Fig. 3.

Bayesian model predictions of trial-to-trial perceptual shifts and timing intervals. a Our Bayesian model predicts that (shown schematically) if τ_O − τ_A > μ_AO action and outcome binding will happen. Otherwise, i.e., τ_O − τ_A < μ_AO, action-outcome repulsion will occur. In both cases, the perceived timings in the baseline move (compress or stretch) towards the temporal consistency ${\widehat{t}}_{\mathrm{O}}-{\widehat{t}}_{\mathrm{A}}\approx {\mu }_{\mathrm{AO}}$ in the operant condition. b, c When τ_O − τ_A > μ_AO, there is positive perceptual shift in action awareness (${\widehat{t}}_{\mathrm{A}}-{\tau }_{\mathrm{A}}>0$) and negative perceptual shift in outcome awareness (${\widehat{t}}_{\mathrm{O}}-{\tau }_{\mathrm{O}}<0$). The opposite happens when τ_O − τ_A < μ_AO. Both binding and repulsion occur in both voluntary and involuntary conditions, but very little effect in the sham condition. d The Bayesian estimates follow the sensory inputs in the baseline condition, i.e., ${\tau }_{\mathrm{O}}-{\tau }_{\mathrm{A}}\approx {\widehat{t}}_{\mathrm{O}}-{\widehat{t}}_{\mathrm{A}}$, where all trials are acausal ($\widehat{\xi }$ = 0) by definition. e The Bayesian estimate shifts towards the prior assumption, ${\widehat{t}}_{\mathrm{O}}-{\widehat{t}}_{\mathrm{A}}\approx {\mu }_{\mathrm{AO}}$, when the sensory inputs are highly consistent with the prior, τ_O − τ_A ≈ μ_AO, and therefore when causality is detected ($\widehat{\xi }$ = 1). Otherwise, the estimate of action and outcome timings follow the sensory inputs. The fitted causal prior P(ξ = 1) is 0.9, 0.9, and 0.1 for the voluntary, involuntary, and sham conditions, respectively (as in Fig. 2). The per-trial results are grouped accordingly into bins of width 200 (randomly chosen), and the mean and SD for each bin are plotted. This format is followed each time a quantity of interest is plotted as a function of τ_O − τ_A