The subjects used both sensory and prior information to guide their perceptual reports on the dynamic task. We measured performance in terms of the variability of the distribution of trial-by-trial errors (quantified as the standard deviation, or STD, and denoted as σ). This variability was lower for perceptual reports on the dynamic task than for either: 1) predictions from that task (σ_prior; Fig. 2h), or 2) perceptual reports on the control task that lacked sequential predictability and thus reflected more purely sensory processing (σ_sensory; Fig. 2g). Moreover, for individual subjects, these different measures of variability were related to each other, such that perceptual errors from the dynamic task were well approximated using the optimal, reliability-weighted combination of prior and sensory information ( σoptimal-2=σPrior-2+σsensory-2; Fig. 2i). This result implies that, on average, the subjects tended to not only use these two sources of information, but also combine them according to their relative reliabilities to optimize perceptual performance on the dynamic task.

This integration of prior and sensory information took into account the changes in the relevance and reliability of the priors that occurred throughout the dynamic task. These changes are illustrated in Fig. 3a, which shows prediction-error STDs averaged across subjects as a function of the number of sounds after a clearly noticeable change-point, or SAC (see legend for details), separately for the two noise conditions. Figure 3b shows linear contrasts that captured the salient, dynamic aspects of these changes for each subject (see inset in Fig. 3e illustrating the three contrasts: CP, describing the effects of a noticeable change-point; Exp, describing the effects of the number of sounds experienced following a noticeable change-point; and Noise, describing the high or low noise condition). Specifically, on change-point trials, predictions were irrelevant and hence most variable with respect to the subsequent sound-source location (signed-rank test for H₀: the median of the distribution of per-subject CP contrasts, which compared change-points to other trials=0, p<10⁻⁵). After change-points, predictions became steadily more reliable as the number of sound sources experienced from the new distribution increased in both noise conditions (p<10⁻⁴ for Exp_low and Exp_high contrasts, which identified linear trends across SAC 2–6 for each of the two noise conditions). The predictions were also more reliable overall in the low-versus high-noise condition (Noise contrast, p<10⁻⁵). These dynamic trends were consistent with predictions from a normative model of predictive inference that had full knowledge of the generative statistics ¹⁰. The model, which produced simulated predictions that were analyzed in the same way as the data, had task-dependent effects that were in the same directions and of roughly the same magnitude as the data, although the subjects tended to produce more variable predictions than the model (Fig. 3a,b diamonds).

These task-dependent changes in the subjects’ predictions were associated with similar changes in the variability of their perceptual reports (Fig. 3c,d) and their confidence in those reports, as assessed by the frequencies of high-confidence reports (Fig. 3e,f). Perceptual-error variability tended to be higher for change-point trials, when predictions were irrelevant (CP contrast, p<10⁻⁵), and for the high-versus low-noise condition (Noise contrast, p<10⁻⁵). Perceptual-error variability also tended to decrease on experiencing more samples from the new distribution, with a reliable effect across individuals in the low-noise condition (Exp_low contrast, p<0.005) but not the high-noise condition (Exp_high contrast, p=0.4). These dynamics were also apparent in the subjects’ confidence report trends (Fig. 3e,f), which reflected trial-by-trial awareness of the changes in perceptual variability and included similar dependencies on CP (p<10⁻⁴), Noise (p=0.032), and Exp_low (p=0.03) and less reliable dependencies on Exp_high (p=0.07). Both the perceptual and confidence report effects were qualitatively similar, in direction and magnitude, to theoretical values computed from optimal combinations of each subjects’ changing priors (circles in Fig. 3a,b) and their fixed sensory reliability estimated from the control task (Fig. 2g; see also Fig. 1d–f). These theoretical values also showed strong effects of CP, Noise, and Exp_low, and smaller effects of Exp_high (Fig. 3c–f, diamonds).

