Fig 1.
Bayesian integration for two-dimensional Gaussian priors under different feedback conditions A–C.
The uncorrelated case is shown in the top row and the correlated case is shown in the bottom row. Prior and posterior are represented through iso-probability contours, the visual feedback is depicted in red and the true shift is marked as a black X. The black dotted lines indicate the prior mean. A Due to the very reliable feedback in the full feedback condition, the posterior is peaked very sharply—regardless of the correlation in the prior. B The partial sh-feedback is reliable in the sh dimension but provides no information about the shift in the sv dimension. This leads to an important difference in the posterior between the correlated and uncorrelated group: knowing the correlation structure reduces uncertainty about the sv dimension of the shift, leading to a more concentrated posterior. C In no-feedback trials, participants can only rely on their prior experience. This feedback condition allows to test for the prior beliefs directly.
Fig 2.
Schematic of the experimental design.
A Participants were recorded over four sessions spread across four days. The first session included an additional training phase (first 200 trials) to allow participants to get used to the experimental setup. After this initial training phase, the different trial types were presented randomly according to the specified proportions. B Experimental conditions for the correlated and uncorrelated group.
Fig 3.
Data of a typical participant (no. 6, correlated group).
The plots show the deviation of the final virtual hand position from the target as a function of the true shift. Dots represent individual trials and the lines show robust fits through the corresponding dots. Different colors indicate different feedback conditions. The crossing of the black dashed lines indicates the optimal pivot-point. Top row A,B: horizontal deviation as a function of the horizontal shift sh. Bottom row C,D: vertical deviation as a function of the vertical shift sv. The left column A,C shows results recorded in the first session of the experiment, the right column B,D shows results from the last session. In early trials, the participant’s reaction to partial feedback trials in the noninformative dimension is very similar to behavior in no-feedback trials. Importantly, across sessions there is a significant reduction in slope in the noninformative dimension of the partial feedback trials, indicating learning of the correlation structure (compare changes in lines highlighted with arrows, that is the yellow dashed lines in panels A and B and cyan dashed lines in panels C and D).
Fig 4.
Performance of all participants in the last session of the experiment.
The performance in the horizontal dimension is shown in panel A, performance in the vertical dimension in panel B. Performance is measured by slopes as in Fig 3 comparing no-feedback trials (purple) and partial feedback trials (yellow and cyan). Learning of the correlation structure is evident whenever the slope in the uninformative dimension of the partial feedback trials is significantly smaller than the slope in no-feedback trials (see also Fig 3). The perfect Bayesian response for no-feedback trials is characterized by a slope of one indicated by the thin black line, the Bayes-optimal slope for partial feedback trials would be zero—assuming that the Bayesian actor perfectly knows the statistics of the task. In both panels, error bars show standard errors of the robust fit.
Fig 5.
Means and correlation of the final hand position in no-feedback trials of the last session.
A The mean of the final horizontal hand position for an ideal actor should be +1cm to fully compensate the mean shift. B The mean of the final vertical hand position for an ideal actor should be −1cm to fully compensate the mean shift. C Correlation coefficient between the vertical and horizontal components of the final hand position. Error bars indicate 95% confidence intervals—bars marked with a star show significant correlations (at a 5% level).
Fig 6.
Changes in slope in partial feedback trials.
The slope is a performance measure determined as in Fig 3 but using a sliding window of 100 trials. A Evolution of the horizontal slopes in partial sv feedback trials of the correlated group where horizontal information is not given by the feedback, but can only be obtained through knowledge of the correlation structure. B Same as A but showing data of the uncorrelated group. C Evolution of the vertical slopes in sh feedback trials of the correlated group where vertical information is not given by the feedback, but can only be obtained through knowledge of the correlation structure. D Same as C but showing data of the uncorrelated group. For the analysis only partial sv- or partial sh-feedback trials were taken out from the pooled data across all sessions. Thin colored lines indicate individual participants and can vary in length since the exact number of relevant trials could fluctuate due to the probabilistic generation of trials. The thick black line shows the median over participants—taking only into account trials where data from all participants exists. The bar at the bottom of the figure indicates the corresponding session (on average).
Fig 7.
Learning of mean shift over all sessions revealed by performance in no-feedback trials averaged over a sliding window of 100 trials.
A Learning of the mean in the horizontal dimension of the correlated group. B Same as A but showing data of the uncorrelated group. C Learning of the mean in the vertical dimension of the correlated group. D Same as C but showing data of the uncorrelated group. For the analysis only no-feedback trials were taken out from the pooled data across all sessions. Thin colored lines indicate individual participants and can vary in length since the exact number of relevant trials could fluctuate due to the probabilistic generation of trials. The thick black line shows the median over participants—taking only into account trials where data from all participants exists. The bar at the bottom of the figure indicates the corresponding session (on average).
Fig 8.
Adaptation of correlation between the vertical and horizontal terminal hand position measured in no-feedback trials.
Correlation values are determined in a sliding window of 50 trials across all four sessions. A Correlated group. B Uncorrelated group. For the analysis only no-feedback trials were taken out from the pooled data across all sessions. Thin colored lines indicate individual participants and can vary in length since the exact number of relevant trials could fluctuate due to the probabilistic generation of trials. The thick black line shows the median over participants—taking only into account trials where data from all participants exists. The bar at the bottom of the figure indicates the corresponding session (on average).
Fig 9.
Changes in slope in partial feedback trials across groups.
The slope is determined as in Fig 6. A Evolution of the horizontal slopes in partial sv feedback trials where horizontal information is not given by the feedback, but can only be obtained through knowledge of the correlation structure. B Evolution of the vertical slopes in sh feedback trials where vertical information is not given by the feedback, but can only be obtained through knowledge of the correlation structure. The correlated group shows a gradual and steady improvement across sessions whereas the other groups do not show such a trend. Different colored lines show the median over the different groups of participants and can vary in length since the exact number of relevant trials could fluctuate due to the probabilistic generation of trials. The bar at the bottom of the figure indicates the corresponding session (on average).
Fig 10.
The plots show the simulation results as medians over six different simulation runs. Blue lines show the medians over six runs where the model was trained with correlated full feedback trials. Pink lines show the medians over six run where the model was trained with uncorrelated full feedback trials. A Median for evolution of horizontal slope in partial sv-feedback trials—compare Fig 9A which shows the participants’ results. B Median for evolution of vertical slope in partial sh-feedback trials—compare Fig 9B which shows the participants’ results. C Median for evolution of horizontal mean-response in no feedback trials—compare Fig 7 and Supplementary S2E Fig which shows the participants’ results. D Median for evolution of vertical mean-response in no feedback trials—compare Fig 7 and Supplementary S2F Fig which shows the participants’ results. The parameters of the model (strength of the initial belief over mean-shift and covariance matrix) were chosen in order to minimize the sum-of-squared-differences between the correlated simulation median and the median obtained from participants from the correlated group. The uncorrelated simulation run used the same set of parameters.
Fig 11.
Median for evolution of xy-correlation in no feedback trials—compare Fig 8A and 8B which shows the participants’ results. The plot shows the simulation results as medians over six different simulation runs. The blue line shows the median over six runs where the model was trained with correlated full feedback trials. The pink line shows the median over six runs where the model was trained with uncorrelated full feedback trials. The parameters of the model (strength of the initial belief over mean-shift and covariance matrix) were chosen in order to minimize the sum-of-squared-differences between the correlated simulation median and the median obtained from participants from the correlated group. The uncorrelated simulation run used the same set of parameters.