Dissecting Bayes: Using influence measures to test normative use of probability density information derived from a sample
Fig 7
(A) On each trial, a sample of 30 or 5 points was drawn from an invisible bivariate Gaussian distribution and shown on a visual display. A participant could rigidly shift the sample up and down from the starting point marked by a blue square. In moving the sample a participant also shifted the invisible pdf of the underlying distribution. After a participant set the location of the sample and its underlying pdf, one yellow point was drawn from the shifted distribution. A heat map illustrates a bivariate Gaussian distribution which underlies the sample after it is shifted from the starting point. The horizontal green line is the penalty boundary. If the new point appeared above the penalty boundary, a participant incurred a penalty, accompanied by an aversive sound. There were two penalty conditions, 0 and -500. If the yellow point fell on or below the green line and above the blue square, a participant received a reward proportional to the distance from the blue square of the starting point to the yellow point. If the yellow point fell at or below the blue square a participant received nothing. The rewards ranged from 0 (at or below the blue square) to 100 points (just below the green line). Once the additional yellow point exceeded the green boundary line, the rewards fell to 0 points or -500 points. The red slanted line is a plot of reward as a function of vertical location. The short red line at the top just marks the penalty region. A participant had to trade off the increased probability of a penalty if they moved the sample upwards and a reduction in reward if they moved it downwards. In the figure, a participant receives 74.3 points. (B) We combined the two values of the penalty (0 points and -500 points) with the two sample sizes (30 points and 5 points), resulting in four conditions in total. All of the tasks considered depended only on the vertical coordinates of the sample and we could in principle have used univariate Gaussian sampled distributed along a vertical line. We used bivariate samples simply to reduce the chance that sample points would overlap and occlude one another.