These behavioral dynamics reflected changes in the degree to which the subjects’ priors biased their perceptual reports. We quantified perceptual bias as the slope of the relationship between the prediction error and the perceptual error measured on individual trials (Fig. 4a–c). A slope of zero implies no relationship between the prediction error and the perceptual error, and thus no bias towards the prior. In contrast, slope values that increase towards unity imply increasing biases of the perceptual reports towards the prior. This perceptual bias varied systematically as a function of task conditions. Specifically, perceptual bias was lower on change-points (CP contrast, p<10⁻⁵) and for the high-versus low-noise condition (Noise contrast, p=0.008). Perceptual bias also tended to increase on experiencing more samples, although these effects were variable across individuals and not statistically reliable in the low-noise condition (Exp_low contrast, p=0.1; Exp_high contrast, p=0.004). These task-dependent changes in the biases were comparable in direction and magnitude to theoretically computed values given an optimal, reliability-weighted combination of the task-specific predictions on the dynamic task (circles in Fig. 3a) and fixed sensory reliability estimated from the control task (Fig. 2g), computed separately for each subject (diamonds in Fig. 4d,e). Despite these comparable task-dependent trends (compare circles and diamonds in Fig. 4e), the subjects’ perceptual biases were on average smaller than the theoretical values (compare circles and diamonds in Fig. 4d). This shift was consistent with their overall worse predictions than the model (compare circles and diamonds in Fig. 3a). However, overall performance, measured as perceptual-error variability, was relatively insensitive to this overall shift, as compared to task-dependent adjustments, in the magnitude of the perceptual biases (compare circles and triangles in Fig. 3c,d).

The above analyses demonstrated that for individual subjects, dynamic changes in the relevance and reliability of priors within an experimental session were associated with changes in the degree to which those priors biased perception. We identified similar effects across subjects, implying that individual differences in perception can reflect differences in how priors are updated and maintained in dynamic environments. Specifically, we compared subjects’ overall biases to the variability of their sensory and prediction errors (linear regression of the mean perceptual biases of individual subjects from non-change-point trials as a function of the STD of perceptual errors from the control task and the STD of prediction errors across non-change-point trials from the dynamic task; F statistic=7.39, p=0.002). According to these fits and consistent with Bayesian theory, subjects with higher overall prior-driven perceptual biases tended to have higher sensory variability (β=0.033, t-test for H₀: β=0, p=0.013; Fig. 5a) and lower prediction variability (β=−0.030, p=0.002; Fig 5b). We also found individual differences in how perceptual biases changed as a function of particular task conditions, and that those differences were predicted by subject-specific changes in priors under those conditions. Subjects whose priors improved (i.e., became less variable) the most also tended to have the largest increases in prior-driven perceptual biases: 1) just after a change-point (Fig. 5e), 2) on experiencing samples from a new distribution (in the low-but not high-noise condition; Figs. 5c and d), or 3) between the high- and low-noise conditions (Fig. 5f). Thus, on average, subjects tended to weigh prior and sensory information according to their relative reliabilities, taking into account variability in the priors across task conditions and individual subjects.

To more quantitatively account for the factors that affected perceptual biases across task conditions and individual subjects, we used a linear model that included normative and non-normative terms that each were weighed according to their contributions to each subject’s behavior (Fig. 6). Data generated by a purely normative model could capture some qualitative aspects of behaviour, but it systematically overestimated perceptual biases (Fig 6A). A linear model that included both normative and non-normative terms offered a better description of behaviour (Fig. 6B). The normative terms were extracted from a Bayesian model of perception, which generated perceptual biases that minimized simulated perceptual errors, given each subject’s variable predictions and sensory estimates. These terms were: 1) prior relevance, which reflected the probability that the current sound came from the same generative distribution as the previous sound (and thus is related to the CP effects illustrated in Figs. 3 and 4; Fig. 6c); and 2) prior reliability, which reflected changes in the total width of the predictive distribution relative to the likelihood, given new samples (and thus is related to the Exp and Noise effects illustrated in Figs. 3 and 4; Fig. 6d). The non-normative terms included one describing a fixed bias as a function of the prediction error, one to allow the strength of perceptual bias to depend on reported confidence (i.e., whether the subject reported high confidence or not), and spatial terms to account for the subjects’ overall tendency to give perceptual reports that were biased slightly towards straight ahead (Fig. 2f). On average, the linear model captured the behavioral trends well (Fig. 6b), based on contributions of each of the terms described above that tended to vary in magnitude across subjects (Fig. 6e). By comparison, a parameter-free normative model captured some of the behavioral trends (Fig. 6a) but reported higher perceptual biases than subjects (compare red points and bar in Fig. 6e), particularly on change-points (compare green points and bars in Fig. 6e).

