Dissecting Bayes: Using influence measures to test normative use of probability density information derived from a sample
Fig 8
Measured influence and the influence ratio.
A. Normative BDT influence and measured influence. We order the sample points for each sample point from 1 (the lowest) to either 5 or 30 (the highest) depending on sample size. We plot the mean across participants of the estimated influence of each sample point on the participant’s actual set point versus its order index (colored circles). We plot the influence expected for the normative BDT decision-maker versus order index (black squares). The weights of each sample point were estimated using a ridge regression (see Methods) for each participant. The regression coefficients were then averaged across the participants. The error bars indicate ±2 s.e.m. Negative influence indicates that a set point is set further away from a penalty boundary when a sample point is generated close to a penalty boundary relative to a starting point. The influence measures for the normative BDT model are skew-symmetric and a thin red line marks the axis of symmetry. The highest points (near the penalty region) and the lowest points (farthest from the penalty region) have influence equal and opposite in sign. The middle points have less influence. In contrast, the human decision maker has influence measures that roughly decrease in magnitude as we go away from the penalty region. The lowest points in the sample have little or no influence. Sample points distant from the penalty region have little influence. B. Influence ratios. The average across participants of influence ratios (measured influence divided by normative BDT influence) for each sample point is plotted versus the order index of the sample point. Error bars denote ±1 s.e.m. A value of 1 indicates that measured influence was identical to the normative BDT influence The influence ratios deviate from 1 (marked by a thin red line). For the sample points nearest the penalty region the influence ratios are too large but they approach 0 for sample points far from the penalty region. The gray-shaded central range could not reliably be estimated due to the denominator (i.e., normative BDT influence) being near zero.