Bayesian Observer Fitting

To compare human performance to that of the Bayesian observer, we first determined the performance of the Bayesian observer on the same perceptual tasks completed by the human participants for a wide range of parameter values. We next found the parameter values for the Bayesian observer that best fit each human participant.

The Bayesian observer's psychometric function for each t_com is the proportion of trials on which it perceives l_com to be greater than l_ref, plotted against l_com: ψ_model(l_com). The Bayesian observer's percepts for reference and comparison distances are normally distributed, with means and SDs resulting from the perceptual length contraction formula:

Note that l* can take on negative or positive values. A negative value would indicate a perceptual spatial reversal in which the more distal tap was perceived to be more proximal. However, the participants' task was to judge only which tap pair had longer spatial separation. Therefore, to determine the Bayesian observer's psychometric function, we used the percept distributions (Eqs. 2) to calculate the probability that the absolute value of lcom* was greater than the absolute value of lref:*

The psychometric function, and consequently the PSE, depended on the Bayesian observer's parameter settings—its spatial uncertainty (σ_s) and low-speed expectation (σ_v)—as well as t_com (Fig. 3, C and D).

We next asked whether the Bayesian observer, with a single σ_s and σ_v, could match a human participant's performance across all t_com blocks of one tap strength. For each human, we modeled the psychometric function, ψ(l_com), as a mixture of the Bayesian observer's psychometric function and a lapse rate term:

For each combination of 50 σ_s values (range 0.1–5.0 cm), 100 σ_v values (range 1–100 cm/s), and 8 ε values (range 0.01–0.08), we calculated the likelihood:

This is the probability, given σ_s, σ_v, and ε, of the participant's complete behavioral data set (d) at a given tap strength: for every comparison length and time, the number of trials in which the comparison length was perceived to be longer than the reference (n_com) and the number of trials in which the reference was perceived to be longer than the comparison (n_ref). Beginning with a uniform prior over σ_s, σ_v, and ε, we used Bayes' rule to find the joint (σ_s, σ_v, ε) posterior. We marginalized this over ε to obtain the joint (σ_s, σ_v) posterior. We read out the mode to obtain our best estimates for these parameters, from which we calculated the best estimate τ = σ_s/σ_v.