Figure 6. Trial-by-trial estimation errors: normative model vs. human observers.

The diagonal structure in the plots indicates that trial-by-trial errors are correlated. (A) Raw trial-by-trial errors with natural stimuli between model and human observers. (B) Correlation coefficients (circular) for trial-by-trial errors between model and each human observer. The error bars represent 95% confidence intervals from 1000 bootstrapped samples of the correlation coefficient. The dashed line shows the mean of the correlation coefficients of errors between human observers in natural stimuli (Figure 6—figure supplement 1). (C) Bias-corrected errors in natural stimuli. (D) Correlation coefficient for bias-corrected errors.

Figure 6—figure supplement 1. Trial-by-trial estimation errors between humans.

The diagonal structure in the plots indicates that trial-by-trial errors are correlated. (A) Raw trial-by-trial errors with natural stimuli between humans. (B) Correlation coefficients (circular) for trial-by-trial errors between humans. The error bars represent 95% confidence intervals from 1000 bootstrapped samples of the correlation coefficient. (C) Bias-corrected errors in natural stimuli. (D) Correlation coefficient for bias-corrected errors.

Figure 6—figure supplement 2. Six alternative models for predicting human tilt estimation performance.

Each model sets its estimates equal to (i) the minimum mean squared error (MMSE) estimates based on three cues (i.e., the normative model used in the main text); (ii–iv) the MMSE estimates based on each single cue alone (Luminance; Texture; and Disparity); (v) random tilt samples from the tilt prior (Prior); and (vi) random tilt samples from a uniform distribution of tilts (Random) (A) Human trial-by-trial errors plotted against the errors made by each of the models. Upper rows show raw errors; lower rows show bias-corrected errors. (B) Circular correlation coefficient for each of the models considered in (A). (C) Choice probability for each of the models considered in (A). Here, we define choice probability as the proportion of trials in which the sign of the model error predicts the sign of the human errors. The pattern is similar to that of the circular correlation coefficient. The MMSE model based on three image cues predicts humans tilt estimation errors better than all other models. In addition to the six models shown here, we assessed a number of ad hoc models. Three single-cue models that set the tilt estimate equal to each of the single cue values (i.e., cue-gradient orientation) predict human performance more poorly than the three-cue normative model used in the main text, but better than the prior or the random model. A model that averages the single-cue values (with equal weights) predicts human estimation better than the single-cue MMSE estimators, but worse than the three-cue normative model used in the main text.