Fig 1.
(A) Typical stimulus presentation for the size-weight illusion (SWI). The subject lifts two objects of equal weight, but different size. The smaller object is perceived to be heavier than the larger one. See Section 4: Comparison with existing literature, for variants in SWI methodology. (B) The SWI is ‘anti-Bayesian’. Stimulus pairs are plotted in a two-dimensional stimulus space: log volume ratio (x-axis) and log weight ratio (y-axis). A value of zero indicates that the two objects have equal volume (or weight). The green markers show SWI stimuli (equal weight but different volume) and arrows show the empirical bias, i.e., the perceived weight ratio, for the same stimuli (data taken from the current experiment). The solid red line represents an equal density prior (log density ratio = 0). The dotted red line indicates another plausible prior based on object statistics [5]: the larger object is heavier but slightly less dense. In the SWI, perceived weight is biased away from these priors. (C) SWI and three hypothetical priors (R1–R3) over log volume ratio and log weight ratio, as suggested by Peters and colleagues [6]. Each prior corresponds to a different density/volume relationship. R2 (smaller is denser and heavier: dashed black line) would predict the SWI. R1 = equal-density prior (red line); R3 = smaller is less dense and lighter (dotted blue line).
Fig 2.
(A) Full stimulus set, with 500 cm3 stimulus for scale on the left. (B) Stimulus volume as a function of weight. (C) Stimulus density as a function of weight.
Fig 3.
Results for within-subset stimulus pairs.
Perceived weight (A, C, E) and difference in perceived weight (B, D, F) for within-subset stimulus pairs, averaged across participants, error bars show ± 1 SEM. Solid lines/open symbols show the model predictions (described below, Equations 1 and 2). The dashed lines indicate veridical perception. Grey markers indicate pairs that included the reference stimulus (either as judged or other object). (A–B) Equal density pairs: (A) Weight estimates as a function of stimulus weight; symbol size indicates the weight of the other stimulus in the pair. (B) Data re-expressed as difference in perceived weight, as a function of weight difference. (C, D) Equal weight (SWI) pairs: (C) Weight estimates as a function of stimulus density; symbol size indicates the density of the other stimulus in the pair. Fig A (panel A) in S1 Supporting Information shows weight estimates as a function of stimulus volume. (D) Data re-expressed as difference in perceived weight as a function of density difference; symbol size indicates the average object density. Fig A (panel B) in S1 Supporting Information shows the difference in estimated weight as a function of volume difference. (E, F): Increasing density pairs: (E) Symbol size indicates the other stimulus’ weight; (F) Data re-expressed as difference in perceived weight, as a function of weight difference; symbol size indicates the average object density.
Fig 4.
(A) Difference in perceived weight as a function of weight difference and density difference in the full dataset. Markers show the data, solid lines show the model predictions (see Equation 2) at equally spaced density differences. The vertical spread of the data with density difference (marker colours) shows the effect of density in typical SWI stimuli (i.e., equal weight: vertical dotted line) and across the full set. The dashed diagonal line shows veridical perception. (B) The influence of density (i.e., density coefficients) as a function of weight difference in the model.
Fig 5.
(A) Schematic of the model’s 5 components. The perceived weight of an object is positively correlated with both objects’ weights (components 1 and 3), and the judged object’s density (component 2). When the objects’ weights are similar (captured by a Gaussian over their difference) the two objects’ densities have an additional influence (components 4 and 5). The importance of each component is conveyed by the thickness of the corresponding arrow. (B) Variance in perceived weight explained by the addition of each component to the constant in the model (bars) and corresponding SSR (magenta lines and markers) (C). The order of components follows the maximum increase in R2 with each addition (see Table A in S2 Supporting Information). X = .
Fig 6.
Efficient coding Bayesian ratio model.
(A) Bi-variate prior over log weight and log volume ratio, optimised to minimise errors in predicted log weight ratio. Red asterisk and dashed black line indicate an example stimulus condition and corresponding slice through the prior that are shown in part B. The solid red line is the identity line (i.e., equal density), and the dashed red line shows the orientation of the fitted prior (plotted over the stimulus range) as defined by its peak values. (B) Likelihood repulsion predicted by Wei and Stocker [29] for one stimulus condition with equal weight, unequal volume stimuli. Schematic of a median (unbiased/noise free) observer likelihood (red) and posterior (blue) for an example stimulus pair (red asterisk in A), given the 1D slice through prior (solid black line). Efficient coding of stimulus properties results in skewed likelihoods with a long tail away from the peak of the prior. The posterior is also skewed. When a symmetric loss function is chosen, e.g., the mean of the posterior (L2, blue dashed line), a bias away from the prior is predicted, i.e., the posterior estimate is shifted away from the prior peak, relative to the peak of the likelihood. Measurement noise (expected to be skewed in stimulus space) further increases the predicted repulsive bias [29]. (C-D) Biases in perceived log weight ratio. Contour lines show the prior; black dots show the stimulus pairs; arrows show the magnitude and direction of the bias for each pair, with SWI conditions in green; the solid red line shows equal density. (C) Empirical biases; (D) predicted biases.
Fig 7.
Peters and colleagues’ model of perceived weight.
(A) Empirical biases in perceived log weight ratio. Dotted lines show the optimal three sub-priors from Peters and colleagues’ [6] model when fitted to our data. Black dots show the stimulus pairs; arrows show the magnitude and direction of the bias, with SWI conditions in green. (B) Predicted biases according to Peters and colleagues’ [6] model.