Fig 1.
Applications of Bayesian decision theory.
A. The decision maker is rewarded for making a speeded reaching that terminates in the white target region T. They may choose any aim point (red diamond). Because of motor uncertainty his actual end point E = (Ex,Ey) is distributed as a bivariate Gaussian centred on the aim point, represented here as a heat map. B. The decision maker shifts his aim point and the bivariate Gaussian distribution shifts with respect to the aim point. The probability of hitting the target is larger in the panel B than in the panel A. C. The target region is divided into two disjoint regions, T1 and T2. Touching either target earns a reward. The decision maker’s aim point is shown in red. Bayesian decision theory allows us to calculate the aim point that maximizes the probability of reward [10,11,25].
Fig 2.
Parametric decision making based on a sample.
A. Bivariate Gaussian PDF (referred to as the "population"). The population PDF is not known to the decision maker. B. The decision maker is given only a sample P1,⋯,PN of size N drawn from the Gaussian. C. The Gaussian parametric decision maker reduces a large number of sample values to the values of a small number of parameters. For the bivariate Gaussian, the sample is often reduced to five parameters that are estimates of population parameters
(referred to as "statistics"). D. The normative decision maker then makes decisions based only on these statistics, ignoring any "accidental" structure in the sample not captured by the parameters. For convenience in presentation we assume throughout that the Gaussian pdf is elongated (anisotropic) so that there there are exactly two orthogonal axes of symmetry. The excluded possibility is that the Gaussian is isotropic (circularly symmetric). All the Gaussian pdfs used in the experiment were anisotropic, vertically elongated.
Fig 3.
A. A hypothetical experiment. A sample is drawn from a bivariate Gaussian pdf marked by a heat map and contours of equal probability density. The blue bar represents the decision maker’s estimate of the probability that an additional point drawn from the same underlying pdf will be in the region above the green line T. The precise task is not important. B. Measuring influence. The (vertical) influence of one point in the sample can in principle be measured by perturbing it slightly in the vertical direction and measuring the effect of the perturbation on the decision maker’s estimate P[T]. The ratio of the change in estimate to the magnitude of perturbation is the influence of the point on the setting. We do not use this method (single point perturbation) but instead use a method based on linear regression. See Methods. The influence measures allow us to characterize how each point in the sample affects decision-making.
Fig 4.
Design of the interval estimation tasks.
A. Examples of 30-point and 5-point samples. The horizontal and vertical coordinates of the points are independent random variables drawn from a bivariate Gaussian distribution. B. The trial sequence of the interval estimation tasks. The sample appears and then two horizontal lines. Participants judged the probability that an additional sample from the same distribution would fall into the region delimited by the horizontal lines. C. Three configurations of interval estimation with respect to the center of the screen. The upper half and lower half intervals are non-overlapping; their set-theoretic union is the symmetric interval. The vertical interval distances are expressed as in proportions of σ, the standard deviation of the population pdf in the vertical direction. The vertical distances 0.126σ, 0.674σ, and 1.645σ correspond with 10%, 50%, and 90% probabilities in the symmetric interval, respectively and 5%, 25%, and 45% probabilities in the upper half and the lower half intervals, respectively.
Fig 5.
Results of the interval estimation tasks.
A. Accuracy. The participant’s mean estimates of probability in the symmetric interval are plotted against the correct probabilities P[S]. The black thick curve is the maximum likelihood estimate of a linear-in-log-odds function fitted to the data. See text. B. Additivity. The sum of estimates in the upper and lower halves of the symmetric interval
is plotted against the estimates of the symmetric interval
. The color scale of the circle indicates the correct probability between 0.1 and 0.9. The black thick line is the best-fit estimate by a super additive function. For A & B, data are averaged across the participants, and the error bars indicate ±2 s.e.m.
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.
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.
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.
Fig 9.
A. Illustration of the Max-point model. Two samples drawn from the same pdf are shown, one of size 5 and the other of size 30. The decision maker sets each sample so that the maximum point PMAX falls on a criterion boundary (dashed green line) chosen by the decision maker. The mental reference boundary is identical for 30-point samples and 5-point samples and as a consequence, the mean settings (red diamonds) for the 5-point sample are markedly higher than those for the 30-point samples. B. An AICc model comparison of the normative BDT model and the Max-Point Model. The lower the AICc, the better the model. C. A plot of mean settings for each participant versus the predictions of the normative BDT model. D. A plot of mean settings for each participant versus the predictions of the Max-point model. In sum, the Max-Point Model outperformed the normative BDT and reproduced the participant’s set point fairly well.