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A Bayesian psychophysics model of senseof agency

1234567890():,;

Roberto Legaspi

1,2 & Taro Toyoizumi1,2Sense of agency (SoA) refers to the experience or belief that one’s own actions caused an
external event. Here we present a model of SoA in the framework of optimal Bayesian cue
integration with mutually involved principles, namely reliability of action and outcome sensory signals, their consistency with the causation of the outcome by the action, and the priorbelief in causation. We used our Bayesian model to explain the intentional binding effect,
which is regarded as a reliable indicator of SoA. Our model explains temporal binding in bothself-intended and unintentional actions, suggesting that intentionality is not strictly necessary
given high conﬁdence in the action causing the outcome. Our Bayesian model also explainsthat if the sensory cues are reliable, SoA can emerge even for unintended actions. Our formal
model therefore posits a precision-dependent causal agency.

1 Laboratory for Neural Computation and Adaptation, RIKEN Center for Brain Science, Wako City, Japan. 2 RIKEN CBS-OMRON Collaboration Center, 2-1
Hirosawa, Wako City, Saitama 351-0198, Japan. Correspondence and requests for materials should be addressed to R.L. (email: roberto.legaspi@riken.jp)
or to T.T. (email: taro.toyoizumi@riken.jp)

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ense of agency (SoA) is the registration1 that the self initiates actions to inﬂuence its external environment2. It
therefore accompanies voluntary actions3–6, allows oneself
to feel distinct from others7–9, and be responsible for its own
actions2,6,10,11. Studies show SoA emerges from, and is particularly sensitive to any disruption in, the congruous ﬂow ofintentional actions to expected sensory outcomes12. Crucially, thedegradation of this experience characterizes certain psychiatricand neurological disorders13–15. For example, studies show
schizophrenic patients tend to attribute someone else’s actions tothemselves. Despite its signiﬁcance16–18, the literature still lacks
the computational principles that can elucidate SoA.
We theorize SoA as the conﬁdence in one’s perception of theaction-outcome effect, and that it is consistent (e.g., spatially or
temporally) with the hypothesis that the action caused the outcome. We adapted the model of Sato et al.19 that was originallyused to explain the ventriloquism effect as a Bayesian estimate of
a common source behind the consistency of the audiovisual stimuli, akin to being the common cause20 of the audiovisual
integration. Formalizing SoA by this Bayesian psychophysics
principle distinguishes our theory from existing works.
We compared the predictions of our model with the results of
two pertinent intentional binding studies. Intentional binding,which is the perceived compression of the time interval between
voluntary action and its outcome, has been reported as a reliable
implicit measure of SoA and has been used in a large number ofstudies providing valuable analyses on the temporal perception ofaction-outcome effects and the nature of SoA21. The seminal
experiment of Haggard et al.3 investigated the perceived actionoutcome timing effects in three conditions: voluntary wherein thesubject intentionally presses a button, involuntary wherein muscle twitches of the subject’s hand are induced by a transcranial
magnetic stimulation (TMS) applied to the motor cortex, andsham TMS wherein the TMS on the parietal cortex produces
audible clicks but no movement (hereafter, voluntary, involuntary, and sham conditions, respectively). Haggard and colleagues
computed the time interval between the perceived action timings(with the timings of either voluntary actions, muscle twitches, or
audible TMS clicks as control experiment) and the perceived
timings of subsequent tone stimuli. They showed that voluntaryactions produced intentional binding, involuntary muscle twitches produced repulsion, i.e., prolonged opposite perception of
the action-outcome intervals, and audible TMS clicks produced
neither binding nor repulsion. Hence, they posit intentionality isnecessary to achieve action-outcome binding.
The second pertains to the study of Wolpe et al.22, whichinvestigated the contribution of cue integration to intentionalbinding by manipulating the reliability of the consequent tone
relative to a background white noise. Such manipulation resulted
in three levels of tone uncertainty conditions, namely low,intermediate, and high uncertainty. Their analyses showed thatwhen tone reliability was reduced, the perceptual shift in tone
timing towards the action was increased.
Although Bayesian integration was proposed as a generalprinciple behind SoA14,23, it was unknown whether the observed
action-outcome temporal compression and repulsion effects are
consistent with Bayesian principles, and if indeed the case, thequestion is how. Our Bayesian model reproduces the above
empirical results on intentional binding based on a computational
principle. Further, it goes beyond timing estimations by exposing
the underlying Bayesian mechanisms that possibly drove thetemporal binding. Our Bayesian model explains the perceived
compressed action-outcome time interval is more consistent withthe prior belief of the causal role of one’s action in producing theimmediate outcome and thus increases the conﬁdence in theBayesian estimate assuming the causal case, modeled as SoA.
2

Moreover, our model explains intentional binding as a speciﬁcclass of the more general notion of causal binding. Our Bayesianmodel predicts that intentional binding generally happens on a
per-trial basis, yielding a bimodal distribution of the perceived
action-outcome interval. Lastly, the model also predicts that if thesensory input signals are perceived as reliable (precise), SoA may
arise even for unintended actions, which serves as a testable
theory for future SoA experiments.
Results
Bayesian inference model of action-outcome temporal binding.
We considered the experimental setup of intentional binding
where a subject presses a button (i.e., the action) and a tone (i.e.,the outcome) sounds 250 ms after the button press. The true
action and outcome timings are thus described by tA = 0 ms andtO = 250 ms, respectively, but they are unknown to the subject.
The task for the subject is to accurately report her perceived
timings of the button press and tone. We assume the arrival of
relevant sensory input informing the timing of each of these
physical events involves sensory delay d and jitter of variance σ2due to sensory noise. Thus, the arrival time τA of sensory inputthat signals the action timing
 is assumed
 to be generated from a
Gaussian distribution, N tA þ dA ; σ 2A , with mean tA þ dA andvariance σ 2A . Similarly, the arrival time τO of sensory
input that
signals the outcome timing is generated from N tO þ dO ; σ 2O .
The brain often resolves such ambiguity in sensory inputs byintegrating multiple sensory cues akin to the Bayesian “ideal
observer”24. Hence, we model a Bayesian observer who estimatesaction timing tA and outcome timing tO based on thecorresponding noisy sensory inputs arriving at time τA for theaction and τO for the outcome. The conditional probability
distributions of τA and τO that the Bayesian observer uses aremodeled as Gaussian distributionsðτ  t Þ2Pðτ A jtA Þ / exp  A 2 A
2σ Aðτ  t Þ2;Pðτ O jtO Þ / exp  O 2 O
2σ O

