact than what is expected of the Bayesian model. Although efficient coding would dictate perceptual decisions as well, it impacts the shape of the posterior distributions, so it is possible that it can explain the observed dissociation between first-order decisions and confidence. We implemented an efficient encoding model following the approach of Wei and Stocker [39], which included a parameter capturing the variance of the prior, σ_P, (as in the Bayesian model), a parameter capturing neural noise, σ_N, and a parameter capturing transduction noise, σ_T, in the mapping from external stimulus to internal evidence. Transduction noise is necessary to account for variance in perceptual decisions and confidence values across trials, as otherwise the model would predict identical outcomes every time the same external stimuli are repeated. The model was fit to the perceptual decisions to give the best-fitting σ_P, σ_N, and σ_T parameters for each participant, and we then assessed its ability to capture the observed confidence patterns. While the predictions of this model better match the confidence data than the Bayesian model without efficient coding, it still underpredicts the confidence bias found favoring the Bias condition (Fig 7B). A more naive model that mirrored the Bayesian one but allowed the likelihood precision to differ between conditions also could not explain the observed bias at the confidence level (Fig G in S1 Text). So, while the results still agree with previous findings arguing that the brain may implement efficient coding, they also suggest that efficient coding cannot itself explain the dissociation found between the information used in perceptual decisions versus confidence.

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Fig 7. Efficient Coding model.

A. Sketch of efficient encoding. The shape of the likelihood distribution on a given trial is constrained by how probable the stimulus signal is, based on the prior. Signals closer to the peak of the prior will be encoded more precisely, as indicated by the shaded red likelihood. The grey distributions reflect the encoding in a homogenous sensory space with uniform variance (neural noise). They map to the non-uniform likelihoods in physical space according to an inverse mapping described in Methods. B. Confidence predictions from Efficient Coding (EC) model against data. Confidence forced-choice predictions simulated from the EC model (solid lines) against the confidence data (points). Although this model better predicts confidence compared to the Bayesian model without efficient coding, it still cannot account for the observed bias at the confidence level.

https://doi.org/10.1371/journal.pcbi.1013826.g007

Model comparison

To compare the models quantitatively, we computed the AIC for each, therefore also penalizing the models with more free parameters. This revealed an AIC_Bayesian = 30767.85, AIC_{Distance-to-Criterion} = 38774.61, AIC_DP = 29831.18, AIC_PCE = 31680.98, AIC_WPPCE = 25855.11, and AIC_EC = 25943.83 (Fig 8A). The best model to explain the group confidence forced-choice data is the WPPCE confidence model (Fig 8B), in which confidence choices are based not only on the posterior evidence that formed the first-order decision, but additionally on the prior-congruent evidence as a separate cue. We additionally fit and compared the six models for each individual participant. This revealed the WPPCE confidence model to be the winning model in 15 out of the 24 participants, the EC confidence model to be the winning model in 7, and two participants for whom the model comparison was inconclusive (Fig 8C). Taken together, these results suggest that, relative to their first-order behavior, participants have a confidence bias that favors the condition in which the slow-motion prior more strongly influences decisions. Further, this bias can best be explained by a model in which participants use the prior-congruent information as an additional cue to inform their confidence, in combination with the Bayesian posterior evidence used for the motion-direction decisions.

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Fig 8. Model comparison.

A. AIC group results. AIC comparison of models fit to the pooled group data. B. Model goodness of fit. Predicted against observed confidence choice rates from each of the models. Each point reflects a pair of stimulus settings with one interval in each condition and a given set of orange/blue responses, and the size of the point reflects the number of trials. The closer the points are to the x = y line, the better the model is at modeling confidence choice rates. Confidence choice rates reflect the probability of choosing the Bias condition as the more confident interval. So, points above the identity line reflect the model underestimating the frequency of confidence choices for the Bias condition, and points below the identity line reflect the model overestimating the frequency of confidence choices for the Bias condition. C. Participant-wise AIC results. AIC comparison of models fit to each individual participant. The winning model for each participant is shown outlined in black, when there was a conclusive winner (all except participants 11 and 21). The WPPCE model was the winning model for 15 participants and the EC model was the winning model for the remaining 7.

https://doi.org/10.1371/journal.pcbi.1013826.g008

Limitations

Although this work extends previous findings to what is often considered a different category of prior, it is still only one example of a long-term prior, and it is possible that others act in different ways. In line with this possibility, recent work examining confidence under the influence of natural image statistics found different effects of an orientation prior versus a lighting prior on confidence [19]. It would therefore be valuable to find ways to investigate other examples. Also, although we see a similar confidence-performance dissociation with a long-term prior as was previously found with high-level priors, we cannot draw any conclusions about a shared mechanism of action. Future studies testing for correlations between the strengths of these dissociations across different priors, as well as neuroimaging work exploring this effect in terms of neural implementation would be useful to clarify this.

