The development of Bayesian integration in sensorimotor estimation

Abstract

Examining development is important in addressing questions about whether Bayesian principles are hard coded in the brain. If the brain is inherently Bayesian, then behavior should show the signatures of Bayesian computation from an early stage in life. Children should integrate probabilistic information from prior and likelihood distributions to reach decisions and should be as statistically efficient as adults, when individual reliabilities are taken into account. To test this idea, we examined the integration of prior and likelihood information in a simple position-estimation task comparing children ages 6-11 years and adults. Some combination of prior and likelihood was present in the youngest sample tested (6-8 years old), and in most participants a Bayesian model fit the data better than simple baseline models. However, younger subjects tended to have parameters further from the optimal values, and all groups showed considerable biases. Our findings support some level of Bayesian integration in all age groups, with evidence that children use probabilistic quantities less efficiently than adults do during sensorimotor estimation.

Figure 1

(A) Experimental protocol. Participants were shown a visual cue with experimentally controlled uncertainty or likelihood which was presented as a “splash” created by a hidden target or “piece of candy” drawn from a prior distribution. Participants were told that the splash was created by candy falling into a pond. They were prompted to place a vertical bar (“net”) where the hidden target fell and were then shown feedback on the target location. (B) Relying on the likelihood. A simple strategy would be to rely entirely on likelihood information by pointing at its centroid on each trial. This strategy is close to optimal when the likelihood is precise or narrow. The black bar or net overlaps with the target in the left panel. However, this strategy is less successful when the likelihood is wider, as samples from the likelihood become a less reliable indicator of target location and the optimal estimate shifts closer to the prior mean. The net is far from the target in the right panel. The optimal strategy involves weighting prior and likelihood information according to their relative uncertainties. (C) Experimental design. In order to quantify integration of the prior and likelihood, we measured reliance on the likelihood under different conditions of prior width and likelihood width. The prior could be narrow or wide, and the likelihood could be narrow, medium, or wide.

Figure 2

Task performance and estimation data. (A) Left: The proportion of candy targets caught as a function of age group (median, error bars = 95% confidence intervals [CIs]). Right: The root mean square error relative to the regression line in the Wide Prior/Narrow Likelihood condition gives an indication of how noisy participants were and is shown as a function of age group (median, error bars = 95% CIs). (B) Estimation data overlaid with linear fit for a representative participant age 11 years. The net position as a function of the centroid of the likelihood is shown for each trial (points). The fitted (blue) and optimal (red) functions are displayed. Note that optimal values here and in (C–D) and Figure 3B assume that participants use the experimentally imposed likelihoods and priors. Each panel displays estimation data for one condition, as defined by prior and likelihood width. (C) The median bootstrapped intercept of individual participants is shown as a function of age group (error bars = 95% CI). The optimal intercept at zero is shown (red). (D) The median bootstrapped estimation slope of individual participants is shown as a function of age group (error bars = 95% CI). The optimal estimation slope values are shown (red).

Figure 3

Prior mean, estimation slope, and estimation slope as a function of trial bin. (A) Median prior mean as a function of prior width (NP = narrow prior, WP = wide prior), likelihood width (NL = narrow likelihood, ML = medium likelihood, WL = wide likelihood), and age group (error bars = 95% confidence interval). The optimal value is shown (red). (B) Median estimation slope as a function of prior width, likelihood width, and age group (error bars = 95% confidence interval). Optimal values are shown (red). (C–E) Estimation slopes were computed for separate blocks and bins of 40 consecutive trials, then averaged across likelihood conditions. Median estimation slopes (error bars = 95% confidence) are shown for the three age groups: (C) 6–8 years, (D) 9–11 years, and (E) 18+ years.

Figure 4

Model selection. (A) Confusion matrix showing the proportion of cases where each model was selected, computed from the data of 1,000 participants per simulated model. We can infer the correct model from simulated data with reasonable accuracy. (B) Median mean squared error for each model as a function of age group (error bars = 95% confidence interval). (C) The number of participants for whom each model was selected. The Bayesian model provides an improved fit for most participants (11 out of 16 ages 6–8 years, 15 out of 17 ages 9–11 years, 11 out of 11 adults).

Figure 5

Estimates of model parameters. (A–C) Estimates of model parameters from 1,000 simulated Bayesian participants. (A) Estimates of the Narrow Prior standard deviation (). (B) Estimates of the Wide Prior standard deviation (). (C) Estimates of the standard deviation added to the likelihood (). Simulated participants used the same prior in both conditions (, = 0.065), used partly differentiated priors ( = 0.048, = 0.083), or used the experimentally imposed prior ( = 0.03, = 0.1). We also varied the amount of noise added to the likelihood ( = 0.01, 0.05, 0.1). The median (error bars = 95% confidence interval) is shown in all panels. (D–F) Wide Prior variance as a function of the Narrow Prior variance inferred from the estimation data of participants whose data was best fit by the Bayesian model: (D) ages 6–8 years, (E) ages 9–11 years, and (F) adults. Blue and green lines show the experimentally imposed prior variance in the Narrow and Wide Prior conditions, respectively. (G) Shows the variance added to the likelihood inferred from the estimation data. (H–I) Switch model. (H) Shows the p(likelihood) inferred from the Switch models for 1,000 simulated subjects per p(likelihood) condition. (I) Shows the p(likelihood) for participants whose data was best fit by the Switch model.

Figure A1

Instructions Part 1. Screens 1 and 2. We presented participants with the instructions that someone behind them was throwing candy into a pond, represented by the screen. Screens 3 to 6. We showed participants 200 samples from the prior distribution (“Where the candy lands”). The first 10 samples (two shown here) were shown “falling” into the pond one by one.

Figure A2

Instructions Part 2. Screens 7 to 10. We showed participants an example trial, where they were given noisy feedback on target location in the form of n = 4 samples from the likelihood (Splash). We showed participants a vertical bar and asked them to use it to catch the candy by moving the bar from left to right. After they provided their estimate, we showed them the candy's true location. Screens 11 and 12. We gave participants information on “bonuses” and the duration of the experiment.