ð1Þ

with mean tA and tO, and variance σ 2A and σ 2O for action andoutcome, respectively. It is noteworthy that sensory delays dA anddO are not included in Eq. (1) for the reason we describe in thenext paragraph.
Before studying the binding effect, let us consider simple
baseline conditions. In one baseline condition, the action timing
is reported by the subject without the presentation of an outcome
tone. If no prior knowledge is available, the Bayesian observerreports the action timing that maximizes the conditional
probability distribution in Eq. (1). Hence, the estimated action
timing ^tA ¼ τ A is solely determined by the noisy sensory input
informing the action timing.
 In this case, the model predicts thatthe distribution of ^tA is N tA þ dA ; σ 2A . The mean and SD of ^tA
in the baseline condition were experimentally reported, e.g.,
Haggard’s results in the voluntary condition suggest dA = 6 ms
and σA = 66 ms (refer to Table 1 in Methods for all conditionbased dA and σA values). Importantly, we assume that theobserver does not take into account sensory delay dA in Eq. (1). If
the Bayesian observer included its effect, it could compensate forthis delay and report unbiased timing, which was not the case in
the experiment. Therefore, we assume that the observer was
unable to take into account the sensory delay in Eq. (1). In theother baseline condition, the subject passively listens to a tone
and reports its timing. This case goes parallel to the above caseand the model predicts that the estimated tone timing is ^tO ¼ τ O ,NATURE COMMUNICATIONS | (2019)10:4250 | https://doi.org/10.1038/s41467-019-12170-0 | www.nature.com/naturecommunications

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Table 1 List of Bayesian model parameters and their values
Baseline parameters
All are temporal values in unit ms• Set A: Reported by Haggard et al.3
Voluntary action
Involuntary action (TMS-induced muscle
twitch)Sham TMS (audible click only)
Auditory tone
• Set B: Reported by Wolpe et al.22
Voluntary action
Low tone uncertainty
Intermediate tone uncertainty
High tone uncertainty

dA

σA

6
83

66
83

32

78

dO

15
−8

Operant parameters
μAO = 230 ms σAO = 10 ms• Set C: Obtained by our Bayesian model
Voluntary action
Involuntary action (TMS-induced muscle
twitch)Sham TMS (audible click only)P(ξ = 1)
0.9
0.9
0.1

72
• Set D: Obtained by our Bayesian model

75
35
46
95



which is distributed according to N tO þ dO ; σ 2O . The comparison of this model prediction to Haggard’s experiment, e.g., would
be dO = 15 ms and σO = 72 ms (refer to Table 1).
Next we study the effect of binding when the subject makes anaction and then listens to the outcome tone, commonly referred
to as the operant condition. In this case, the Bayesian observer
makes an inference not only based on the conditional probability
distribution in Eq. (1) but also based on the prior distribution oftA and tO. Adapting the Bayesian model of the ventriloquism
effect19, we assume the prior distribution depends on theobserver’s belief whether the action caused the outcome, i.e.,the causal case: ξ = 1, or the action and the outcome areunrelated, i.e., the acausal case: ξ = 0:
82
< exp  ðtO tA μAO Þ ; ðξ ¼ 1Þ
2
2σ AO
ð2Þ
PðtA ; tO jξ Þ /:1:
ðξ ¼ 0Þ
The action causes the outcome in the causal case (ξ = 1) so thatthe outcome timing involves a typical delay μAO with respect tothe action timing and a Gaussian-distributed jitter of SD σAO. The
outcome is caused by something other than the action in
the acausal case (ξ = 0) so that tA and tO are independent. Lastly,
we deﬁne P(ξ) as the prior for each belief: P(ξ = 1) for the causal
case and P(ξ = 0) = 1 − P(ξ = 1) for the acausal case. We
hypothesize the estimation of ξ to be essential for the perceptionof causality and SoA (explained below).
Given a pair of sensory inputs at τA and τO, the Bayesian
observer estimates the most probable timing for the action andthe outcome, and whether these observations are consistent
with the causal case. According to the Bayesian estimation
theorem, the maximum-a-posteriori (MAP) estimate (^tA ; ^tO ; ^ξ) of
the corresponding pair of physical sensory timing (tA, tO) and thecausal variable ξ is given by^tA ; ^tO ; ^ξ ¼ arg max PðtA ; tO ; ξjτ A ; τ O Þ;ð3ÞtA ;tO ;ξ

where P(tA, tO, ξ|τA, τO) is the posterior probability distribution of(tA, tO, ξ) given the sensory inputs (τA, τO). Hence, whether theBayesian observer estimates the action-outcome effect to becausal or not depends on the posterior ratio comparing the causal
case (ξ = 1) and the acausal case (ξ = 0), namelyr  max PðtA ; tO ; ξ ¼ 1jτ A ; τ O Þ= max PðtA ; tO ; ξ ¼ 0jτ A ; τ O Þ:tA ;tOσOtA ;tO

ð4Þ
Causality is detected if the conﬁdence in the causal estimate is
greater than that in the acausal case, i.e., r > 1. The MAP estimate

61
66
90

Low tone uncertainty
Intermediate tone uncertainty
High tone uncertainty

0.9
0.6
0.5

of Eq. (3) is then given by (see Methods for the derivation)^tA ; ^tO ; ^ξ ¼

(

 
σ2 
σ2 
τ A þ σ 2A τ O  τ A  μAO ; τ O  σ 2O τ O  τ A  μAO ; 1 ;ðr>1Þðτ A ; τ O ; 0Þ;

ðr<1Þ

tot

tot

ð5Þ
σ 2tot  σ 2A þ σ 2O þ σ 2AO .

This indicates, on one hand, thatwithperceptual shift does not happen if the causality is not detected(^ξ ¼ 0)—the time estimates for action and outcome simply reﬂect
the corresponding sensory signals in this case. On the other hand,perceptual shift happens if the causality is detected (^ξ ¼ 1)—the
action and outcome timing attract each other in the form ofbinding if τO − τA > μAO and repel each other in the form of
repulsion if τO − τA < μAO. The magnitude of perceptual shift for
the action and outcome timing depends on coefﬁcients σ 2A =σ 2tot
and σ 2O =σ 2tot , respectively, implying that perceptual shift is greater
for a more unreliable stimulus. This model predicts that theoccurrence of binding, repulsion, or no perceptual shift is trialdependent, inﬂuenced by the noisy sensory signal τO − τA
informing the action-outcome interval. We denote the probabilityof detecting causality (i.e., ^ξ = 1) by Pc (see Methods for its
analytical expression). Pc increases with larger P(ξ = 1) andsmaller σAO if σ AO  σ A ; σ O .
Proposed measure of SoA. Separate from the judgement of
causality described above, we also directly quantify the conﬁdence
in the causal MAP estimate
CCE ¼ max PðtA ; tO ; ξ ¼ 1jτ A ; τ O ÞtA ;tO

ð6Þ

and we postulate this quantity to be a possible indication of the
pre-reﬂective feeling of agency (FoA; see Discussion). The analytical expression of conﬁdence in causal estimate (CCE) in
Methods yields the following requirements to have high CCE: (i)the timing of sensory signals must be consistent with the causation of the outcome by the action, namely τO − τA ≈ μAO; (ii) the
causal prior probability P(ξ = 1) must be high; (iii) the sensory
inputs must be precise, i.e., the amplitudes σA and σO of sensory
jitter must be small enough. Furthermore, by computing for the
peak of the conditional probability distribution, instead of integrating over tA and tO, CCE does not only indicate the causationof the outcome by the action but is also sensitive to the accuracyof the action and outcome timing estimates. We therefore posit
SoA as encapsulation and manifestation of several pertinent
aspects, which include temporal consistency in the actionoutcome effect, the prior belief of an action causing the outcome, and the reliability of the perceived sensory signals. Hence,
our Bayesian model coherently explains not just SoA that arises
from the causation of the outcome by the action but also one that