A key question addressed in this work is whether arousal systems, as reflected in pupil diameter, contribute to the dynamic modulation of perceptual biases. Using linear regression at each time-point relative to sound onset (the average sound-evoked pupil response from all probe trials and subjects is shown in Fig. 7a), we found that pupil diameter varied with several of the factors from the linear model that accounted for behavioral biases (Eq. 6; Fig. 7b). Specifically, prior reliability was reflected in the baseline diameter before presentation of the probe sound, with smaller baselines reflecting more reliable priors (p=0.03; Fig. 7c,h). However, prior reliability did not modulate the magnitude of the stimulus-evoked pupil response, after accounting for the baseline effect (Fig. 7f,i). In contrast, prior relevance was unrelated to baseline diameter but was robustly encoded by the stimulus-evoked pupil diameter, with larger evoked pupil responses reflecting lower prior relevance (Fig. 7b,e). This effect peaked around the time of the maximum sound-evoked pupil response (permutation test for effect duration: duration=1.0 s, p=0.02; Fig. 7i). The pupil response, but not the baseline, also reflected the subjects’ upcoming confidence report, with high confidence corresponding to larger pupil diameters, particularly late in the fixation interval (duration=1.8 s, p=0.01; Fig. 7d,g,i; note that these duration estimates were limited by the size of our measurement window).

If the arousal system is contributing to the dynamic regulation of the influence of priors on perception, then pupil diameter may co-vary with adjustments in prior influence even after accounting for all of the factors in the behavioral linear model (for example, if variability in internal representations of sound-source location affect both behavior and arousal). We therefore included the residual perceptual bias from our model of behavior (Fig. 6) in our model of pupil diameter. A positive/negative value of the residual biases indicates that the subject was more/less biased by the prior on the given trial than predicted by the linear model. There was a trend toward positive coefficients for this term in explaining baseline pupil diameter (larger baseline diameters corresponded to slightly stronger biases than predicted by the behavioral model; p=0.06; Fig. 7h). In addition, there was a robust reflection of the residual bias term in sound-evoked pupil response (smaller responses near the peak of the evoked response corresponded to stronger biases than predicted by the behavioural model; duration=1.2 s, p=0.02; Fig. 7i). This residual bias effect implies that pupil diameter reflects not just particular factors like prior reliability and relevance that can be used to make effective predictions in dynamic environments ¹¹, but also the extent to which those and other factors are actually used to bias perception from one stimulus to the next.

In addition to these average, within-subject effects, there were also across-subject relationships between pupil diameter and perceptual biases. In particular, stimulus-evoked pupil responses tended to be, on average, smaller in subjects with higher overall perceptual biases (PE term in Fig. 6e; Fig. 8c) or relevance-dependent biases (PE*relevance term in Fig. 6e; Fig. 8d). These effects were not evident for baseline pupil diameter (Fig. 8a,b). However, because the behavioral influences of overall perceptual biases and prior relevance covaried considerably across subjects (r=0.77, p<10⁻⁹), we constructed a new linear model that included two individual-difference variables that corresponded to the shared and unique variance of the two behavioral coefficients. The effects of the shared term were negative for most of the measurement window (Fig. 8e; duration=2.2 s, p=0.01). In contrast, the unique-variance term did not show a strong relationship to average pupil traces. This result implies that subjects who had the strongest overall perceptual biases, and modulated them most according to prior relevance, tended also to have the smallest stimulus-evoked pupil responses.

To further quantify these within- and across-subject relationships between pupil diameter and task performance, we used pupil diameter to predict the subjects’ perceptual biases. Specifically, we created three normalized variables to reflect within- and across-subject variability in pupil responses at the time of peak response (1.4 s following stimulus onset) along with their multiplicative interaction. Each pupil-derived variable was included as a modulator of prediction errors in three different linear models of perceptual errors. In the simplest model, pupil-derived measures alone predicted systematic differences in perceptual biases observed in the behavioral data (Supplemental Fig. 3a), such that biases were: 1) larger for trials in which pupil responses were smaller than average (t-test, p<10⁻⁴), 2) larger for subjects who had smaller than average pupil responses (p<10⁻³), and 3) modulated from trial to trial more steeply for subjects with smaller overall pupil responses (p<0.01; Supplemental Fig. 3b). Consistent with these relationships, the pupil-based measures offered a substantial improvement to the base model in terms of predicting behavior (likelihood-ratio test, p<10⁻⁷; Supplemental Fig. 3c). The pupil-based measures also offered an explanatory advantage when added to more complex models that accounted for direct fixed effects (one coefficient for all subjects) or random effects (one coefficient per subject) of relevance, reliability, and confidence reports on perceptual biases (p<10⁻⁴ for both models; Supplemental Fig. 3c). Taken together, these results imply that fluctuations in pupil diameter, particularly those mediated by stimuli and related to context relevance, can be used to determine the extent to which perception is biased towards pre-existing priors.

