Article Figures & Data
Figures
- Figure 1.
A, Illustration of Jazayeri and Shadlen's (2010) ideal observer model. The likelihood function for the current stimulus [P(M|S); Eq. 1] is modeled by a Gaussian of width σL, and windowed by the prior P(S) (red), which represents the previous sensory history, to yield the posteriori estimate P(S|M) in Equation 1 (orange). The reproduced interval is given by the BLS estimate from the posteriori distribution (star). The model introduces some biasing errors (as the BLS estimate no longer corresponds to the stimulus), but it reduces the variance of production times. B, Illustration of Gaussian prior models. In this version, the prior is a Gaussian probability density function derived from past trials. In the fixed prior model, the width of the prior σP is fixed across observers and conditions. In the optimized prior version, σP is calculated to minimize total error (see text; Fig. 4).
- Figure 2.
A, Reproduction distribution of visual stimuli for nonmusical controls for the duration 850 ms during sessions where the intervals were drawn from short (squares, 494–847 ms), intermediate (circles, 671–1024 ms), or long (triangles, 847–1200 ms) intervals. B, C, As in A, for expert drummers (B) or bowstring musicians (C). D, Average reproduction durations of visual stimuli for subjects without musical training as a function of stimulus duration for three stimulus ranges (squares, 494–847 ms; circles, 671–1024 ms; triangles, 847–1200 ms). Straight lines show best-fitting linear regressions over each range. The central tendency index is given as the difference between the slope of these fits and the equality line (dashed). E, F, As in D, for expert drummers (E) and bowstring musicians (F). G–I, As for D–F, except for auditory rather than visual stimuli. Here all subjects performed veridically (on average).
- Figure 3.
A, Regression index plotted against Weber fraction for interval bisection. The regression index is the deviation from veridicality of the linear fit to reproduction data like those of Figure 2 (0 for veridicality, 1 for total regression to the mean). Small symbols are individual subjects; large symbols are groups (blue, drummers; green, string musicians; red, controls). Filled circles show results for the visual task, open triangle for audition. The curves show the no-prior model (dashed black), Jazayeri and Shadlen's (2010) BLS model (gray), our Bayesian model (Fig. 1B) with prior of fixed width of 120 ms (cyan), and the Bayesian model where the width of the prior is chosen to optimize performance following the procedure of Figure 4 (orange). The continuous curve shows the model where the width of the prior is curtailed to remain >90 ms (estimated to best fit the data), the dashed where it is free to become infinitely narrow. B, CV (normalized average root variance of the reproductions) plotted against bias (difference between average production time and physical sample interval) for visual reproductions for the three subject groups (colors as in A). The total error (root mean squared error) is given by the distance from the origin, similar for the three groups. The continuous curves show simulations of the models in A, assuming the existence of a further nonsensory noise source, assumed to be constant for all subjects, and optimized to provide best overall fit of data (in practice a Weber fraction of 0.1). Each curve was created by varying sensory Weber fraction from 0.01 to 0.3: the colored stars superimposed indicate the average Weber fractions for that subject group.
- Figure 4.
A, B, Error landscapes showing relative RMS error for Weber fractions plotted against regression index (A) or width of the prior, σp (B). The points show the average data for our subjects (circles, vision; triangles, audition; blue, drummers; green, string musicians; red, controls). All points lie in the minimal-error valley, indicating that their behavioral choices are near optimal and adaptable. Note that A is a generic solution for any mechanism that causes a regression toward the mean, while B refers specifically to the Gaussian-prior model of variable width.
