Automatic and fast encoding of representational uncertainty underlies probability distortion

Humans do not have an accurate representation of probability information in the environment but distort it in a surprisingly stereotyped way (“probability distortion”), as shown in a wide range of judgment and decision-making tasks. Many theories hypothesize that humans automatically compensate for the uncertainty inherent in probability information (“representational uncertainty”) and probability distortion is a consequence of uncertainty compensation. Here we examined whether and how the representational uncertainty of probability is quantified in the human brain and its relevance to probability distortion behavior. Human subjects kept tracking the relative frequency of one color of dot in a sequence of dot arrays while their brain activity was recorded by magnetoencephalography (MEG). We found converging evidence from both neural entrainment and time-resolved decoding analysis that a mathematically- derived measure of representational uncertainty is automatically computed in the brain, despite it is not explicitly required by the task. In particular, the encodings of relative frequency and its representational uncertainty respectively occur at latencies of approximately 300 ms and 400 ms. The relative strength of the brain responses to these two quantities correlates with the probability distortion behavior. The automatic and fast encoding of the representational uncertainty provides neural basis for the uncertainty compensation hypothesis of probability distortion. More generally, since representational uncertainty is closely related to confidence estimation, our findings exemplify how confidence might emerge prior to perceptual judgment.


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We are aware that representational uncertainty here is not equivalent but 229 proportional to p(1-p). In the neural analysis we will present, we chose to focus on the 230 encoding of p(1-p) (as a proxy for representational uncertainty) instead of , 231 because was highly correlated with numerosity N and thus would be difficult to 232 be separated from the latter in brain activities. In contrast, p(1-p) and N were 233 independent of each other by design. Besides, p(1-p) has a nice connection with the 234 "second-order valuation" proposed by Lebreton et al. [18].

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We also evaluated whether each of BLO's three assumptions was the best among 265 its alternatives on the same dimension. In particular, we used the group-level Bayesian 266 model selection [22][23][24] to compute the probability that each specific model 267 outperforms the other models in the model set ("protected exceedance probability")

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and marginalized the protected exceedance probabilities for each dimension (i.e.,

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adding up the protected exceedance probabilities across the other two dimensions).

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Indeed, all three assumptions of the BLO model in the present study-log-odd,

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Evidence for representational uncertainty in response time 284 The above modeling analysis on subjects' reported relative frequency (i.e. model fits 285 as well as the factorial model comparison) suggests that subjects might compensate 286 for representational uncertainty that is proportional to p(1-p) and the N-large condition 287 was associated with lower uncertainty than the N-small condition.

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The response time (RT) of reporting relative frequency (defined as the interval 289 between the response screen onset and the first mouse move) provides an additional 290 index for representational uncertainty, given that lower uncertainty would lead to 291 shorter RTs. We divided the values of p into five bins and computed RT for each bin 292 and separately for the N-small and N-large trials. According to a 5 (p bins) by 2 (N-

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For example, representational uncertainty for relative-frequency estimation is 308 proportional to p(1-p), which is maximal at p=0.5 and minimal when p is close to 0 or uncertainty modeled in the present study is higher in the N-small condition than in the 313 N-large condition ( Fig 2C), but there seems to be little reason to expect relative-314 frequency estimation to be more difficult for displays with fewer dots.

Neural entrainment to periodic changes of p or p(1-p) 317
After showing that the BLO models can well capture the behavioral results, we next 318 examined whether the brain response could track the periodically changing p values 319 (P-cycle) or p(1-p) values (U-cycle) in the stimulus sequence. Recall that the P-cycle 320 and U-cycle trials had identical individual displays and differed only in the ordering of As shown in Fig 3, the brain response indeed tracked the periodic changes of p 325 and p(1-p). In particular, in P-cycle trials (Fig 3, left), the phase coherence between 326 the periodic p values and the MEG time series reached significance at 3.33 Hz 327 (permutation test, FDR corrected PFDR < 0.05), mainly in the occipital and parietal 328 sensors. Importantly, significant phase coherence was also found at 3.33 Hz in U-cycle 329 trials between the periodic p(1-p) values and the MEG time series (Fig 3, right). That 330 is, the brain activity was also entrained to the periodic changes of the representational 331 uncertainty of p (i.e. p(1-p)). Given that subjects were only asked to track the value of 332 p but not p(1-p), the observed neural entrainment to the task-irrelevant p(1-p)

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suggests an automatic encoding of representational uncertainty in the brain.  trials. Light and dark green curves respectively denote the N-small and N-large conditions. spectrum mark the frequency bins whose phase coherence was significantly above chance 342 level (permutation tests, PFDR <0.05 with FDR correction across frequency bins). Insets:

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Grand-averaged phase coherence topography at 3.33 Hz for each cycle and numerosity 344 condition. Solid black dots denote sensors whose phase coherence at 3.33 Hz was 345 significantly above chance level (permutation tests, PFDR<0.05 with FDR corrected across 346 magnetometers).