We used an auditory-localization task in which subjects heard sounds with varying source locations that were simulated using head-related transfer functions (HRTFs) from the IRCAM database (http://recherche.ircam.fr/equipes/salles/listen/download.html). Each sound was a sequence of five Gaussian noise pulses bandpass filtered between 100 Hz and 15 KHz. The pulses were 50 ms each with a delay of 10 ms following each pulse, for a total stimulus duration of 300 ms. The latency for the sound to reach the ears following the command execution was ~3 ms. For each subject, we tested a number of HRTFs during the initial session by playing sound sequences that circularly traversed the entire horizontal plane in 15° intervals. We picked the HRTF that was reported to give the most uniformly circular percept for the sound sequence. Each subject performed 3–6 total sessions.

Each subject completed two tasks per session. The first was a control localization task that required the subject to indicate the perceived location of simulated sound sources that were sampled independently and uniformly randomly along the frontal, horizontal plane. In each of 72 trials, the subject was required to: 1) fixate for 2.5 s on a central spot while listening to the auditory stimulus; and 2) indicate the perceived location of the sound using a mouse, which controlled a cursor that moved along a semi-circular arc on the computer screen that represented the range of possible sound-source locations (Fig. 1). Failure to maintain fixation resulted in a warning sound and trial break. Feedback was displayed on the screen after the subject reported the perceived location.

The second task was a dynamic localization task that required the subject to report predictions, perceptions, and confidence judgments of sound-source locations that were generated from a change-point process along the same horizontal plane. For this task, the subject listened to extended sequences of sounds generated by the change-point process, with an interval of 150 ms between each sound presentation. Each sound was paired with a visual cue indicating its simulated source location on the semi-circular arc. During the presentation of these sequences, no action was required. Occasionally, however, the sequences stopped, indicating the start of a “probe trial” with the following structure (Fig. 1c). First, the subject was required to predict the angular location of the next, upcoming probe stimulus on the arc using a mouse. Second, following the prediction, the subject was required to maintain fixation for 2.5 s on the same central spot used in the control task. The auditory probe stimulus, with no corresponding visual cue, was presented at the beginning of this fixation period. Fourth, after the fixation period ended, the subject indicated the perceived location of the probe stimulus using the mouse and the visual display. Fifth, the subject then reported confidence (high/low) on the accuracy of the perceptual report (Fig 1). Each subject performed four blocks of the dynamic task per session, which included ~30 probe trials each. Each session lasted ~90 min, with some variability due to the self-paced nature of the prediction, perceptual report, and confidence reporting periods of the task (median [IQR] reaction times were: 1.72 [1.49–2.35] sec for the prediction, 2.02 [1.50–2.30] sec for the perceptual report, and 1.18 [0.94–1.43] sec for the confidence report).

The sequence of simulated sound-source locations for the dynamic task was determined according to a process that included both irreducible variability (noise) and abrupt discontinuities (change-points). Our goal was to test if and how these manipulations, which can affect the reliability and relevance, respectively, of new information on existing predictions, also affect perceptual reports that can, in principle, use such predictions to improve the perception of ambiguous stimuli ¹⁰. Source locations were sampled from a Gaussian distribution with a standard deviation (STD) that was held constant within a block of 600 trials (10° or 20° for the low- or high-noise condition, respectively) and a mean that underwent abrupt change-points with a fixed probability, or hazard rate (H), of 0.15 for each sound sample. At each change-point, the mean of the generative distribution was resampled uniformly across the sound space, such that the newly generated source locations were independent from previous ones. The sequence was interrupted for probe trials at random using a procedure that ensured: 1) a roughly even distribution of probes occurring 1–6 sounds after a change-point (SAC); 2) that probes were separated by at least 8 sounds; and 3) the number of sounds between any two probe trials was the same, on average, regardless of the nature of the two probe trials (SAC 1–6). Thus, on some trials the probe sound-source location was independent of the previous stimulus sequence (SAC=1). On other trials, the probe location was generated from the same distribution that produced the immediately preceding locations (SAC=2–6).