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is inﬂuenced by the reliability of the different agency cues—aprecision-dependent causal agency.
Simulation results and model predictions. Here we brieﬂy
describe how we obtained the parameter values used in our
simulation (but see Methods for more details about the modelﬁtting and simulation). Fitting of dA, dO, σA, and σO is straightforward; they are suggested by the means and SDs of the reported
subjects’ baseline estimation errors (Table 1-Sets A and B in
Methods). After ﬁxing these parameters, the model is left withthree free parameters, μAO, σAO, and P(ξ = 1). As described in
Eq. (5), μAO has an important role in determining whether
binding or repulsion happens in each experimental condition. A
ﬁxed value of μAO = 230 ms successfully accounts for this qualitative behavior in all the six experimental conditions (three from
Haggard et al.3 and three from Wolpe et al.22) that we study. The
analytical expressions in Methods suggest that σAO and P(ξ = 1)
have a largely overlapping role in detecting causality. Causality ismore likely detected if σAO is small or P(ξ = 1) is large, although
the exact mechanisms are slightly different. At least one of thesetwo parameters needs to be adjusted according to the conditions
to account for the experimental observations. For simplicity, we
ﬁx σAO = 10 ms to be a small enough constant to permit
noticeable perceptual shift and adjust P(ξ = 1) (see Table 1 for theparameter values in six experimental conditions) to account for
two observations in each condition, namely the perceptual shifts
in the action timing and the outcome timing.
Our results show that our simple Bayesian model qualitatively
reproduces the perceptual shifts that were reported in the study
by Haggard et al.3 (Fig. 1). Consistent with their ﬁndings, ourBayesian observer inferred the perceived action and outcome
timings to shift towards each other in the voluntary condition,
resulting in compressed temporal intervals between the action
and outcome perceptual shifts. However, reversed and prolonged
perceptual shifts were observed in the involuntary condition. The
model also reproduced no appreciable perceptual shifts in thesham condition.
Our Bayesian model predicts binding and repulsion to increase
with stronger causal prior (Fig. 2). From Eq. (5), the amount ofbinding or repulsion is given by ðτ O  τ A Þ  ^tO  ^tA , which is(A) Judgments in baseline conditionsτ O  τ A  μAO ðσ 2A þ σ 2O Þ=σ 2tot in the causal case ð^ξ ¼ 1Þ andnone otherwise ð^ξ ¼ 0Þ.
signals are As the sensory distributed
according to τ A  N tA þ dA ; σ 2A and τ O  N tO þ dO ; σ 2O ,theaverageofτO − τA − μAO
factoris
m ¼ tO  tA þ dO  dA  μAO . Hence, the sign of m determines
whether binding or repulsion is predicted on average. With thecurrent set of parameters, m is positive in the voluntary
condition, yielding binding, and negative in the involuntary
condition, yielding repulsion (schematically drawn in Fig. 3a).
Perceptual shift is almost zero regardless of the causal prior P(ξ =1) in the sham condition, because m ≈ 0. We chose P(ξ = 1) = 0.1
for this under-constrained sham condition, assuming thatcausality would not be frequently detected.
Our Bayesian model provides interesting insights on what
possibly drives the perceived action-outcome temporal compression and repulsion effects. We empirically observed sensory delay
d to increase with larger SD σ of the Gaussian-distributed jitter(observed in both Haggard et al. 3 and Wolpe et al.22; see Table 1
in Methods). This may imply that, as action or outcome
ambiguity is increased due to noise (greater σ) for increased
sensory uncertainty, more time would be needed (greater d) for asensory input to reach the subject’s perceptual threshold for
temporal awareness in the baseline condition. Thus, because ofm’s dependency on dO − dA, binding more likely happens when
the outcome is unreliable (i.e., with large dO) and repulsion more
likely happens when the action is unreliable (i.e., with large dA).
To further illustrate the model prediction from our simulations, we plotted separately the action and outcome perceptual
shifts for the three conditions as functions of the temporal
disparity τO − τA (c.f. Eq. (5)). Indeed, our data show that for
instances in which τO − τA > μAO, action awareness is delayed(positive action shift, Fig. 3b) and outcome tone is anticipated(negative outcome shift, Fig. 3c), thereby demonstrating binding.
The opposite happens when τO − τA < μAO, thereby demonstrating repulsion in both action and outcome awareness (Fig. 3b, c,
respectively). We then plotted how the model’s MAP estimates on
the action-outcome interval are affected by the sensory timedifference τO − τA in the baseline (here, ^ξ = 0 is forced; Fig. 3d)and operant (Fig. 3e) conditions. We observe from the baseline
condition that the MAP estimates follow sensory inputs,(A) Judgments in baseline conditions

Tone
only

Action
only



Tone
only

Actiononly

Time(B) Judgments in operant conditions
Action + Tone

(B) Judgments in operant conditions
Action + Tone

(1) Voluntary condition(1) Voluntary condition

189 ms

222 ms–46 ms+15 ms–15 ms+13 ms(2) Involuntary condition(2) Involuntary condition

308 ms–27 msTime

295 ms+31 ms(3) Sham condition+19 ms–26 ms(3) Sham condition

249 ms

250 ms–7–8
Results from Haggard et al. (2002)

0 ms
0 ms
Simulation results with Bayesian modelFig. 1 Qualitative replication of the empirical results reported by Haggard et al.3. Each subject’s mean judgment error in the single-event baseline condition
was subtracted from the mean judgment error for the corresponding event in the operant condition. This resulted in the values underneath the blocks thatindicate the magnitude and direction to which the temporal perceptions shifted. A positive perceptual shift informs delayed awareness and a negative shift
informs anticipated awareness. The action and outcome timings are perceived to shift towards each other in the voluntary condition. In contrast, they areperceived to repulse in the involuntary condition. There is no discernible perceptual shift in the sham condition
4

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300
Voluntary condition
Involuntary condition
Sham condition

Perceptual shift intervals
(ms)

