Dissecting Bayes: Using influence measures to test normative use of probability density information derived from a sample
Fig 6
Hypothetical costs due to failures of accuracy and additivity in two previous experiments.
A. A stimulus from Trommershäuser et al. [10]. We chose the median stimulus from their experiment. In their task, the participant made speeded reaching movements to the reward region (green circle). The red circle denotes the penalty region. The distance between the two circles is 1.5 times the radius of the circle. The radii of circles were 8.97 mm. A touch within the green region earns +100, within the red, -100, and within the green and red, 0. Hitting outside of both regions earns nothing. Black circles denote a possible isotropic bivariate Gaussian distribution of end points around the aim point (SD 3.89 mm, the average SD in Trommershäuser et al [10]). Given the standard deviation of the bivariate Gaussian distribution, the optimal aim point maximizing the expected reward was calculated and is shown as a red diamond. The objective probabilities of hitting each region with possible end points are plotted against the subjective probabilities as circles. We consider a hypothetical participant who overestimates small probabilities by the linear-in-log-odds function in Fig 6A (30 points) with γ = 0.88,p0 = 0.76. The probability distortion slightly shifts the optimal aim point (with the new aim point shown as a green diamond almost completely covered by the red diamond). B. A stimulus similar to the stimuli in Experiment 1 used by Zhang et al [22]. They used a two-alternative forced-choice task. One of the options was a large, single rectangle target and the other comprised three disjoint smaller rectangles. To simplify our example, we replace the triple target with a double target. Hitting in either colored bar of the double target earned a full reward. Participants decided which target (single or double) to attempt to hit and made speeded reaching movements to the center of the chosen target. Hitting within the rectangle earned the same reward. The standard deviation of the reaching movement was chosen to be 3.05 mm (the average of the participants’ measured SDs in experiment 1 in Zhang et al [22]). The widths of the two rectangles are 1.5 times the SD and the gap between two rectangles is 0.75 times the width of that rectangle. These widths and gap correspond to a median value of the targets used in Zhang et al [22]. The heights of the rectangles are set so that the virtual participant’s end points do not fall outside the vertical boundaries. The width of the single rectangle is adjusted so that the objective probability of hitting the single target is the same as that of hitting the double target. The normative decision maker would pick each target 50% of the time. As a consequence of distortion of probability and super-additive, the decision maker instead picks the double target more often. If the single target is slightly increased in width by 14.7% (shown in a light red border), the decision maker would pick them equally often though his chances of hitting the single target are objectively greater.