Subjects were motivated to make accurate perceptual reports on each probe trial through an incentive system. They were instructed to report high confidence if they were confident that the true location was within a 16° window centered on their second (perceptual) report, and to report low confidence otherwise. A correct/incorrect high confidence report resulted in a score of (15/−10), and a correct/incorrect low confidence resulted in a score of (5/−3). Subjects were paid a bonus that depended on their total score.

To provide an intuitive understanding of how behavior was affected by particular task conditions, we sorted probe trials into twelve conditions according to the recency of the previous change-point (SAC=1–6) and noise condition (high/low). To emphasize the effects of change-points on the behavioral reports, these analyses included data only from sequences following easily noticeable change-points, corresponding to changes in generative mean of at least twice the generative STD for SAC=1 (note that the linear model described below was used to analyze the full data sets, including all change-points). Perceptual and prediction errors were computed by subtracting reported percepts and predictions from the true (simulated) sound source location for each trial. For each condition, the STD of prediction and estimation errors was used as a metric of average absolute error magnitude.

To quantify how prediction errors, estimation errors, and average confidence reports depended on specific task conditions, we performed four orthogonal linear contrasts over our twelve task conditions. Each contrast was computed by multiplying a weight matrix by the measured prediction errors, estimation errors, or average confidence reports, aggregated according to the task conditions for a single subject. Weight matrices were mean centered and chosen to identify: 1) differences between change-point and non-change-point trials (CP); 2) linear increases with increases in the number of sounds experienced (Exp) following a change-point, from SAC=2 to SAC=6, in the high-noise condition (Exp_high); 3) comparable linear increases in the low-noise condition (Exp_low); and 4) differences between the high- and low-noise conditions (Noise). Thus, the contrasts provided a per-subject measure of how much each behavioral measurement was modified according to these factors. For Figs. 3–5, we considered only sound sequences following relatively large change-points, corresponding to at least twice the generative STD. Contrasts were normalized for presentation in Figs. 3 and 4 by dividing the contrast value for each subject by the standard deviation of that contrast across all subjects. This procedure allowed all contrasts to be meaningfully displayed on the same set of axes.

To quantify the influence of the prior on the perceptual report, we measured the slope of the best-fit line to the relationship between prediction errors (prediction–true location) and perceptual errors (percept–true location) for the given task condition. Slopes close to one indicate a high perceptual bias, and slopes close to zero indicate low perceptual bias. To measure how the perceptual bias evolved as a function of the number of sounds after a change-point (SAC), we used the following linear model and included only data from sequences following noticeable change-points (jumps of at least twice the generative STD): [1]ERRpercp=β0+β1ERRpred,1high+⋯+β6ERRpred,6high+β7ERRpred,1low+⋯+β12ERRpred,6low+β13Biasspatial where ERR_percp is the perceptual error and ERRpred,1high is the prediction error on change-point trials (SAC=1) in the high-noise condition, and so on. The last term, Bias_spatial, captures the slight bias in the perceptual reports towards center of the screen.

The theoretically expected overall perceptual-error STD per subject (abscissa in Fig. 2i) was computed from an optimal, reliability-weighted combination of prior and sensory information: σtheoretical-2=σpredictions-2+σsensory-2. The theoretically expected perceptual-error STD per subject (diamonds in Fig. 3c,d), given their corresponding predictions for each SAC condition, was computed using (σtheoreticalSAC)-2=(σpredictionsSAC)-2+σsensory-2. The theoretically expected frequency of high-confidence reports (diamonds in Fig. 3e,f) was computed as the probability mass contained in a 16° window centered at the mean of a Gaussian with a STD of the theoretically expected perceptual errors, σtheoreticalSAC. Thus, the proportion of expected high-confidence reports increased with narrower perceptual error distributions. The theoretically expected perceptual bias per subject (diamonds in Fig. 4d,e) was computed as σsensory2/[(σpredictionsSAC)2+σsensory2]. In all of the above, σ_predictions is the STD of prediction errors on non-change-point trials, σpredictionsSAC is the STD of prediction errors for the specified number of sounds after a change-point, and σ_sensory is the STD of perceptual errors on the control task, computed per subject.

