Figure 1. Illustrations of PLW displays, stimulus distribution, and behavioral results. (a) Each point-light walker (PLW) consisted of 16 white dots. The white lines were invisible in the experiment. (b) Response display in which a blue probe was positioned on a white circle. We instructed the participants to manipulate the mouse to shift the probe's location on the circle, thereby indicating their perceived PLW direction. Before conducting the experiment, we informed the participants that the bottom, top, left, and right midpoints of the response circle corresponded to the 0°, 180°, -90° (270°), and 90° PLW directions, respectively, as indicated by the numbers that were invisible in the experiment. (c) Illustrations of stimulus distributions (light solid lines) used in the current study. The left panel shows a distribution in which four peaks are at the cardinal directions (0°, ±90°, and 180°); the right panel shows the peaks at the oblique directions (±45° and ±135°). The dots indicate the PLW directions used in the current study. (d-e) Estimation error and standard deviation are against the PLW direction. The dots and shaded areas indicate the mean estimation error and standard error averaged across all participants.

Figure 2. Illustration of the extended efficient Bayesian observer model and ideal long-term priors used in the current study. (a-b) Illustration of an efficient Bayesian inference model scaled by the perception-motor mapping process developed by Sun, Xu, et al. (2024). The model consists of two parts: (a) Bayesian decoding constrained by efficient encoding, and (b) motor mapping. (a) The efficient Bayesian observer model assumes that the stimulus feature ($\theta $) (e.g., heading direction) is first encoded in a sensory signal (m) by maximizing the mutual information between $\text{θ}$ and m ($F(\theta )$), which is distorted by the external (δE) and internal (δI) noises. The two types of noises are assumed to follow a normal distribution with the distribution center 0°and standard deviations σE and σI. Then, the sensory signal (m) is transformed back to the stimulus space by inverting the F process (${F}^{-1}(\theta )$), which generates the likelihood distribution (p(m|$\theta $)). Lastly, the sensory system conducts a Bayesian inference process by combining the prior (p($\theta $)) and the likelihood distribution (p(m|$\text{θ}$)), which computes an optimal estimate of the feature value $\widehat{\theta }\text{(}m\text{)}$. (b) Sun, Xu, et al. (2024) found that the reported estimates (${\widehat{\theta }}_{r}(m)$) that are recorded in the experiment are not exactly equal to the estimates of Bayesian decoding ($\widehat{\theta }(m)$). They are linearly related to each other, given by ${\widehat{\theta }}_{r}(m)=\alpha \widehat{\theta }(m)$. The value of $\text{α}$ is determined by the response range. The larger the response range is, the larger the $\alpha $is. (c) A quad-modal long-term prior distribution used in the current study, in which the four peaks were located at the 0°, ±90°, and 180°.

Figure 3. Results of predicted estimation errors and standard deviations in Experiment 1. (a) and (b) show the estimation error results of the 800-ms and 250-ms conditions. (c) and (d) show the standard deviation results of the 800-ms and 250-ms conditions. The dashed line illustrates the mean estimation error or standard deviation averaged across all participants. The solid line illustrates the mean predicted estimation error or standard deviation averaged across all participants. The shaded area illustrates the standard error of predicted estimation errors or standard deviation across all participants. In addition, the panels in the left, middle, and right column illustrate the results of models with a weighted integration prior (Equation 4.3), the quad-modal long-term prior (Equation 4.1), and the short-term prior (Equation 4.2).

Figure 4. The values of model parameters against different models. (a), (b) and (c) corresponds to the {$\omega \_stim,\kappa \_sen,\alpha $} that indicate the weight of stimulus distribution in the weighted- integration prior, the magnitude of sensory noise, and the scaling factor of the motor-action mapping. Please check the Appendix Figure 2 for the ${\kappa }_{prior}$s of different models. Please check Table 1 for the parameters’ significances.

Figure 5. Five new long-term priors used in the models.

Figure 6. Results of Experiment 2. (a) Estimation error and (c) standard deviation is against the PLW direction. The dots and shaded areas indicate the mean estimation error and standard error averaged across all participants. Note that the baseline condition is the same as the 800-ms condition in Experiment 1. (b) Predicted estimation error and (d) predicted standard deviation are against the PLW direction. Each panel corresponds to one model with a certain prior. The dashed line illustrates the mean estimation error and standard deviation averaged across all participants. The solid line illustrates the mean predicted estimation error and standard deviation averaged across all participants. The shaded area illustrates the standard error across all participants.

Figure 7. The weight of stimulus distribution {${\omega }_{stim},$} in the weighted- integration prior against different stimulus distributions. Note that the baseline condition is the same as the 800-ms condition in Experiment 1.