290
280

Action-outcome
repulsion

270
260
250

Action-outcome
binding

240
230
220

0

0.2

0.6
0.4
P ( = 1)0.8

1Fig. 2 Bayesian model predictions of the inﬂuence of causal prior strengthon action-outcome perceptual shifts. The best estimates of the Bayesianmodel (in Fig. 1) were obtained from different causal priors, speciﬁcally, P(ξ = 1) is 0.9, 0.9, and 0.1 (marked by the colored dots) for the voluntary,
involuntary, and sham conditions, respectively. The intervals between theaction and outcome perceptual shifts shrink in the voluntary, but widen in
the involuntary, condition with a strong causal prior. Minimal changes inperceptual shifts are predicted for the sham condition even with a strongcausal prior^tO  ^tA ¼ τ O  τ A , whereas the perception of action and outcome timings shifted towards the prior mean, ^tO  ^tA  μAO , inthe voluntary and involuntary conditions but not so much in thesham condition with weak causal prior. Therefore, our model isagnostic as to whether the action is self-intended or unintended.
Binding towards ^tO  ^tA  μAO will happen, be it in the opposite
direction, as long as the action is believed to have caused theoutcome. This suggests that causality is the phenomenon that
underlies intentional binding, and likely SoA, with self-intended
causality being a speciﬁc case. The temporal window of τO − τA
for detecting causality is wider in the voluntary and involuntary
conditions than in the sham condition.
We then examined how the prior belief in causation affects our
proposed measure for SoA in Haggard’s experimental setup. Ourmodel predicts CCE to strengthen together with the causal prior
but its strength differs depending on the conditions even at thesame strength of the prior (Fig. 4a). Interestingly, dA and σA are
the only parameters of our Bayesian model that differentiate thethree conditions in this ﬁgure. As we described above, these twoparameters are empirically correlated such that the delay dA
increases with larger σA. Hence, the difference in CCE in the three
conditions can be attributed to the inequalities in SDs of thesubjects’ action timing estimation errors in the three conditions:Sham<σ Involas per the data of Haggard et al.3 Haggard et al.
σ Vol
A <σ A
A
speculated that the unexpected and surprising quality of the
TMS-induced movement could account for the repulsion effect in
the involuntary condition. We suggest that this surprise might
have introduced uncertainty in the perception of action input
signals. Hence, although subjects were certain of the nature of
their voluntary actions, they could be less certain of the
proprioception signals induced by TMS, which could explain
the inequalities in σA. As a result, the model gives CCEVol >CCESham > CCEInvol according to requirement (C), i.e., reliable
sensory inputs, for having high CCE when compared at the same
strength of the causal prior.
The relation between CCE and SoA becomes clear when weanalyze them with the ﬁtted values of the causal prior (P(ξ = 1) =0.9 for the voluntary and involuntary conditions and P(ξ = 1) =0.1 for the sham condition as indicated in Table 1). Figure 4b
plots CCE on a per-trial basis as functions of the temporal

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disparity τO − τA (c.f. the analytical expression for CCE in
Methods). CCE in the voluntary condition has a higher peak than
the involuntary condition as we described above (due to small σAin the voluntary condition for the requirement (C)). In bothvoluntary and involuntary conditions, CCE diminishes as τO − τAmoves farther from μAO because of the requirement (A) of small |τO − τA − μAO| for having high CCE. Finally, CCE for the shamcondition takes much lower values than the voluntary or
involuntary conditions because of the requirement (B) of large
P(ξ = 1) for having high CCE.
In a similar fashion, we then examined the underlying
psychophysical mechanisms that could account for the temporalbinding observed by Wolpe et al.22, in which three uncertainty
levels (high, intermediate, and low uncertainty) of the outcome
stimulus were tested. We use the Bayesian model that was used toreproduce the Haggard’s experiments with the same values of μAO
and σAO but adjusted the strength of the causal prior P(ξ = 1) to
ﬁt the reported action timing and outcome timing in eachcondition. We used P(ξ = 1) = 0.9, 0.6, and 0.5 for low,intermediate, and high tone uncertainty conditions, respectively(see Table 1 and Methods). This means that the prior belief incausation decreases with the tone uncertainty, which is plausible.
(Alternatively, we could increase σAO, which produces similar
results; see above discussion on model ﬁtting.)
Our model reproduces the experiments of Wolpe et al.22(Fig. 5a), qualitatively explaining the temporal binding theyobserved in terms of a single, coherent cue integration
formulation. The Bayesian estimate of the action-outcome
intervals shift towards ^tO  ^tA  μAO , as per the causal
temporal prior in Eq. (2) when causality is detected. On the
one hand, the magnitude of the shift is greater when theoutcome uncertainty is high (c.f. Eq. (5)). However, on
the other hand, causality is less frequently detected when theoutcome uncertainty is high with the reduced causal prior.
These two opposing effects are summarized in Fig. 5b. The
model can qualitatively reproduce the experiments if the former
effect is more dominant. Quantitatively, however, the latter
effect is necessary to mitigate the former effect.
Next, we plot how the Bayesian estimate of the action-outcome
interval, ^tO  ^tA , depends on the sensory inputs τO − τA. The
perceived intervals faithfully follow the sensory inputs in thebaseline condition (Fig. 5c), where all trials are acausal (^ξ ¼ 0) by
deﬁnition. In the operant condition (Fig. 5d), the Bayesianestimate shifts towards the prior assumption ^tO  ^tA  μAO when
the sensory inputs are highly consistent with the prior τO − τA ≈μAO and, thus, when the causality is detected (^ξ ¼ 1). Otherwise,
the estimate of action-outcome intervals follows sensory inputs.
The temporal window of τO − τA for detecting causality is wider
when the outcome uncertainty is lower.
Next, we quantify again CCE as a possible measure of SoA.
CCE diminishes with outcome uncertainty even when compared at the same level of causal prior (Fig. 6a). Hence, CCE
explicitly depends on the outcome uncertainty. When plotted asfunctions of temporal disparity, with the speciﬁc causal priors
obtained for each outcome uncertainty condition, the peakvalues of CCE noticeably differ across the uncertainty conditions (Fig. 6b). This is because of the different values of theoutcome uncertainty σO but also partly because of the different
values of the causal prior. In all conditions, CCE falls off withthe disparity of sensory inputs from the prior mean, |τO − τA −μAO|. This fall-off is milder when the uncertainty is lower.
These results clearly manifest again three basic requirements of
CCE as follows: (i) the consistency of sensory inputs with thecausal prior; (b) strong prior belief in causality; and (c) reliable
sensory inputs.

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NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-12170-0aO − A > AO
Actionbinding

OutcomebindingAOttˆO OA tˆAO − A < AO

Outcome
repulsion

Actionrepulsion

AO
tˆA

A

O

t *O

100
150
0
–50
–100
–150
–200
–200

AO Action binding

Action repulsion

0200
O – A (ms)

400

600

e

400
300

e

as

(

0)

lc

a
us

a

Ac

100
0

150
0
–50
–100
0

AO Outcome binding200
O – A (ms)

400

600

600
Voluntary condition
Involuntary condition
Sham condition

e
as

0)

lc

400
300

=
(ˆ

sa

au

AO

Ac

200

Causal case (ˆ = 1)

100
0

–100
–200
–200

100

500

ˆ =

Voluntary condition
Involuntary condition
Sham condition

150

–150
–200

Voluntary condition
Involuntary condition
Sham condition

200

200

Outcome repulsion

600
500

Baseline tˆO − tˆA(ms)

Voluntary condition
Involuntary condition
Sham condition

Outcome shift: tˆO − O
(ms)

Action shift: tˆA − A
(ms)