Auditory localization in a dynamic environment can be posed as a perceptual inference problem with the goal of inferring the location of the sound source on trial t (X_t) according to a noisy internal sensory representation of that sound source (λ_t) and the history of previously experienced sound sources (X_1:t−1). This problem can be simplified by exploiting the conditional independencies in the Markov change-point process through which sound sources are selected (see Supplementary Fig. 1). In particular, the sound sources locations on the current trial (X_t) are independent of those on previous trials (X_1:t−1) conditioned on the mean of the generative distribution on the current trial (μ_t). In turn, the mean of the generative distribution on the current trial (μ_t) is also independent of previous observations conditioned on: 1) the mean of the generative distribution on the previous trial (μ_t−1), and 2) a latent change-point variable that determines whether the mean is resampled on the current trial (s_t). Applying Bayes rule to invert the generative graph in Supplementary Fig. 1 yields the following inference equation: [2]P(Xt∣λt,X1:t-1)=P(λt∣Xt)∑μtP(Xt∣μt)∑st∑μt-1P(μt∣μt-1,st)P(st)P(μt-1∣X1:t-1)∑XtP(Xt,λt∣X1:t-1) where the likelihood P(λ_t|X_t) reflects the conditional distribution of internal representations across true sound source locations; P(X_t|μ_t) reflects the conditional probability of a sound source location being generated from a particular generative mean; P(μ_t|μ_t−1, s_t) reflects the process through which means are resampled on change-point (s_t =1) trials; and P(s_t) is the hazard rate (H), which was fixed to 0.15 for the task and all simulations. The likelihood P(λ_t|X_t) was modeled as a normal distribution centered on X_t with a variance that was fixed for each subject to the variance of perceptual reports made by that subject on the control task ( σsensory2). P(μ_t−1|X_1:t−1 is the distribution over possible generative means, which can be updated recursively. Although exact Bayesian solutions to this recursive problem exist ^13,30, we use a normal approximation to the Bayesian mixture distribution with a mean (μ̂) and variance (σ̂²) that capture the key dynamics of normative inference and offers better descriptions of human behaviour ¹⁰. As in previous work, predictions made using this approximation were more accurate than subject predictions. To account for this discrepancy, we created a subjective prediction μ̂_subj by sampling a random normal variable with mean equal to μ̂ and a variance that was incremented on each trial according to the difference in variance of subject and normative prediction errors: [3]σ^subj2=σ^2+Var(X-subjectpredictions)-Var(X-μ^)

Thus, the overall model incorporates the prospect of sub-optimal predictions about the sound-source location but implements Bayesian optimal combination of these predictions with incoming sensory information according to environmental dynamics.

Perceptions and predictions from the normative model were simulated by sampling internal representations (λ_t) and subjective predictions (μ̂_subj) for each trial according to the actual sequence of sound source locations experienced. Descriptive statistics for model simulations were averaged across 100 such simulated runs.

In addition to simulating behavioral data, the normative model also allowed us to extract latent variables that guide normative adjustments in perceptual bias. In particular, the model adjusts bias towards prior expectations in accordance with the relevance and reliability of those expectations. The relevance of prior expectations (π_t) is, in our generative framework, equal to the probability that a change-point did not occur on this trial given all previous data. This probability was computed on each trial by marginalizing Eq. 2 over all dimensions other than s. The impact of normative priors also depends critically on their reliability relative to that of the likelihood distribution capturing the noisy internal sensory representation (λ_t): [4]priorreliability:τt=σsensory2σsensory2+σ^subj2+σnoise2 where τ_t is prior reliability, σsensory2 is the variance of perceptual reports made by that subject on the control task, σ^subj2 is the variance on subjective assessments of the underlying mean, and σnoise2 is the expected variance of sound source locations about that mean. The sum of the latter two terms reflects the total variance on the model’s predictive distribution over possible sound locations. To ensure that these latent variables best reflected circumstances experienced by the subject, we fixed the model predictions (μ̂_subj) to the actual subject predictions from each trial and computed each measure as the expected value across all possible values of λ_t using a grid-point approximation.

To provide a more complete description of how behavioral data from all conditions, including all generative change-point and non-change-point trials, depended on both normative and non-normative factors, we fit the following linear model to data from all trials in each session: [5]ERRpercp(t)=β0+β1ERRpred(t)+β2ERRpred(t)·πt+β3ERRpred(t)·τt+β4ERRpred(t)·bet+β5Biascenter+β6Biasspatial where β₁ describes the overall prior bias; β₂ and β₂ describe the extent to which the overall bias is dynamically modulated by the prior relevance and reliability, respectively (see above); β₄ describes the interaction of prior bias with confidence report (a binary variable); β₅ describes the bias towards the center of the screen; and β₆ captures the angular spatial bias (mean perceptual error at the given angle) measured in the control task. Residuals from the model fit were signed according to the prediction error on each trial to create a residual bias term for use in pupil analysis.

