90 research outputs found

    Response slopes for the training session.

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    <p>Response slope as a function of the SD of the Gaussian prior distribution, , plotted respectively for trials with low noise (‘short’ cues, red line) and high noise (‘long’ cues, blue line). The response slope is equivalent to the linear weight assigned to the position of the cue (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003661#pcbi.1003661.e038" target="_blank">Eq. 1</a>). Dashed lines represent the Bayes optimal strategy given the generative model of the task in the two noise conditions. <i>Top</i>: Slopes for a representative subject in the training session (slope SE). <i>Bottom</i>: Average slopes across all subjects in the training session (, mean SE across subjects).</p

    Best observer model's estimated parameters.

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    <p>Group-average estimated parameters for the ‘best’ observer model (SPK-P-L), grouped by session (mean SD across subjects). For each subject, the point estimates of the parameters were computed through a robust mean of the posterior distribution of the parameter given the data. For reference, we also report the true noise values of the cues, and . (<sup>*</sup>) We ignored values of .</p

    Schematic of the task.

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    <p>Subjects attempted to move a virtual ball (represented by the green circle) to the center of a target line (represented by the black horizontal line). The ball moved with constant y-velocity and hit the target after 1 s, whereas it moved with Brownian motion in the x-direction. Final positional errors were penalized by a quadratic cost function that was displayed as a parabola and the error cost was displayed at the end of the trial (blue bar). Subjects could exert control on the x position of the ball by moving their hand to the left or right (gray solid and dashed arrow lines). This incurred a control cost which was the quadratic in the control signal and the cumulative across a trial (yellow bar) was constantly displayed. At the end of the trial subjects received feedback of the total cost, the sum of control and error cost (yellow-blue bar). Subjects were required to minimize the total cost on average and were tested on four conditions (2 noise levels×2 control cost levels). The path taken by the ball is shown for a typical trial.</p

    Experiment 2: Medium Uniform and Medium Peaked blocks.

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    <p><i>Very top:</i> Experimental distributions for Medium Uniform (light brown) and Medium Peaked (light blue) blocks, repeated on top of both columns. <i>Left column:</i> Mean response bias (average difference between the response and true interval duration, top) and standard deviation of the response (bottom) for a representative subject in both blocks (light blue: Medium Uniform; light brown: Medium Peaked). Error bars denote s.e.m. Continuous lines represent the Bayesian model ‘fit’ obtained averaging the predictions of the most supported models (Bayesian model averaging). <i>Right column:</i> Mean response bias (top) and standard deviation of the response (bottom) across subjects in both blocks (mean s.e.m. across subjects). Continuous lines represent the Bayesian model ‘fit’ obtained averaging the predictions of the most supported models across subjects.</p

    Participants’ mean absolute residual errors and mean scores.

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    <p><i>A:</i> Mean absolute residual error (mean ± SE across subjects; residual errors are computed after removing the residual error for the 0 perturbation condition) by perturbation size (0, ±0.5, ±1.5 cm) and trial uncertainty (Low, High). These data are the same as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0170466#pone.0170466.g002" target="_blank">Fig 2A</a>, here shown in absolute value and aggregated by perturbation size. <i>B:</i> Participants’ mean scores (mean ± SE between subjects) by perturbation size and trial uncertainty. Even though the residual errors (panel A) are significantly different from zero and significantly modulated by perturbation size (<i>p</i> < .001) and the interaction between the uncertainty and perturbation size (<i>p</i> < .001), the scores (panel B) are significantly affected only by the trial uncertainty (<i>p</i> < .001).</p

    Main statistics of the experimental distributions and nonparametrically inferred priors (Experiment 3 and 4; Standard feedback).

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    <p>Comparison between the main statistics of the ‘objective’ experimental distributions and the ‘subjective’ priors nonparametrically inferred from the data. The subjective moments are computed by averaging the moments of sampled priors pooled from all subjects ( s.d.); see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002771#pcbi-1002771-g008" target="_blank">Figure 8</a>, right column and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002771#s4" target="_blank">Methods</a> for details.</p

    Experiment 5: Medium Bimodal and Wide Bimodal blocks, mean bias and nonparametrically inferred priors.