150

d

c

200

Operant tˆ0 − tˆA
(ms)

b

t

O tˆ

t *A

–100
0

200
O – A (ms)

400

600

–200
–200

0

200
O – A (ms)

400

600

Fig. 3 Bayesian model predictions of trial-to-trial perceptual shifts and timing intervals. a Our Bayesian model predicts that (shown schematically) if
τO − τA > μAO action and outcome binding will happen. Otherwise, i.e., τO − τA < μAO, action-outcome repulsion will occur. In both cases, the perceived
timings in the baseline move (compress or stretch) towards the temporal consistency ^tO  ^tA  μAO in the operant condition. b, c When τO − τA > μAO,
there is positive perceptual shift in action awareness (^tA  τ A >0) and negative perceptual shift in outcome awareness (^tO  τ O <0). The opposite happens
when τO − τA < μAO. 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., τ O  τ A  ^tO  ^tA , where all trials are acausal (^ξ = 0) by deﬁnition. e The
Bayesian estimate shifts towards the prior assumption, ^tO  ^tA  μAO , when the sensory inputs are highly consistent with the prior, τO − τA ≈ μAO,
and therefore when causality is detected (^ξ = 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 afunction of τO − τA

Discussion
We formalize SoA by drawing parallels from a Bayesian inferenceof the ventriloquism effect that estimates a common cause behind
its multisensory integration. Understanding causality has beenviewed to facilitate predictive, adaptable, and goal-directedactions25–27; hence, this may bring about SoA. Our Bayesianmodel integrates the action-outcome signals, compares them withthe prior expectation, and infers the causality between them aswell as the timing of these sensory signals. Our model could
concisely reproduce the intentional binding experiments by
Haggard et al.3 and Wolpe et al.22. Whether intentional binding
effects indeed follow Bayesian principles remained obscure.
6

Speciﬁcally, this was raised as an open question by Moore andFletcher14, pointing out only indirect empirical evidence existed
in support of Bayesian cue integration, and Wolpe et al.22 evenposited that Bayesian cue integration does not explain outcome
binding. Our model explains the temporal binding and repulsion
phenomena as compromise between the noisy sensory observations and the prior belief of the action-outcome timing. Importantly, our Bayesian model predicts that the perceptual binding is
generally trial-dependent and it must be correlated with theestimated causality ^ξ between the action and outcome. Thisprediction can be tested when the probability Pc for detecting
causality is not close to 0 or 1, by examining whether the

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NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-12170-0a3.5 ×10b–4

Voluntary condition
Involuntary condition
Sham condition3

2.5

Voluntary condition
Involuntary condition
Sham condition

2

2

CCE

CCE

2.5

1.5

1.5
1

1

0.5

0.5
0

–4

3 ×10

0

0.2

0.4

0.6

0.8

1

P ( = 1)

0
–200

0

200
O –  (ms)

400

600

Fig. 4 Bayesian model predictions of the conﬁdence in causal estimate (CCE), which is our proposed measure for SoA. a Our Bayesian model predicts CCE
to increase with a stronger causal prior. Furthermore, CCE differs for each condition even with equal prior strengths. This can be attributed to the differencein the amplitude of the jitter in the self-generated vs TMS-induced movement (muscle twitches) and audible clicks. b When plotted as functions of thetrial-to-trial temporal disparity τO − τA, with the speciﬁc causal priors obtained for each condition, marked in a, CCE has a higher peak in the voluntary
condition, but much lower values in the sham condition. Furthermore, CCE diminishes as the temporal disparity in sensory inputs moves further away fromthe prior mean |τO − τA − μAO|. This falling of the CCE is faster when the causal prior is weaker and the uncertainty in the action input signal is higher

distribution of action-outcome intervals is bimodal and whether
the intervals correlate with the reported causality between theaction and outcome. We have therefore shown how Bayesianmechanism may underlie intentional binding. This is a signiﬁcant
contribution, as no previous Bayesian proposals accounted forexperimental data on intentional binding and repulsion.
In addition, we theorize SoA as the CCE. CCE is high when theaction-outcome timing is consistent with the causal prior, thecausal prior is strong, and the action and outcome signals arereliable. This notion is consistent to what have been propoundedas demonstrations of SoA: SoA arises from the causal relation
between performed actions and their consequences1,21,27,28, andfrom the integration of different agency cues whose individualinﬂuences are determined by their reliability14,15,29–32. Hence, we
posit CCE to be a plausible measure of SoA. Here, Bayesian cue
integration in terms of CCE is derived based on the computational principle of optimal inference in contrast to empiricalobservations that causality and reliability are involved. Further,CCE can explain outcome binding in terms of cue reliability thatwas previously considered non-Bayesian22. CCE is not an indicator of intention or a simple estimate of whether the action
caused an outcome, but a new proposal of how SoA may emerge
from the conﬁdence in the estimate of the causality and timing(see discussion below on CCE against intention-based temporalbinding).
Speciﬁcally, we postulate CCE ﬁts the notion of a pre-reﬂective,
implicit FoA. Synofzik et al.1,30 provide a compelling account ofsuch feeling: FoA is best accounted for by multimodal weighting
and integration of different agency cues, and consists of an
automatic registration of whether an action or sensory event iscaused by the self or not. They posit FoA is nothing other thanﬁrst-person in that the self is implied; hence, no external attribution (e.g., to TMS that caused the action) is possible. In theevent that there is a feeling of exogenous causation, this will beoverwritten by an explicit, interpretative judgment of agency(JoA) based on contextual beliefs or rationalizations. Similarly,
the analytical expression of CCE shows that it is a multimodal
weighting and integration process that lies at the center ofobtaining a Bayesian causality inference. Furthermore, CCE itself
does not attribute causality to any external agent, such as in thecase of strong causal prior for TMS-induced movements. The
judgement of the causality, ^ξ, is then made based on the posteriorratio r that compares CCE with the conﬁdence in the acausal
estimate. Perceptual timing in our model simply reﬂects thesensory signals if the causality is not detected (^ξ = 0), whereas
they are overwritten by the inﬂuence of the prior if the causality isdetected (^ξ = 1). For example, in the involuntary condition of
Haggard et al.3, the estimated action and outcome timing by themodel repulse reﬂecting the judgment of the causality. A compelling speculation in the paper by Haggard et al.3 suggests thisnotion: the repulsion in the involuntary condition “reﬂects a
mental operation to segregate, and thus to discriminate, pairs ofevents that cannot plausibly be linked by our own causal agency”(p. 384). We suggest such mental operation ﬁts the notion of JoA,
as quantiﬁed by the time shifts in Eq. (5) with the detectedcausality, and the peculiar feeling of causation by the involuntary
movement to be FoA, quantiﬁed by CCE.
Following the above explanation, our theory therefore has adifferent take of the binding effect by Haggard et al.3, which
requires intentionality. Although intentional binding has been
repeatedly observed in the context of voluntary action, it remains
contentious in the literature whether it is indeed speciﬁc tovoluntary action, or causality contributes to this effect33. Ourmodel argues that the judgment of the causality is central to theperceived temporal action-outcome binding, consistent withcurrent evidence that competes with the intentional account: thetemporal binding is actually causal, not intentional21,27,34. For
example, our model judges the causation of the tone even by the
TMS-induced action in the involuntary condition. Hence, ourBayesian model predicts this unintended causality. Furthermore,
our Bayesian model predicts that the action-outcome timing
shifts toward the prior belief, ^tO  ^tA  μAO , when the causality isperceived irrespective of the nature of the action, whether selfgenerated (i.e., the voluntary condition) or unintended (i.e., theinvoluntary condition). Interestingly, this temporal binding
toward the same prior belief produces the compression andrepulsion effects if the perceptual delay in the action timing (dA)
is small and large, respectively. What causes this difference in theperceptual delay? We found that unreliable senses (with large σAor σO) tend to involve long perceptual delays (with large dA ordO). Hence, the observed large perceptual delay in the TMSinduced action timing may be caused by the internal prediction
error due to the absence of efference copy35–37 and artiﬁciallyperturbed neural activity. In this sense, intentionality is notstrictly necessary for the sense of causality but inﬂuences theprecision-dependent action-outcome timing shifts in our model.
This is consistent with a recent empirical ﬁnding of intentionalbinding-like effects that emerged without intentional actions33.

