Fig 1.
Experimental design A Trial description in likelihood-only task, coin task—learning and coin task—transfer phases. In the beginning of each trial, participants saw 5 dots which represented splashes caused by a thrower throwing a coin into the pond (grey screen). Participants moved a net (blue vertical bar) to the position where they think the centre of splashes (likelihood-only task), or the coin (coin task) is. After participants submitted a response, feedback information was displayed for 1 second on the screen. The next trial then started automatically. Feedback information differed between tasks and phases. In the likelihood-only task, the horizontal position of the real centre of splashes was displayed as an orange vertical line. In the learning phase of the coin task, the true coin position was displayed as a yellow dot every trial. An accumulated score was displayed when participants hit the coin. In the transfer phase, only an accumulated score was displayed. B Task design: Likelihood uncertainty was manipulated through the dot dispersion. Prior uncertainty was manipulated through the accuracy of the thrower, which participants learned from coin position feedback during the learning phase of coin task. Specific prior and likelihood combinations of the experiment 1 and 2 are explained in detail as follows. C Design of experiment 1. There were two priors (narrow Pn σ = .025 and wide Pw σ = .085) and two likelihoods (narrow Ln σ = .06 and wide Lw σ = .15). Participants underwent two orthogonal prior/likelihood combinations in the learning phase and then one prior coupling with both likelihoods in the transfer phase. In the figure example, the learning conditions are PnLn and PwLw (boxes with “L” ticks) while the transfer conditions are PwLn and PwLw (boxes with “T” ticks), with PwLn being the new combination of the transfer phase. For the serial group, combinations in the learning phase were delivered block-wise. For the parallel group, combinations in the learning phase were delivered in an interspersed way. Trials were always administered in an interspersed way in the transfer phase. D Design of experiment 2. In the learning phase, participants underwent combinations having one prior (wide Pw σ = .085) paired with two out of three (narrow Ln σ = .024, medium Lm σ = .06, wide Lw σ = .15) likelihoods. During learning, the interpolation group experienced PwLn and PwLw trials while the extrapolation group experienced PwLn and PwLm trials. All participants then undertook PwLn, PwLm and PwLw trials in the transfer phase. For the interpolation group, PwLm was the new combination. For the extrapolation group, PwLw was the new combination. All trials were administered in an interspersed way.
Fig 2.
visualisation of quantitative performance measures including slope (A), transfer score (B) and optimality index (C). A Slope in the coin task was calculated by linearly regressing participants’ estimated coin position over the centre of splashes. The values of slope vary between 0 and 1. A higher slope means a higher weighting on likelihood, with 1 meaning total reliance on likelihood and 0 meaning no reliance on likelihood. B A transfer score was calculated by normalising a measured slope change by a predicted slope change based on subject-specific prior and likelihood estimations. A transfer score of 1 means transferring following an optimal Bayesian observer model. A transfer score equals or smaller than 0 means no transfer. C optimality index. For each trial given the true posterior, we can compute the probability that a coin would be within the net from the chosen position (Xnet), i.e., the success probability. We defined the optimality index for a trial as the success probability normalised by the maximal success probability. In the figure, this equals to the blue area divided by the yellow area. Note that here for visualisation purpose only, there is no overlap between the two areas, which may not and does not have to be the case in real measurement.
Fig 3.
Slope and transfer score in the experiment 1. slope A Discovery & B Validation set. Violin plots show the distribution of slopes in the learning, transfer-old and transfer-new trials of experiment 1, separated by prior, likelihood and cognitive load group (upper serial, lower parallel). The central blue line shows the median. The error bar represents the interquartile range. Filled dots represent each participant. Each prior/likelihood combination is represented by orange PnLn, green PnLw, pink PwLn, and violet PwLw. Bayes-optimal values are presented as coloured dashed lines, with colours corresponding to prior/likelihood types. transfer score B Discovery & D validation sets. The distribution of transfer scores in the serial and parallel groups. The upper panels show distribution of transfer scores of each prior/likelihood combination, separated by cognitive load group. Lower panels merge data of different prior/likelihood combinations. The central black line and error bar in the discovery set (Fig 3B) represent the mean and standard error while the central black line and error bar in the validation set (Fig 3D) represent the median and interquartile range. *p< = .05; ** p< = .01; ***p< = .001; n.s. non-significant.
Fig 4.
Slope and transfer score of experiment 2, A discovery & B validation set Violin-plots show the distributions of slope in the learning and transfer phases of experiment 2, separated by likelihood and group. The central green line represents the median. The vertical bar within the violin spans between the second and third quartiles. Each filled dot represents one participant. Three optimal slopes given likelihoods are presented as colour dashed lines, with pink representing PwLn, blue representing PwLm, and violet representing PwLw respectively. C discovery & D validation set Distributions of transfer scores in the interpolation and extrapolation groups. The central green line is the median of each group, and the vertical bar is the interquartile range. Insets showed predicted slope (gray bar) along with measured slope (olive bar) in the transfer-new trials. Note that there was no significant difference in the interpolation group. *p< = .05; ** p< = .01; ***p< = .001; n.s. non-significant.
Fig 5.
Optimality index. A discovery experiment 1 B validation experiment 1 C discovery experiment 2 D validation experiment 2 Bar graphs show the mean optimality index and error bars represent standard errors. In the left panel of each subplot, the mean of first 10 trials of the transfer-new (light green bar) is compared against the mean of the last 10 trials (dark green bar). In the right panel, the mean of the last 10 trials of the learning phase (light green bar) is compared against the first 10 transfer-old trials (dark green bar).
Fig 6.
Computational models and comparison of model fitness with Bayesian decision model A schematic illustration of a linear regression model for the coin task. An optimal linear mapping between scatters of splashes and slopes (solid line) will produce optimal extrapolation even if no true generative process is represented. In reality, because of noisy slope estimations (dotted line), interpolation (darker green zone) would be closer to optimal than extrapolation (light green zone). B schematic illustration of an exemplar model for the coin task. Instead of representing the parameters of prior distribution, participants retrieved exemplar samples (violet circles) acquired from the learning phase. Samples are then weighted (bars representing weighting) by the gaussian likelihood of splashes to infer the coin position of a given trial. C (experiment 1) & D (experiment 2) Model comparison using Bayesian information criteria (BIC). Colour code: grey Bayesian model, blue linear regression model, green likelihood-only regression model, and violet exemplar model. The smaller BIC values indicate the better models. Insets showed differences between modelled and measured transfer scores.