**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.**

*M*= 6.94,

*SD*= 0.77), 17 children (eight boys, nine girls) ages 9–11 years (

*M*= 10.06,

*SD*= 0.75), and 11 adults (five men, six women) over 18 years old (

*M*= 27.27,

*SD*= 5.31). The data of two participants were excluded due to their looking away from the screen during the experiment. Participants overlapped with the sample of a previous study, where some participants from the current study were included as age-matched controls and compared with a clinical population (Chambers, Sokhey, Gaebler-Spira, & Kording, 2017).

*n*= 4 samples (white dots on a blue background, diameter = 2% screen width) from a Gaussian likelihood distribution that was centered on the target location

*c*of the splash. This consists of full reliance on the likelihood and works well when the likelihood distribution is narrow, because the closely spaced points of the splash are an accurate indicator of target location (Figure 1B, left). However, full reliance on the likelihood would cause a participant to miss targets more frequently as the likelihood distribution widens (Figure 1B, right). When sensory information is unreliable, rather than relying on the likelihood completely we maximize performance by giving more weight to our prior belief on target location. More generally, the optimal strategy involves weighting sources of information according to their relative uncertainties.

*c*as the independent variable. The slope provides an estimate of the reliance on the likelihood, which we term the

*estimation slope*. If participants relied only on the likelihood to generate their estimate, then they should point close to the centroid of the splash

*c*on all trials, leading to an estimation slope ≈ 1. If instead participants ignore the likelihood and use only their representation of the prior, then their estimates should not depend on

*c*, leading to an estimation slope ≈ 0. The intercept/(1 − estimation slope) computed from the fitted function reflects the subjective prior mean used to provide estimates. Therefore, from participants' estimates we obtain a measure of their reliance on the likelihood and the subjective prior mean.

*p*(likelihood), was a free parameter. In this model, participants ignored the uncertainty of prior and likelihood information and used a fixed

*p*(likelihood) for all conditions.

*p*(correct), and the root mean square error (RMSE) with respect to the regression line in each condition. We note that the RMSE contains contributions from both additional noise

*p*(likelihood) parameter, using bounds of 0.1 and 0.9. We ran the optimizer five times with randomly selected initial parameters. We performed leave-one-out cross validation on the data of each participant by fitting model parameters to five conditions and computing the MSE for the left-out condition. For the Prior-only model, the MSE was computed by comparing estimation slopes to 0; for the Likelihood-only model, it was computed by comparing estimation slopes to 1. We selected the model with the lowest MSE summed across left-out conditions. We estimated the parameters of the Bayesian and Switch models by fitting the models to 1,000 data sets resampled with replacement from each participant's data. We used estimation slopes measured from participants' data for model selection and parameter estimation.

*SD*= 5% in screen units). If estimates fell outside screen limits, they were set to the screen limit. Prior-only and Likelihood-only participants were generated by excluding the likelihood and prior terms from the posterior distribution, respectively. Switch subjects alternated between Prior-only and Likelihood-only strategies. In order to illustrate that we can infer model parameters with reasonable accuracy, we performed parameter estimation for 1,000 simulated Bayesian subjects whose subjective prior variance was at intermediate values between the theoretical variance parameters. We simulated cases where subjective prior variances were undifferentiated (

*p*(likelihood) parameter of the Switch model (0.2, 0.4, 0.6, 0.8) from the data of 1,000 simulated subjects per condition. Simulations allowed us to ensure that our model-selection and parameter-estimation procedures produced unbiased results.

*p*(correct) of candy caught (Figure 2A, left panel). Performance increased significantly with age (one-way analysis of variance [ANOVA]),

*F*(2, 41) = 27.44,

*p*< 0.0001, with significant differences between age groups—6–8 years versus 9–11 years:

*t*(31) = 3.86,

*p*< 0.001; 6–8 years versus 18+ years:

*t*(25) = 7.72,

*p*< 0.0005; 9–11 years versus 18+ years:

*t*(26) = 3.83,

*p*< 0.001 (corrected

*α*= 0.0167). We compared the proportion of candy targets caught,

*p*(correct), to chance level for each age group (Figure 2A). The performance of all age groups far exceeded chance level, as tested with one-sample

*t*tests—6–8 years:

*t*(15) = 33.77,

*p*< 0.0001; 9–11 years:

*t*(16) = 32.26,

*p*< 0.0001; 18+ years:

*t*(10) = 29.31,

*p*< 0.0001 (corrected

*α*= 0.0167). In addition, we examined the RMSE of estimates relative to the regression line in the Narrow Likelihood/Wide Prior condition, which gave an indication of the variability of sensorimotor estimates (Figure 2A, right panel). We found a significant effect of age group on RMSE (one-way ANOVA),

*F*(2, 41) = 10.76,

*p*< 0.0005, and significant differences between age groups—6–8 years versus 9–11 years:

*t*(31) = 2.80,

*p*< 0.01; 6–8 years versus 18+ years:

*t*(25) = 4.31,

*p*< 0.001; 9–11 years versus 18+ years:

*t*(26) = 2.19,

*p*= 0.0378 (corrected

*α*= 0.0167). The RMSE (in units of screen width) was within an acceptably low range for all age groups—6–8 years:

*M*(

*SD*) = 0.07 (0.03); 9–11 years: 0.04 (0.02); 18+ years: 0.03 (0.01). This shows that all age groups understood the candy-catching task and carried out the task above chance level. Although we do observe differences between age groups, these differences between cannot be attributed to a lack of understanding of the task.

*R*

^{2}(

*SD*) = 0.32 (0.22); 9–11 years: 0.54 (0.20); 18+ years: 0.70 (0.12). This procedure allowed us to quantify the nature of prior and likelihood integration for each participant.

*F*(1, 42) = 0.37,

*p*= 0.55; age group:

*F*(1, 42) = 2.53,

*p*= 0.12—nor their interaction,

*F*(1, 42) = 0.76,

*p*= 0.39. However, there was a significant effect of likelihood width,

*F*(2, 84), 3.94,

*p*= 0.03, and a significant Likelihood width × Age interaction,

*F*(2, 84) = 3.69,

*p*= 0.04). Post hoc

*t*tests comparing levels of the likelihood-width variable and the interaction between likelihood width and age group did not show significant differences between conditions (see Appendix, Tables A1 and A2). Therefore, age group and stimulus factors did not play a strong role in influencing the prior mean used by participants.

*F*(1, 42) = 5.66,

*p*< 0.05, and likelihood width,

*F*(2, 84) = 53.78,

*p*< 0.0001, as well as a nonsignificant main effect of age group,

*F*(1, 42) = 0.42,

*p*= 0.52. Therefore, there is evidence that the sample as a whole integrated both the prior and the likelihood into their judgments.

*F*(1, 42) = 26.34,

*p*< 0.0001, and no significant Age group × Likelihood width interaction,

*F*(2, 84) = 3.39,

*p*= 0.06. The use of likelihood may not change considerably over the course of development. However, there is evidence for an influence of age group on use of prior statistics. We therefore examined the influence of the prior width on the estimation slope for different age groups, using paired

*t*tests—6–8 years:

*t*(30) = 0.54,

*p*= 0.59; 9–11 years:

*t*(32) = 3.76,

*p*< 0.001; 18+ years:

*t*(20) = 7.08,

*p*< 0.0001 (corrected

*α*= 0.0167). At a group level, 6- to 8-year-olds do not distinguish between Narrow and Wide Prior conditions, with this difference becoming significant at 9–11 years. Therefore, young children ages 6–8 years show the ability to incorporate likelihood into their judgments as adults do, but there is no evidence that they make use of the prior width until 9–11 years.

*F*(1, 41) = 0.01,

*p*= 0.94; prior width,

*F*(1, 41) = 1.96,

*p*= 0.17; or trial bin,

*F*(2, 82) = 0.60,

*p*= 0.55; but we did find a significant main effect of experimental block,

*F*(1, 41) = 7.79,

*p*< 0.01. As before, we found a significant Prior width × Age group interaction,

*F*(1, 41) = 14.58,

*p*< 0.001. However, interactions between age group, prior width, trial bin and experimental block were not significant. The significant effect of experimental block revealed a subtle tendency to rely more on the likelihood in Block 1—

*M*(

*SD*) = 0.55 (0.25)—than in Block 2—0.46 (0.22). However, we did not find evidence for a change over the course of the experiment in how participants distinguished between conditions based on uncertainty. Therefore, we did not find evidence that young children learned the prior variance over the course of the experiment.

*R*

^{2}(

*SD*) = 0.74 (0.26); 9–11 years: 0.94 (0.07); 18+ years: 0.96 (0.02). Our findings suggest that some combination of prior and likelihood was present in children as young as 6 years.

*p*(likelihood) of trials where participants use the likelihood from simulated data (Figure 5H). The

*p*(likelihood) is variable across participants who use this strategy (Figure 5I). This simple strategy provides an alternative to Bayesian integration but is not prevalent in our sample.

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