Pupil diameter was sampled from both eyes at 60 Hz using an infrared eye-tracker built into the monitor (Tobii T60-XL). Pupil analyses used the mean value from the two eyes at each time point measured during fixation. We excluded from further analyses trials with blinks, which we identified using a custom blink-detection routine on the basis of missing pupil diameter measurements and/or vertical and horizontal eye position that deviated from fixation for at least 10 contiguous samples (median [IQR] percentage of excluded trials across sessions = 5.54 [3.16–9.16]%). For the remaining trials with <10 missing contiguous pupil samples, we linearly interpolate the data before low-pass filtering. Low-pass filtering was done using a Butterworth filter with a cut-off frequency of 4 Hz. These filtered measurements were then z-scored in each session. We also removed a linear trend in the average pupil diameter over the whole session to account for any slow drift. The pupil baseline for each probe trial was defined as the mean of the first three samples immediately preceding the measurement period for that probe trial.

To measure how the variables driving behavior were encoded in pupil diameter, we used the following linear model to explain the fluctuations in: 1) the baseline pupil diameter, and 2) stimulus-evoked pupil response across the 2.5 s following the auditory stimulus: [6]Pupildiameter=β1πt+β2τt+β3Betn+β4Biasresidual+β5Previousbaselinediameter+β6TimesincePreviouspprobe+β7-9(lowfreq.terms) where τ_t and π_t are the reliability and relevance, respectively; β_1–4 capture relationships between pupil diameter and the computational and behavioral variables of interest; β_5–6 capture persisting fluctuations in pupil diameter that are attributable to the previous trial; and β_7–9 includes a set of three low-frequency cosine components for each session that capture task-irrelevant variability due to slow modulations or session-based differences in pupil diameter. The exact form of the cosine components was (cos(π · k (2n − 1)/2N)), where k=0,1,2; n is the trial number within the session; and N is the total number of trials in the session. When this model was fit to evoked pupil responses, an additional nuisance variable was added to the explanatory matrix that accounted for trial-by-trial differences in baseline diameter.

Significance testing for evoked pupil coefficients was done through cluster-based permutation testing to account for multiple comparisons over time. In short, t-tests were performed on each set of coefficient values across subjects separately for each time point. Cluster size was determined according to the number of contiguous time points for which this t-test yielded p<0.05. Cluster corrected p-values were determined by comparing cluster sizes attained in this way to those from a permutation distribution of maximal cluster sizes ⁶⁰.

Trial-by-trial pupil measurements were extracted for the time of peak pupil response (1.4 seconds) from the behavioral model. Trial-by-trial measurements from each subject were regressed onto a set of nuisance variables that included explanatory variables β₅₊ from Eq. 6 to remove variance attributable to potentially confounding factors. Residual pupil measurements were concatenated across subjects and then divided into two separate variables: one variable accounted for average measurements for each subject and one that reflected normalized deviations from the average measurement within each subject. An additional term was created through the multiplicative interaction of each subject’s mean pupil response and pupil-response variability, to account for the possibility that individual differences in the overall arousal response modulate the extent to which trial-to-trial fluctuations in arousal modulate perceptual bias. The three resulting variable arrays were z-scored and multiplied by trial-by-trial prediction errors to create a predictor matrix. Trial-by-trial perceptual errors were regressed onto three separate models with and without the inclusion of the pupil predictor matrix: 1) a null model including an intercept term and a prediction error term to capture fixed effects of perceptual bias across all subjects as well as the spatial bias terms described above [NE]; 2) a fixed-effects model that also included interaction terms accounting for modulation of perceptual bias by prior relevance and reliability the subjects’ confidence report [FE]; and 3) a random-effects model that included all terms in model 2 separately for each subject [RE]. Since the random-effects model used dummy variables to account for individual differences in perceptual bias, the pupil predictor matrix included only within-subject variability and thus only one additional parameter rather than three. The marginal benefit of pupil-predictor terms was evaluated through likelihood-ratio tests (evaluating the additional explanatory power offered by pupil predictors) and quantified using AIC, a likelihood-based measure of goodness-of-fit that applies a penalty for each model parameter.