While the conditions we created and models we tested allowed us to quantitatively compare prior-use in decisions and confidence, they also came with some limitations. Our use of a binary discrimination task for the perceptual judgments allowed us to build the conditions such that the prior only influenced decision rates in the Bias condition. However, there could still be residual effects of the prior on perception in the No-Bias condition that do not impact performance, which could be explored in future work using a reproduction task. It is also possible that participants formed other short-term priors about the task structure. We tried to avoid this by having the line orientations and absolute motion directions sampled nearly uniformly across the full 360° range, making conditions and stimulus settings difficult to detect. But, future work could instead use a more conspicuous task structure and then model participants’ learning of it. This would make a more complete Bayesian Optimal Observer model, whereas here in our Bayesian model we focus specifically on a Bayesian use of the slow-motion prior in decisions and confidence. We also note that here we chose to model the prior in terms of the directional bias that is created in these stimuli, rather than modeling the prior over speed. This was for computational tractability of the confidence models as well as simplicity of interpretation, since the directional prior reflects the focus of the motion direction task better. Further, the effect of the slow speed prior on the perceived motion direction bias has been explored extensively in previous work [26,27,32,60] and was not central to our research questions, which only required that we could well capture the resulting bias effect in decisions and confidence. Still, it could be interesting for future work that is more focussed on understanding the nature of the slow speed prior to explore these questions in a speed estimation task. Finally, though the WPPCE model could best explain the data, and predicts the confidence bias of interest, it still does not perfectly capture the observed confidence patterns in their entirety. This may be in part due to noise in the data, particularly since the confidence forced-choice paradigm was complex for participants and we could not force balanced trial numbers across all combinations of stimuli and responses, so some settings were very sparsely sampled. However, it would be valuable to continue exploring extensions to the model that may allow it to better predict confidence choices, such as quantifying metacognitive noise as an additional parameter.

Ethics statement

Participants were compensated with 10€ per hour and gave signed, informed consent before starting the experiment. The local ethics committee (Comité d’Ethique pour les Recherches en Santé) approved the study, which conformed to the Declaration of Helsinki.

Participants

We pre-registered that we would test 25 participants, ensuring at least 20 clean datasets that met the prespecified inclusion criteria, which most importantly included that they showed the basic effects of the prior – that the perceptual decision rates were biased towards the orthogonal motion direction in the Bias condition. We tested 25 participants and later excluded the data from one of them because they did not show the basic effect of the prior, leaving data from 24 participants (7 male, 17 female) included in our analyses. Participants were between 21 and 39 years of age (M = 24.76, SD = 4.28), reported to have normal or corrected-to-normal vision and were fluent in English.

Setup

The experiment was programmed using MATLAB [61] and Psychtoolbox-3 [62–64]. Participants sat in a dark room with their head on a chin and forehead rest placed 57 cm away from a CRT monitor (Vision Mater Pro 454) with a display resolution of 1,920 x 1,200 (refresh rate = 60Hz).

Procedure

ASA procedure.

At the beginning of the first session, after receiving verbal instruction about the stimuli, participants completed a short stimulus training consisting of 80 trials across five mini-blocks (16 trials each) in which the line motion stimulus was shown and they input their answer about whether the lines were moving towards the orange or blue region using the mouse. For the first mini-block, the true line motion direction was displayed. In addition, the lines started at high contrast so that participants could clearly see and understand the stimuli, and got lower contrast in later blocks (Weber contrasts: 22%, 22%, 6%, 6%, 6%), and their movement was displayed for a longer duration at first and got progressively shorter in each block (durations: 1000ms, 266ms, 133ms, 133ms, 133ms). After this training, participants completed a staircasing procedure, which took approximately 10 minutes. We used this to select the appropriate θ values for each individual that would target the desired perceptual decision rates in each condition, given their performance and the strength of their bias. We ran an accelerated stochastic approximation (ASA) staircasing procedure [65] once in the No-Bias condition, with the reference point exactly 90° away from both preferred, orthogonal line orientation directions. For the Bias condition, we ran two staircasing procedures, to capture the bias effect from the influence of the prior in both directions (a blue-bias and an orange-bias), by setting the preferred, orthogonal line orientation either 35° towards the blue region or towards the orange region from the reference (Fig 1A). For each of these staircasing procedures, four staircases controlled the motion direction (relative to the reference) and were set to converge to two decision probability thresholds of 0.25 and 0.75. The staircasing procedures were all interleaved, meaning that the conditions were also interleaved. We then fit a cumulative normal distribution function (Ф) with a free noise (σ_asa) and bias (µ_asa) parameter to the data from each of these three separate ASA procedures:

