Fig 1.
(A) Single trial illustration. Target area and puck are presented on a monitor from bird’s-eye perspective. Releasing the pressed button accelerates the puck by applying a force, which is proportional to the press-time. In trials without feedback the screen turned black after button release, while in feedback trials participants were able to see the puck moving according to simulated physics. (B) Four phases of the experiment. In the ‘prior’ phase, no feedback about puck motion was available, whereas in the ‘feedback’ phase subjects obtained visual feedback about the pucks’ motion. Two pucks with different colors and correspondingly different masses were simulated. In the ‘no feedback’ phase subjects obtained a new puck as indicated by a new color and obtained no feedback. In the last phase, subjects first watched 24 collisions between the new puck and the pucks they had interacted with in the ‘feedback’ phase before interacting again with the puck. Note that the puck of the ‘no feedback’ and ‘collisions + no feedback’ phase are identical.
Fig 2.
Press-times as function of initial distance to target.
Press-times for all participants by condition and experimental phase are shown with data points in black and Newtonian relationship with perfect knowledge about the involved parameters in blue. The top row shows the data of subjects in the light-to-heavy condition and the bottom row shows the data of subjects in the heavy-to-light condition. (A) Press-times of participants in the first phase (“prior”), (B) second phase (“feedback”) for the yellow puck, (C) second phase (“feedback”) for the red puck, (D) third phase (“no feedback”), and (E) last phase (”collisions and no feedback”) after having seen 24 collisions.
Fig 3.
Task performance and pucks’ traveled distance for three phases of experiment.
(A) Participants’ performance by experimental phase as quantified by pucks’ average absolute error in final position. The number of the ring at which the center of the puck stopped was used for coding performance, e.g. 1 and 3 in the shown cases. (B) Aggregated final positions of pucks versus initial distance of pucks to target. Phases of the experiment are separated by columns and conditions are separated by rows. The line of equality representing final positions prescribed by the Newtonian model with perfect knowledge of all parameters is shown in blue.
Fig 4.
Hierarchical Bayesian network for the Newtonian interaction model.
The model expresses the generative process of observed press-times across trials i, participants j, and pucks k including Weber-Fechner scaling given perceptual uncertainties of distance xi,j and mass mj,k of the pucks and subjects’ press-time variability. The parameter values refer to the prior probability distributions. See the text for details.
Fig 5.
Residuals of estimated press times and inferred masses in phase two for three cost functions.
(A) Residuals were calculated for each participant and each puck in phase two (”feedback”) given the actual press-times and the best fits for the linear heuristics and the Newtonian model. Residuals for both models were calculated for all three cost functions. (B) MAP estimates of the masses used by individual subjects inferred according to the Newtonian model for the the three cost functions. Red and yellow pucks had different masses for subjects in the two conditions “heavy-to-light” and “light-to-heavy”.
Fig 6.
Posterior estimates of perceptual uncertainty and press-time variability inferred with data from phase two “feedback”.
(A) Inferred posterior distributions of perceptual uncertainty for the linear heuristics model and the Newtonian physics model. Dark green distributions display posterior distributions for the Newtonian model class, dark blue ones for the linear model class. A separation into cost functions is not included since the different cost functions did not lead to significant differences. (B) Inferred posteriors for individual press-time variability varied significantly between subjects between the two models. All but one participant show lower or equal values of variability regarding the press-time for the Newtonian model class.
Fig 7.
Bayes factors calculated from posterior odds sampled using the product space method.
Bayes factors are displayed for different phases and combinations of phases. Blue line at 1 marks the point where neither model is stronger supported by evidence. Red line at 3.2 marks the transition from Bayes factors being only worth mentioning to substantial evidence in favor of one the models. Colors of bars indicate the model favored by the Bayes factors.
Fig 8.
MAP values of inferred latent mass in Newtonian model class with quadratic loss function for each participant and condition.
Fig 9.
Bayesian model for learning through observing collisions with prior and posterior mass beliefs.
The left panel shows inferred posterior mass beliefs for the pucks from feedback phase 2 for each participant. All 100 trials were used to infer the mass beliefs. These posteriors were used as priors for the inference from observations. The graphical model for learning by observing collision is shown in the middle panel. Uncertainty about the pucks’ velocities is introduced for the initial velocities vF and vNF as well as for the resulting velocities uF and uNF after the elastic collision. Utilizing the physical relationship of velocities and masses in an elastic collision enables inferring beliefs about the unknown puck based on previous mass beliefs of pucks in phase 2. Resulting posterior mass beliefs are shown in the right panel for inferences based on 6 and 24 observations of collisions.