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NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-12170-0a(A) Judgments in baseline conditions
Action
Tone
only
only
Time
(B) Judgments in operant conditions
Action + Tone

(A) Judgments in baseline conditions
Tone
Action
only
only
Time
(B) Judgments in operant conditions
Action + Tone
(1) Low uncertainty tone condition(1) Low uncertainty tone condition
190 ms

160 ms–24 ms+36 ms+39 ms–51 ms(2) Intermediate uncertainty condition(2) Intermediate uncertainty condition
191 ms154 ms–65 ms+31 ms–26 ms+33 ms(3) High uncertainty tone condition(3) High uncertainty tone condition

170 ms

113 msResults from Wolpe et al. (2013)b
Perceptual shift intervals(ms)

Low uncertainty condition
Intermediate uncertainty condition
High uncertainty condition

220
200
Actionoutcome
binding

180
160
140
0

0.2

0.6
0.4
P ( = 1)

d

800
600

Low uncertainty
Intermediate uncertainty
High uncertainty

400

ˆξ =
e(

0.8

1

800
700

Low uncertainty
Intermediate uncertainty
High uncertainty

se

al

su

a
Ac

0)

s

ca

200
0

=
(ˆξ

0)

a
lc

600
Operant tˆO – tˆA(ms)

Baseline tˆO – tˆA(ms)

Simulation results with Bayesian model

260
240

120

c

–47 ms+33 ms–105 ms+32 ms

sa

au

Ac

500
400

AO

300
200

Causal case (ξˆ = 1)

100
–200

0

200

400
O – A (ms)

600

800

0
0

200

400
O – A (ms)

600

800

Fig. 5 Qualitative replication of, as well as predictions related to, the results reported by Wolpe et al.22. a Qualitative replication of the experimental results(left panel) by our Bayesian model (right panel). b The action-outcome binding increases under heightened uncertainty. However, causality is less detected
when the causal prior is lower, which decreases the action-outcome binding effect. The best estimates of the Bayesian model in a were obtained fromdifferent causal prior strengths, speciﬁcally P(ξ = 1) is 0.9, 0.6, and 0.5 (marked by the colored dots) for the low, intermediate, and high tone uncertainty
conditions, respectively. c, d The causal prior strengths that correspond to each condition were used for the Bayesian estimate of the action-outcome
timing interval ^tO  ^tA in the baseline and operant conditions. The Bayesian estimate follows the sensory inputs in the baseline condition where all trials are
acausal, but shifts towards the prior assumption, τO − τA ≈ μAO, when causality is detected. The temporal window of τO − τA for detecting causality is wider
when the outcome uncertainty is lower, which means more instances demonstrate binding

We predict that experimental manipulations that reduce σAwould increase perceived SoA even for unintended artiﬁcial
actions. The prediction is therefore distinct from what was previously considered and can therefore serve as testable prediction
for future experiments on causal agency.
Our theory also has a different take of the binding effect of
Wolpe et al.22. Wolpe et al.22 showed intentional binding as cue
integration with uncertainty in outcome signals. They speculated
that action and outcome bindings are driven by two distinct
mechanisms: action binding is predicted by cue integration but
outcome binding supports the predictive pre-activation hypothesis38, i.e., the neural representation of the sensory outcome is8

activated prior to it. Hence, the outcome signals are perceived
faster with less jitter than when it is not predicted to occur after
the action. This could explain why the subjects’ timing estimations are largely erroneous in the baseline condition and why theoutcome binding is greater than the action binding. Our theory,
although qualitative, explains both action and outcome bindings
by a single Bayesian cue integration mechanism. Our model
explains that the magnitudes of the action and outcome perceptual shifts, τO − τA − μAO, are inﬂuenced primarily by the
ambiguity of the outcome sensory signals, ðσ 2A þ σ 2O Þ=σ 2tot , andalso in part by the strength of the causal prior that diminisheswith outcome uncertainty.

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NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-12170-0a
3.5

Low uncertainty
Intermediate uncertainty
High uncertainty

3
2.5

2

2

CCE

2.5

1.5

1.5

1

1

0.5

0.5

0

×10–4

3.5

Low uncertainty condition
Intermediate uncertainty condition
High uncertainty condition

3

CCE

b

×10–4

0

0.2

0.4

0.6

0.8

0

1

0

200

P ( = 1)

400
O – A (ms)

600

800

Fig. 6 Bayesian model predictions of CCE as function of the causal prior and temporal disparity τO − τA. a The different effects of the causal prior on CCE
across the three conditions is evident even with equal causal priors, which means that CCE depends on outcome uncertainty. b When plotted as functionsof the temporal disparity τO − τA, given the condition-dependent causal priors (marked by the colored dots in a), CCE falls off with the disparity of sensory
inputs from the prior mean, |τO − τA − μAO|, faster when the outcome uncertainty is higher

The intentional binding paradigm has also been used to study
pathological SoA39–41. Patients with schizophrenia tend to havemuch stronger temporal binding than healthy volunteers.
Moreover, unlike healthy volunteers, their temporal binding ofaction timing does not depend on the probability of the outcome
tone presentation41. These results are explained by our Bayesianmodel by assuming that schizophrenia patients cannot easily
adapt their abnormally strong belief in causality (i.e., too large P(ξ = 1)) and the uncertainty in the outcome (i.e., σO). Anotherimportant point is that, unlike healthy volunteers, patients withschizophrenia exhibit temporal binding of action timing that
depends on the presence or absence of the outcome. It will be aninteresting future study to model this result by explicitly incorporating the probabilistic occurrence of the outcome in ourBayesian model.
In summary, we posit that as the Bayesian cue integration is
primarily precision-dependent so is our theory of SoA. Ourmodel predicts and awaits conﬁrmation that if the uncertaintyof the sensory input signals could be maintained small, evenunintended causal action may give rise to high CCE (hence,strong SoA)—hence, our notion of precision-dependent casual
agency. We posited the precise estimation that gives rise to SoA
encapsulates consistency in the perceived action-outcome
effect, the prior belief of the causation of the outcome by theaction, and the reliability of the perceived sensory signals. This
theory may shed light on the mechanism of reduced SoA in
psychosis, the understanding of the difference between FoA and
JoA, and the design of prosthetic devices that heighten SoA.
Furthermore, the challenge for future experiments that aim tolink intentional binding to SoA is to demonstrate effects beyond
what our model has already predicted: with the reliability of
sensory inputs and strength of causal prior diminished, intentionality should be sufﬁcient for strong intentional binding toemerge or not.
Methods
Analytical expressions for the Bayesian estimates. The MAP estimate (Eq. (3))
of the Bayesian observer has a simple analytical expression. The MAP estimation iscomputed based on the posterior probability P(tA, tO, ξ|τA, τO) = P(τA, τO, tA, tO,
ξ)/P(τA, τO), where the peak location only depends on the joint distribution P(τA,τO, tA, tO, ξ) in the numerator. The joint distribution is decomposed as P(τA, τO, tA,
tO, ξ) = P(τA|tA)P(τO|tO)P(tA, tO|ξ)P(ξ), where the conditional p
distributionsfor
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