where P is the decision probability and θ is the stimulus value used, which was the angle between the reference and the motion direction (Fig 1C). There was no lapse rate parameter to secure robust fitting during the experiment. These fit psychometric functions were then used to select the stimulus values that targeted probabilities of choosing blue (P(‘Blue’)) of 0.15 and 0.35 when there was an orange bias, and 0.65 and 0.85 when there was a blue bias in the Bias condition, and probabilities of choosing blue of 0.15, 0.35, 0.65, and 0.85 in the No-Bias condition. These stimulus values were used for the main task. We did not include all four decision probabilities with both bias directions (blue and orange) because we did not want to examine cases in which the bias worked against the stimulus and hence the Bias condition actually required more stimulus information than the No-Bias condition. Including those trials would have substantially increased the length of the experiment, and would have caused an imbalance in the number of trials across conditions, without adding much to our analyses. Such extreme Bias condition cases (in terms of θ) may also have risked the participants noticing the two conditions.

Analysis

As pre-registered, we removed any trials with reaction times that were longer than 8 seconds for any decision, or shorter than 100ms for any motion direction decision. It was also possible that some participants failed to demonstrate the expected perceptual bias, at least for some stimulus settings (such as for a particular direction of bias, or at some difficulty levels), due to individual differences in the strength of the prior. So, we also removed the data from any stimulus settings for any given participant in which the basic bias effect was not present. This was determined to be the case if, as pre-registered, the selected θ value (from the ASA procedure) targeting a given perceptual decision rate in the No-Bias condition was not more extreme than the θ value targeting the same perceptual decision rate in the Bias condition. Trials with the same condition in both intervals were also primarily included to constrain the fitting of the cfc-model, and were removed from several of the other analyses, such as the non-parametric confidence analysis and the modeling building on the Bayesian model.

Our behavioral analysis testing the effect of condition on motion direction decisions was done using a logistic mixed-effects model with the ‘lme4’ package [67] in R [68]. The test was two-tailed and used an alpha value of 0.05. We did not assess confidence choices using a regression approach because we prespecified that we would only do this if the first-order decision probabilities were adequately matched, which they were not. Additionally, we realized from further model exploration that the Bayesian model does not predict equal confidence given matched first-order decision rates, so this analysis was not the best approach to assess deviations from the Bayesian benchmark in any case. Instead, and as planned given that first-order decision rates may not be perfectly matched, to examine confidence bias we used a non-parametric analysis following a similar approach to that in previous work [35]. We transformed the perceptual decision probabilities to put them in the (- ∞ ,∞) domain by taking their log-odds, and then computed the differences between the two conditions for each stimulus pair setting, excluding pairs with the same condition in both intervals. For each of these transformed choice probability differences (cp_diff), we then computed the corresponding rate of choosing the Bias condition as the more confident one (P(“Bias”)) and fit a cumulative normal distribution function (Ф) with a free noise (σ_cb) and bias (µ_cb) parameter to this relationship:

Modeling

Quantifying confidence bias.

In order to further quantify the confidence bias, before exploring the different process models that might explain it, we fit the cfc-model developed in previous work [33]. This model is based on elements of signal detection theory [69] and includes two main parameters. The confidence noise parameter accounts for inefficiencies in using all the sensory information for the confidence judgment, and the confidence boost parameter reflects the extent to which confidence is based on the same information as that used to commit on the perceptual decision or new sensory information. For our purposes, we focus on a third parameter, the confidence bias term, β, which captures the propensity to choose one condition as more confident when it is competing with another condition, for instance, central versus peripheral vision [35]. Importantly, this parameter is computed after accounting for the stimulus strengths, first-order responses in each interval, and a potential interval bias (propensity to prefer a particular interval as the more confident one). In a single condition, confidence bias would reflect the extent to which the sensory sensitivity is correctly estimated, such that if this parameter is equal to 1, sensitivity is perfectly estimated. If this parameter is below 1, sensitivity is underestimated indicating underconfidence and if this parameter is above 1, sensitivity is overestimated indicating overconfidence. While we cannot estimate confidence bias in a single condition (the bias would affect equally both intervals and thus cancel out any effect), we can compare confidence biases across two conditions, as in our case. Here, the cfc-model fits β as the ratio of confidence biases between conditions with the No-Bias as the baseline, such that β  = 1 indicates no confidence bias. If β > 1, this would indicate a confidence bias favoring the Bias condition, and β  < 1 would indicate a confidence bias favoring the No-Bias condition. In order to collapse the Bias condition into one psychometric function for the cfc-model, we projected the orange bias trials onto the blue bias space by taking:

and

We fit the model to the pooled data, using normalized stimulus intensity values (θ_norm) that were adjusted based on each participants’ fit sensory noise (σ_No-Bias) and bias (µ_No-Bias) from the No-Bias condition:

The model freely fit the sensory noise and criteria in each condition, along with the confidence parameters: confidence noise in each condition, confidence boost (reflecting additional information used in confidence) in each condition, interval bias, and the confidence bias parameter, β. We did this for 100 bootstrapped runs of the model in order to extract the confidence intervals surrounding the estimate of β. The other fitted confidence parameter values are reported in Supporting Information (Table A in S1 Text). We implemented and fit this model using the previously developed ‘cfc’ software [70]. More on the details underlying the model and the other parameters fit can be found in the original work [33].

Bayesian decision model.

For the remaining modeling, we wanted to assess whether the information used in confidence computations deviates from that used in perceptual decisions, and in particular whether the use of the slow-motion prior deviates. Hence, we took the approach of fitting a model to the first-order perceptual decision data, and then exploring whether deviations from that model are needed to explain confidence. So, the first-order parameters were fit exclusively to the perceptual decisions, and any further confidence parameters (w in the Decoupled Prior model and ɑ in the WPPCE model) were fit exclusively to the confidence choice rates (per set of perceptual responses). To capture the first-order motion direction decisions, we built a Bayesian observer model (Fig 3A). It has a Gaussian mixture prior distribution of motion directions (in degrees away from the reference) with means at the two preferred, orthogonal motion directions (µ_orth1, µ_orth2),

where σ_P is the standard deviation of each Gaussian making up the mixture prior. Due to low σ_P values and a decision region restricted to between -90° and 90°, wrapping was not an issue for these Gaussians. In the No-Bias condition, µ_orth1 and µ_orth2 are at + and -90°. In the Bias condition, µ_orth1 is 35° away from the reference in the bias direction, and µ_orth2 is 145° away from the reference in the non-bias direction. The Gaussian likelihood is formed from the stimulus corrupted by sensory noise,

where σ_L is the sensory noise and r is the internal signal caused by the stimulus. These combine to form the posterior distribution, and the perceived probability of the motion direction being in the orange (encoded as the negative direction) or blue (encoded as the positive direction) region is computed by the area under this posterior distribution across the entire orange or blue region, normalized:

(1)(2)

where N is equal to the total area under the posterior in the allowed decision region:

(3)

The decision is then based on the max of these two values,

We fit this model to the motion-direction decisions of each individual participant in order to find the σ_P and σ_L values that would best explain their first-order performance. However, in order to deal with the computational demand of this we required a slight simplification. For the Bias condition, we considered the prior to be a single Gaussian centered around µ_orth1. Because the Gaussian centered around µ_orth2 (+-145°) is so far outside of the allowed decision region (from -90° to +90°), this had a negligible impact on the model and made it possible to fit with the amount of data available to us. The ability of the model to still account well for first-order performance can be seen in Fig 3C. We then used the fit σ_P and σ_L parameters for the rest of the confidence modeling described below, with the exception of the Efficient Coding model, which fit a different perceptual decision model as a baseline, based on the idea of efficient coding.

Bayesian confidence model.

We then considered the expected confidence patterns of an observer that follows the Bayesian confidence model. This model is in line with the Bayesian confidence model discussed in the literature in which confidence reflects the perceived posterior probability of being correct about a decision [5–10]. In our case, confidence then corresponds to the perceived posterior probability that the orange/blue decision is correct, which is described above in Equation (1) if orange is chosen, and [2] if blue is chosen. The confidence forced-choice after both intervals is then based on whichever of these two confidence values from the two intervals is higher:

(4)

where the conf value is the perceived posterior probability correct. This model has no free parameters beyond the first-order sensitivity parameters, σ_P and σ_L.

Distance-to-Criterion confidence model.

