Task timing. (A) Temporal-order-judgment task administered in the pre- and post-tests. In each trial, participants made a temporal-order judgment in response to an audiovisual stimulus pair with a varying stimulus-onset asynchrony (SOA). Negative values: auditory lead; positive values: visual lead. (B) Oddball-detection task used in the exposure phase and post-test top-up trials. Participants were repeatedly presented with an audiovisual stimulus pair with a SOA that was fixed within each session but varied across sessions. Occasionally, the intensity of either or both of the auditory and the visual stimuli was increased. Participants were instructed to press a key whenever such an oddball stimulus occurred.

Behavioral results. (A) The probability of reporting that the auditory stimulus came first, the two arrived at the same time, or the visual stimulus came first as a function of SOA for a representative participant in a single session. The adapter SOA was -0.3 s for this session. Curves: best-fitting functions estimated jointly using the data from the pre-test (dashed) and post-test (solid). Shaded areas: 95% bootstrapped confidence intervals. (B) Mean recalibration effects (shifts in the point of subjective simultaneity from the pre- to the post-test phase) averaged across all participants as a function of adapter SOA. Error bars: ±SEM.

Illustration of the model for cross-modal temporal recalibration. (A) The probability density function of the arrival latency of the auditory and visual signals relative to the physical onset of each stimulus. (B) The resulting probability density function of the measured SOA, m, before (dashed) and after (solid) recalibration. The measurement distribution peaks at the physical SOA plus an audiovisual temporal bias. Temporal recalibration is modeled as cumulative changes in the audiovisual bias, Δ, across the exposure phase. (C) Two recalibration models. The fixed-update model updates the audiovisual bias so that subsequent measurements of the audiovisual SOA approach zero. The causal-inference model updates the audiovisual bias based on the perceived SOA, ŝ, i.e., taking different causal scenarios into account. The percept ŝ is computed as a weighted average of estimates inferred from the scenarios of a common cause, C = 1, and separate causes, C = 2. α: learning rate. See text for details.

Model predictions. (A) Data and model predictions of recalibration as a function of adapter SOA. (B) Model prediction of the asymmetry index, the summed recalibration effect across adapter SOA. Dots: individual participants. Error bars: 68% bootstrapped confidence intervals. Identity line: perfect model prediction.

Model simulations. (A) Effect of the prior probability of a common cause on cross-modal temporal recalibration. (B) Asymmetry due to differing uncertainty of auditory vs. visual arrival latency. Left panel: When auditory uncertainty is smaller than visual uncertainty, reducing auditory uncertainty leads to less recalibration in response to visual-lead adapter SOA. Right panel: when visual uncertainty is smaller than auditory uncertainty, the opposite effect results. Top-left insets: corresponding SOA likelihood functions for a measured SOA of zero as auditory or visual uncertainty is varied.

Causal-inference model of the temporal-order-judgment task. (A) An example measurement distribution for a SOA of zero. The measurement distribution peaks at the audiovisual bias β, whose left and right slopes reflect the visual and auditory uncertainty, respectively. The point of subjective simultaneity is marked by the dashed line. (B) Simulated estimate distribution for a SOA of zero. The dashed lines represent the criteria placed symmetrically around zero, forming a temporal window of SOA estimates treated as simultaneous. The areas under the estimate distribution partitioned by the criteria indicate the probabilities of the three possible responses for a stimulus pair with a SOA of zero. (C) Simulated psychometric function computed by repeatedly calculating the probability of each possible response for all SOAs.

Model parameters. Check marks signify that the parameter is used for determining the likelihood of the data from the temporal-order judgment task in the pre- and post-test phase and/or for the Monte Carlo simulation of recalibration in the exposure phase.