action and outcome are Pðτ A jtA Þ ¼ exp ðtA  τ A Þ2 =ð2σ 2A Þ = 2πσ 2A andpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Pðτ O jtO Þ ¼ exp ðtO  τ O Þ2 =ð2σ 2O Þ = 2πσ 2O , respectively, and the priordistribution isPðtA ; tO jξ Þ ¼

82< exp  ðtO tA μAO Þ =Z:

2σ 2AOðξ ¼ 1Þ

1

ðξ ¼ 0Þ

1=Z0



2
pﬃﬃﬃﬃﬃRðt t μ Þ
with normalization constants Z1  R exp  O 2σA 2 AO dtA dtO  2π σ AO T
AORand Z0  R dtA dtO ¼ T 2 .
The prior probability distribution P(tA, tO|ξ) cannot be normalized unless aﬁnite range of (tA, tO) is deﬁned. Therefore,we only consider it in the range
R ¼ tA ; tO jtA 2 tA  T=2; tA þ T=2 ; tO 2 tO  T=2; tO þ T=2 and assume
that it is zero outside R, where again tA ¼ 0 ms and tO ¼ 250 ms are the true action
and outcome timings, unknown to the observer, and T = 250 ms is a large enoughbut ﬁnite constant that specify the interval lengths in consideration. Hence, theprior probability distribution P(tA, tO|ξ) must be normalized within R. Our results
are robust to a shift in the center of R.
We separately compute the peak location ð^tA ; ^tO Þ for the causal case ξ = 1 andthe acausal case ξ = 0 and, then, compare these two peaks. In the acausal case,because P(τA|tA) and P(τO|tO) take the maximum
 values at tA = τA and tO = τO,
respectively, the location of the acausal peak is ^tA ; ^tO jξ¼0 ¼ ðτ A ; τ O Þ and the peakPðξ¼0Þ
value is max Pðτ A ; τ O ; tA ; tO ; ξ ¼ 0Þ ¼ 2πσσ Z . In the causal case, the peak of thetA ;tOA O 0

joint distribution is founda quadratic function. The peak locationis by minimizing




σ2 
σ2 
given by ^tA ; ^tO jξ¼1 ¼ τ A þ 2A τ O  τ A  μAO ; τ O  2O τ O  τ A  μAO ,
σ tot

σ tot

where σ 2tot  σ 2A þ σ 2O þ σ 2AO is the total variance, and the peak value is computed2ðτ O τ A μAO ÞPðξ¼1Þ
as max Pðτ A ; τ O ; tA ; tO ; ξ ¼ 1Þ ¼ 2πσ σ Z exp . We deﬁne the log
2σ 2tA ;tOA O 1

tot

ratio of the posterior peaks for ξ = 1 and ξ = 0 by
2 !
max PðtA ; tO ; ξ ¼ 1jτ A ; τ O Þτ  τ A  μAO
t ;t¼ exp θ  O
r A O2σ 2tot
max PðtA ; tO ; ξ ¼ 0jτ A ; τ O ÞtA ;tO

h

i


ÞZ0
^ ^
with θ  log PPððξ¼1
ξ¼0ÞZ1 . If r > 1, the MAP estimate is given by tA ; tO j^ξ¼1 and^ξ ¼ 1, which predicts perceptual shifts. If r < 1, the MAP estimate is given by^t ; ^t j^ and ^ξ = 0, which predicts no perceptual shifts. The probability for
A

O

ξ¼0

detecting causality (i.e., ^ξ = 1) is also easily computable, because τO − τA − μAO is
distributed according to the Gaussian distribution N ðm; σ 2A þ σ 2O Þ withm  tO  tA þ dO  dA  μAO . Hence, the causality is detected if jτ O  τ A 
pﬃﬃﬃﬃﬃ
μAO j< 2θσ tot and this happens with probability
2 0
1
13
0
pﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃ
2θσ
1 6 B 2θσ tot  m C
þ
m
C7
B
tot
Pc ¼ 4erf @qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
A þ erf @qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
A5:
2
2 σ2 þ σ2
2 σ2 þ σ2
A

O

A

O

Next, we evaluate the conﬁdence in the causal MAP estimation
CCE  max PðtA ; tO ; ξ ¼ 1jτ A ; τ O Þ, which comprises the numerator of the ratio r.
tA ;tOTo quantify this conﬁdence, we need to ﬁrst evaluate P(τA, τO) = P(τA, τO, ξ = 1) +NATURE COMMUNICATIONS | (2019)10:4250 | https://doi.org/10.1038/s41467-019-12170-0 | www.nature.com/naturecommunications

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P(τA, τO, ξ = 0) withPðτ A ; τ O ; ξ ¼ 1Þ ¼

RRPðtA ; tO ; ξ ¼ 1; τ A ; τ O ÞdtA dtOR2ðτ τ μ Þ
Þσ AO
¼ Pðξ¼1
exp  O 2σA 2 AO
Z σ
1 tot

and
Pðτ A ; τ O ; ξ ¼ 0Þ ¼

tot

RR
R

PðtA ; tO ; ξ ¼ 0; τ A ; τ O ÞdtA dtOÞ¼ Pðξ¼0
Z0

Combining these expressions together, we obtain
CCE ¼ max Pðτ A ; τ O ; tA ; tO ; ξ ¼ 1Þ=Pðτ A ; τ O ÞtA ;tO2ðτ τ μ Þσ tot
¼ 2πσ σ σ Sigmoid θ  O 2σA 2 AO þ log σσAO
A O AO