It is possible that, instead of computing the area under the posterior distribution as in the Bayesian confidence model, participants instead just base their confidence on one sample from the posterior. In this model, confidence reflects the distance between a sample from the posterior (r_c) and their criterion (crit):

(5)

High confidence decisions then occur when the internal posterior sample is far from the criterion, or is more clearly orange or blue. Again, the confidence forced-choice is then based on which of these confidence values is higher, as in Equation (4). We also assume in this model that the sample on which confidence is based is on the same side of the criterion as the decision, such that confidence will not oppose the decision. Like the Bayesian model, this model has no added free parameters for confidence.

Decoupled Prior confidence model.

The Decoupled Prior confidence model reflects participants computing confidence as in the Bayesian confidence model above, but allowing for a different estimated precision of the prior relative to the likelihood such that the prior may have a stronger or weaker effect in confidence compared to the first-order decisions. In this sense, the prior used in confidence decouples from that used in perceptual decisions. The relative over- or underweighting of the prior is implemented by an over- or underestimation of the sensory noise of the likelihood through a scalar weighting parameter w, such that the estimated posterior probabilities of each choice at the confidence level are:

(6)(7)

and N is:

(8)

If w = 1, the sensory precision is correctly estimated in the confidence computation and the prior and likelihood are precision-weighted in the same way as in the first-order decisions. This, additionally, would be identical to the Bayesian confidence model. If w < 1, the sensory noise is underestimated, and the prior is underused relative to the likelihood. On the contrary, if w > 1, the sensory noise is overestimated, and the prior is overused relative to the likelihood. In this model, the scaling parameter w was fit as a free parameter. Confidence was allowed to vary between 0 and 1, so there could be cases in which, due to the different prior-use at the confidence level, it was below 0.5 and hence did not agree with the perceptual decision. Such cases reflect the possibility of a change of mind following the perceptual decision, with confidence in the originally chosen option then being particularly low.

Prior-Congruent Evidence (PCE) confidence model.

In the PCE confidence model, confidence is not based on the Bayesian posterior distribution but is instead based on the amount of prior-congruent information in the stimulus. For these line motion stimuli, the prior-congruent information refers to the information in the preferred, orthogonal motion direction (Fig 6A). So, the confidence is proportional to the length of the vector component in this orthogonal direction,

(9)

where θ_r is the angle between the reference and the motion direction of the internal signal generated by the stimulus, and θ_P is the angle between the reference and the preferred, orthogonal direction. The confidence choice is then based on which of these PCE values is larger, or which of the two intervals had more prior-congruent evidence, independently of the posterior evidence and the first-order responses.

Weighted Posterior and Prior-Congruent Evidence (WPPCE) confidence model.

The WPPCE confidence model suggests that the prior-congruent evidence described above serves as an additional signal, but not the sole signal, to confidence. In this model, confidence is based on the weighted combination of this PCE and the Bayesian posterior evidence (BPE), weighted according to a weighting parameter ɑ,

(10)

where the PCE term is computed as in Equation (9), and the BPE term is computed as in Equation (1) or (2), depending on the perceptual decision. Larger values of ɑ indicate more use of the prior-congruent evidence in confidence, and smaller values of ɑ indicate more use of the posterior evidence. When ɑ = 0, this model is identical to the Bayesian confidence model and when ɑ = 1, this model is identical to the PCE confidence model. The weighting term, ɑ, was a free parameter.

Model-fitting and comparison.

The confidence models with additional free parameters for confidence – the Decoupled Prior model and the WPPCE model – were fit using a maximum likelihood estimation (MLE) approach to find the w and ɑ parameter values respectively that could best explain the confidence forced-choice data. Due to model complexity, we fit the three free parameters of the EC model to the perceptual decisions of each participant using a grid-search across a parameter grid spanning all combinations of σ_P from 20 to 100 in steps of 5, σ_N from 0.02 to 0.5 in steps of 0.02, and σ_T from 0.02 to 0.1 in steps of 0.02, to find the maximal likelihood. We used a simulation-based approach to approximate the likelihoods for the EC, Decoupled Prior, and WPPCE models, which were analytically intractable. This was also done to compute the likelihood of the other models, to assess model fit. Model fitting was done in R using the ‘stats4’ package. In fitting the Decoupled Prior model, we also did an initial grid search across values of w from 0.1 to 4, in steps of 0.1, and used the w value that corresponded with the highest likelihood as the start value of the MLE procedure. This was done to avoid convergence issues. For a model comparison, we compared AIC between the models, computed according to

where k is the number of free parameters and is the maximized value of the likelihood function. We considered a model to be conclusively better if its AIC was lower by at least 2, following convention in the literature. For completeness, we also compared the models in terms of BIC, which did not change the results (Fig H in S1 Text). We ensured the models to be distinguishable from one another in a model recovery analysis (Fig I in S1 Text).