Continuous Bayesian model: simulation and fitting

Results in Experiment 3 are compared with the performance of a Bayesian ideal observer for the continuous task. The ideal observer infers the perceived displacement as a reliability-weighted combination of the sensory likelihood and prior distributions,

where Dperceived is the mean of the inferred posterior distribution, wprior is the weight assigned to the prior, Dprior is the mean of the prior distribution, wlikelihood is the weight assigned to the likelihood, and Dlikelihood the likelihood distribution. When both Dprior and Dlikelihood are Gaussian distributions, the weight terms are given by:

and

That is, the more reliable (i.e., less variable) estimate is weighted higher. This reliability-weighted inference is additionally the Bayes optimal estimate because the variance of the estimate, σperceived2, is lower than the variance of both the prior and the likelihood distributions:

We simulated the final response as the mean of the posterior distribution, i.e., its maximum value. The values of the parameters used to simulate the ideal observer responses shown in Figure 5c were the same as the ones used in the experiment. To fit the model to data, we minimized squared error between each participant’s responses and those of the Bayesian ideal observer model to identify the best-fit values for their internal prior and likelihood distributions. Best-fit parameters were identified on a participant-by-participant basis. Parameter optimization was performed using MATLAB’s fmincon function.

Continuous displacement perception is Bayesian. a, Task schematic. Participants performed the same SSD task as in Experiments 1 and 2 but provided a continuous estimate of where the target landed after the saccade using a mouse cursor (+). b, Distributions used in the experiment. Distributions for the three noise levels are centered on displacement = 1° for illustration. c, Bayesian predictions for the experimental parameters in b. d, e, Results from n = 11 participants for displacements in the direction of the saccade (d) and opposite to the direction of the saccade (e). Bins were averaged across participants and connected with lines. Error bars: SEM. f, Presented versus reported displacements relative to the direction of the saccade (positive = in saccade direction, negative = opposite to saccade direction). Lines were fit to individuals and averaged across participants. Shaded region: SEM. Participants exhibited a response bias opposite to the direction of the saccade. g, h, Bayesian predictions with biased priors (against the direction of the saccade, as observed in f for displacements in the saccade direction (g) and opposite to the saccade direction (h). i, j, Model fits for participants’ internal likelihood distribution SDs (i) and prior means (j). *p < 0.05, **p < 0.01, ****p < 0.0001. Extended Data Figure 5-1 shows the results of incorporating the observed bias into the categorical Bayesian ideal observer model.