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Notably, the significant phase coherences between the periodic p or p(1-p) values

Fig 4. Neural responses to p and p(1-p) predict the slope of probability distortion.
We defined and to respectively quantify how much the 387 behavioral measure of the slope of distortion, , and the relative strength of p to p (1-p) in neural responses, , changed across the two numerosity conditions. Main plot:

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Correlation between and the at the parietal magnetometer channel MEG0421.

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The logarithm transformation was used only for visualization. Each dot denotes one 391 subject. Inset: Correlation coefficient topography between and , on which 392 MEG0421 is marked by a solid black dot.

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Neural encoding of p and p(1-p) over time and the associated brain 395 regions 396 After establishing the automatic tracking of cyclic changes in p(1-p) and its behavioral 397 relevance, we next aimed to delineate the temporal dynamics of p and p(1-p) in the 398 brain signals. Recall that on each trial one of the two variables of our interest (p or p(1-399 p)) changed periodically and the other was aperiodic. Therefore, we could decode the 400 temporal course of aperiodic p(1-p) in P-cycles trials and aperiodic p in U-cycle trials.

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In particular, we performed a time-resolved decoding analysis based on all 306 402 sensors (see Methods) using a regression approach that has been employed in including the parietal region where ∆ and were positively correlated (Fig 4).

Representational uncertainty and confidence 494
The representational uncertainty that concerns us lies in the internal representation of 495 probability or relative-frequency. One concept that may occasionally coincide with such epistemological uncertainty but is entirely different is outcome uncertainty, which is

Methodology implications 546
The "steady-state responses" (SSR) technique [48]-using rapid periodic stimulus 547 sequence to entrain brain responses-has been widely used with EEG/MEG to 548 investigate low-level perceptual processes, which increases the signal-to-noise ratio 549 of detecting the automatic brain responses to the stimuli by sacrificing temporal 550 information. In our experiment, we constructed periodic p or p(1-p) sequences and 551 showed that SSR can also be used to reveal the brain's automatic responses to these 552 more abstract variables. Moreover, we demonstrate the feasibility to perform time-  Task. Each trial started with a white fixation cross on a blank screen for 600 ms,

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following which a sequence of displays of orange and blue dots was presented on a and subjects were required to click on the scale to indicate the relative-frequency of 584 orange (or blue) dots on the last display. Half of the subjects reported relative frequency 585 for orange dots and half for blue dots.

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To encourage subjects to pay attention to each display, one out of six trials were 587 catch trials whose duration followed a truncated exponential distribution (1-6 s, mean 588 3 s), such that almost each display could be the last display. The duration of formal 589 trials was 6 or 6.15 s. Only the formal trials were submitted to behavioral and MEG 590 analyses.

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On each display, all dots were randomly scattered without overlapping within an 592 invisible circle that subtended a visual angle of 12°. The visual angle of each dot was 593 0.2° and the center-to-center distance between any two dots was at least 0.12°. The

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Design. We adopted a steady-state response (SSR) design, which could achieve a 600 higher signal-to-noise ratio than the conventional event-related design [48]. The basic 601 idea was to vary the value of a variable periodically at a specific temporal frequency 602 and to observe the brain activities at the same frequency as an idiosyncratic response to the variable. The variables of most interest in the present study were relative 604 frequency p and its representational uncertainty quantified by p (1-p).

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where ; )) and ; << respectively denote the trial-averaged power spectrum for 703 stimulus and response time series, and | ; <) | denotes the magnitude of the trial-704 averaged cross-spectrum between stimuli and responses. The value of for any 705 specific frequency is between 0 and 1, with larger indicating stronger phase-locked 706 responses.

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For a specific variable (p or p(1-p)  Ferrari-Toniolo S, Bujold PM, and Schultz W, Probability distortion depends on that the model on its row was identified as the best model for the 50 datasets generated by the model on its column. Synthetic datasets that were generated by BLO were all value, we used numerical simulations to compute the correlation (Pearson's r) between 1100 the true value and the sampled value, based on the BLO parameters estimated from