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    <p><i>Very top:</i> Experimental distributions for Medium Bimodal (dark purple, left) and Wide Bimodal (light purple, right) blocks. <i>Top:</i> Mean response bias across subjects (mean s.e.m. across subjects) for the Medium Bimodal (left) and Wide Bimodal (right) blocks. Continuous lines represent the Bayesian model ‘fit’ obtained averaging the predictions of the most supported models across subjects. <i>Bottom:</i> Average inferred priors for the Medium Bimodal (left) and Wide Bimodal (right) blocks. Shaded regions are s.d. For comparison, the experimental distributions are plotted again under the inferred priors.</p

    Risk-sensitivity.

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    <p>A. Results of the multilinear regression analysis of the low control cost conditions for subject number 5. The line shows the average motor command that the subject produces for a given position (blue - low noise level, yellow - high noise level). The slope of the line is a measure for the position gain of the subject. B. same as in A. but for the high control cost conditions (green - low noise level, red - high noise level). C.–F. Compares various measures between the high and low noise conditions. A risk-neutral controller predicts values to be the same for both condition (dashed line), a risk-averse controller predicts values to fall above the dashed line and a risk-seeking controller below it. C. Negative position gain for the high noise condition plotted against the low noise condition for all six subjects in the low control cost conditions (subject 5 in black, ellipses show the standard deviation). The dashed line represent equality between the gains. D. as C. but for the high control cost conditions. E. Negative velocity gain for the high noise condition plotted against the low noise condition for all six subjects for the low control cost conditions (ellipses show the standard deviation). F. as E. but for the high control cost conditions.</p

    Reconstructed prior distributions.

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    <p>Each panel shows the (unnormalized) probability density for a ‘prior’ distribution of targets, grouped by test session, as per <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003661#pcbi-1003661-g002" target="_blank">Figure 2</a>. Purple lines are mean reconstructed priors (mean 1 s.d.) according to observer model SPK-L. <b>a: Gaussian session.</b> Recovered priors in the Gaussian test session are very good approximations of the true priors (comparison between SD of the reconstructed priors and true SD: ). <b>b: Unimodal session.</b> Recovered priors in the unimodal test session approximate the true priors (recovered SD: , true SD: screen units) although with systematic deviations in higher-order moments (comparison between moments of the reconstructed priors and true moments: skewness ; kurtosis ). Reconstructed priors are systematically less kurtotic (less peaked, lighter-tailed) than the true priors. <b>c: Bimodal session.</b> Recovered priors in the bimodal test session approximate the true priors with only minor systematic deviations (recovered SD: , true SD: screen units; coefficient of determination between moments of the reconstructed priors and true moments: skewness ; kurtosis ).</p

    Decision making with stochastic posterior distributions.

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    <p><b>a–c</b>: Each panel shows an example of how different models of stochasticity in the representation of the posterior distribution, and therefore in the computation of the expected loss, may affect decision making in a trial. In all cases, the observer chooses the subjectively optimal target (blue arrow) that minimizes the expected loss (purple line; see Eq. 4) given his or her current representation of the posterior (black lines or bars). The original posterior distribution is showed in panels b–f for comparison (shaded line). <b>a</b>: Original posterior distribution. <b>b</b>: Noisy posterior: the original posterior is corrupted by random multiplicative or Poisson-like noise (in this example, the noise has caused the observer to aim for the wrong peak). <b>c</b>: Sample-based posterior: a discrete approximation of the posterior is built by drawing samples from the original posterior (grey bars; samples are binned for visualization purposes). <b>d–f</b>: Each panel shows how stochasticity in the posterior affects the distribution of target choices (blue line). <b>d</b>: Without noise, the target choice distribution is a delta function peaked on the minimum of the expected loss, as per standard BDT. <b>e</b>: On each trial, the posterior is corrupted by different instances of noise, inducing a distribution of possible target choices (blue line). In our task, this distribution of target choices is very well approximated by a power function of the posterior distribution, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003661#pcbi.1003661.e185" target="_blank">Eq. 7</a> (red dashed line); see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003661#pcbi.1003661.s003" target="_blank">Text S2</a> for details. <b>f</b>: Similarly, the target choice distribution induced by sampling (blue line) is fit very well by a power function of the posterior (red dashed line). Note the extremely close resemblance of panels e and f (the exponent of the power function is the same).</p
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