tot

tot

where Sigmoid(x) = 1/(1 + e−x) is the sigmoid function.
In this work, we focus on the timing to investigate the intentional binding
effects but the mathematical elucidations above can permit other modalities (e.g.,visual or haptic) and structural properties (e.g., inter alia, location, size, shape, andtexture).
Model ﬁtting. The simple analytical expression for the Bayesian timing estimate
has an intuitive form and exposes all parameter dependencies explicitly. This
allowed us to perform a theoretically guided parameter search to reproduce theexperiments. We posit the perceptual delay d and jitter of SD σ due to sensory noise
explain the reported means and SDs of the baseline event timing. Hence, we could
immediately ﬁx the values of parameters dA, σA, dO, and σO (Table 1-Sets A and B).
This leaves us with three free parameters, μAO, σAO, and P(ξ = 1), where ﬁtting is
not direct. Equation (5) shows that μAO alone can determine the qualitative difference between action-outcome binding (τO − τA > μAO) and repulsion (τO − τA <μAO). This immediately gives us a possible range of μAO that couldaccount for bothbinding and repulsion, which is 182 ms < μAO < 259 ms, because tO þ dO tA þ dA  μAO must be positive and negative in the voluntary condition andinvoluntary condition, respectively, from Eq. (5). We therefore tested
μAO ϵ½190; 200; ¼ ; 240; 250ms with 10 ms increments. Our model also explainsthat both σAO and P(ξ = 1) can similarly inﬂuence the magnitude of binding andrepulsion (c.f. Eq. (5) and formula for Pc). To obtain discernible perceptual shifts,
σAO should be small and P(ξ = 1) should be large. As their effects are similar, we
ﬁxed σAO = 10 ms and we varied P(ξ = 1) later on, and observed how different
causal prior strengths inﬂuenced action-outcome binding and repulsion.
The principal measure of intentional binding is the mean perceptual shift of
temporal awareness of action and sensory outcome. A perceptual shift is the change
in the subjective estimation of action or outcome
 timing from the baseline to theoperant condition. This can be computed as E ^tA  E½τ A  and E ^tO  E½τ O  (c.f.
Eq. (5)) for action and outcome timings, respectively. A positive shift therefore
informs the perception of timing shifted later in time and a negative shift informs
the perception of timing shifted earlier in time. We could then compute for themodel estimation error as absolute difference between our Bayesian model’sestimates of the mean action and outcome perceptual shifts and the corresponding
perceptual shifts reported in the experiments. We then selected the parametervalues that best minimized the model estimation error.
Simulation details. Table 1 lists all the parameters of our Bayesian model. We
performed different simulations to reproduce the action and outcome perceptual
shifts reported by Haggard et al.3 and Wolpe et al.22, and to explain their underlying psychophysical mechanisms in Bayesian terms.
In the ﬁrst simulation, our objective was to determine μAO, to reproduce theperceptual shifts reported by Haggard et al3. We generated 35,000 instances of τAand τO pairs for each experimental condition using the baseline parameters in
Table 1-Set A. Testing each value in the set of possible values for μAO, and withσAO = 10 ms, we obtained the model estimation errors for the reported action andoutcome perceptual shifts listed in Table 2-Set A. We took the average of the model
estimation errors for the voluntary, involuntary, and sham conditions to obtain asingle model estimation error. We looked at the model estimation errors for (a)action perceptual shifts only, (b) outcome perceptual shifts only, and (c) actionoutcome perceptual shifts. Our results showed the best estimates of the model to be
at μAO = 230 ms. Furthermore, we observed our Bayesian model’s estimates of theperceptual shift in action timing alone was sufﬁcient to indicate the optimal
parameters of the model.
Our objective in the second simulation was to obtain the speciﬁc strength of thecausal prior that reproduces Haggard et al.’s results. With μAO = 230 ms andσAO = 10 ms, we tested for P(ξ = 1) in the range 0 to 1 with increments of 0.1. We
used the same pairs of τA and τO from the ﬁrst simulation, and we computed once
again the model estimation errors for the empirical results listed in Table 2-Set A.
We selected the P(ξ = 1) that best minimized the model estimation errors for thevoluntary, involuntary, and sham conditions, and ﬁt the experimental data.
Table 1-Set C includes the parameters that yielded the best model estimates.
10

Table 2 Reported perceptual shifts in action and outcome
temporal awareness
Judged event (operant
Mean estimation
condition)
error (in ms)• Set A: Reported by Haggard et al.3
Voluntary action, then
21
Tone
−31
Involuntary action, then
56
Tone
46
Sham TMS, then
25
Tone
7
• Set B: Reported by Wolpe et al.22
Action, then31
Low uncertainty tone
−16
Action, then23
Intermediate−19
uncertainty tone
Action, then24
High uncertainty tone
−10

Mean perceptual
shift (in ms)
15
−46
−27
31
−7
−8
39
−51
31
−65
32
−105

Figure 1 shows the action and outcome perceptual shifts, as well as the intervals
between perceptual shifts, which were obtained by our Bayesian model using theseparameters.
In the third simulation, we aimed to reproduce the perceptual shifts reported by
Wolpe et al.22, listed in Table 2-Set B. We generated another set of 35,000 τA andτO pairs using this time the baseline parameters listed in Table 1-Set B. We
performed simulations with μAO = 230 ms, σAO = 10 ms, and P(ξ = 1) in the range
0 to 1 with increments of 0.1. We did not perform additional simulations toredetermine μAO, as our aim is to reproduce qualitatively all the experiments withthe same μAO and σAO as possible in order to have simple yet consistent
explanations by our Bayesian model. Although we did not modify here μAO andσAO, our analyses and results can show their effects can be predicted and explainedby our model. The model estimation errors once again indicate the estimates ofaction perceptual shifts led to the best estimates of the model. We list under
Table 1-Set D the P(ξ = 1) that yielded the best estimates of the model for the low,intermediate, and high uncertainty tone conditions. Figure 5a shows the action andoutcome perceptual shifts, and intervals between shifts, predicted by our Bayesianmodel for this experimental setup.
In the fourth simulation, our objective was to determine the inﬂuence of thecausal prior and the temporal difference τO − τA (that varies in every trial) on thevarious predictions of our Bayesian model for Haggard et al.’s experimental setup.
We used the model parameters and τA and τO pairs from the ﬁrst and second
simulations. We obtained our Bayesian model’s predictions of the intervals
between action and outcome perceptual shifts, binding and repulsion effects,
action-outcome timing interval, ^tO  ^tA , in the baseline and operant conditions,and CCE. The results are shown in Figs. 2–4.
Our objective and target results in the ﬁnal simulation were the same as the
fourth simulation, but we used the model parameters and τA and τO pairs from thethird simulation to account for the experimental setup of Wolpe et al.22. The
resulting plots are shown in Figs. 5 and 6.
Reporting summary. Further information on research design is available in
the Nature Research Reporting Summary linked to this article.

Data availability
All relevant data are within the manuscript, which can be immediately generated using
the supplementary MATLAB source codes.

Code availability
The MATLAB source codes that were used to generate the simulated datasets andanalyze the simulation results are appended as Supplementary Source Codes.

Received: 18 January 2019 Accepted: 22 August 2019

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Acknowledgements
This study was supported by Brain/MINDS from AMED under Grant Number
JP19dm020700 and JSPS KAKENHI Grant Number JP18H05432.

Author contributions
R.L. and T.T. planned the project, conceived the Bayesian model, performed the simulations, analyzed the data, and wrote the manuscript.

Additional information
Supplementary Information accompanies this paper at https://doi.org/10.1038/s41467019-12170-0.
Competing interests: The authors declare no competing interests.
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