On the Origins of Suboptimality in Human Probabilistic Inference
a: Setup. Subjects held the handle of a robotic manipulandum. The visual scene from a CRT monitor, including a cursor that tracked the hand position, was projected into the plane of the hand via a mirror. b: Screen setup. The screen showed a home position (grey circle), the cursor (red circle) here at the start of a trial, a line of potential targets (dots) and a visual cue (yellow dot). The task consisted in locating the true target among the array of potential targets, given the position of the noisy cue. The coordinate axis was not displayed on screen, and the target line is shaded here only for visualization purposes. c: Generative model of the task. On each trial the position of the hidden target was drawn from a discrete representation of the trial-dependent prior , whose shape was chosen randomly from a session-dependent class of distributions. The vertical distance of the cue from the target line, , was either ‘short’ or ‘long’, with equal probability. The horizontal position of the cue, , depended on and . The participants had to infer given , and the current prior . d: Details of the generative model. The potential targets constituted a discrete representation of the trial-dependent prior distribution ; the discrete representation was built by taking equally spaced samples from the inverse of the cdf of the prior, . The true target (red dot) was chosen uniformly at random from the potential targets, and the horizontal position of the cue (yellow dot) was drawn from a Gaussian distribution, , centered on the true target and whose SD was proportional to the distance from the target line (either ‘short’ or ‘long’, depending on the trial, for respectively low-noise and high-noise cues). Here we show the location of the cue for a high-noise trial. e: Components of Bayesian decision making. According to Bayesian Decision Theory, a Bayesian ideal observer combines the prior distribution with the likelihood function to obtain a posterior distribution. The posterior is then convolved with the loss function (in this case whether the target will be encircled by the cursor) and the observer picks the ‘optimal’ target location (purple dot) that corresponds to the minimum of the expected loss (dashed line).