……………………………………………………..
Bayesian integration in
sensorimotor learning
Konrad P. Körding & Daniel M. Wolpert
Sobell Department of Motor Neuroscience, Institute of Neurology,
University College London, Queen Square, London WC1N 3BG, UK
………………………………………………………………………………………………………………………………………………………..
When we learn a new motor skill, such as playing an approaching
tennis ball, both our sensors and the task possess variability. Our
sensors provide imperfect information about the ball’s velocity,
so we can only estimate it. Combining information from multiple
modalities can reduce the error in this estimate1–4. On a longer
time scale, not all velocities are a priori equally probable, and
over the course of a match there will be a probability distribution
of velocities. According to bayesian theory5,6, an optimal estimate
results from combining information about the distribution of
velocities—the prior—with evidence from sensory feedback. As
uncertainty increases, when playing in fog or at dusk, the system
should increasingly rely on prior knowledge. To use a bayesian
strategy, the brain would need to represent the prior distribution
and the level of uncertainty in the sensory feedback. Here we
control the statistical variations of a new sensorimotor task and
manipulate the uncertainty of the sensory feedback. We show
that subjects internally represent both the statistical distribution
of the task and their sensory uncertainty, combining them in a
manner consistent with a performance-optimizing bayesian
process4,5. The central nervous system therefore employs prob-
abilistic models during sensorimotor learning.
Subjects reached to a visual target with their right index finger in a
virtual-reality set-up that allowed us to displace the visual feedback
of their finger laterally relative to its actual location (Fig. 1a; see
Methods for details). On each movement, the lateral shift was
randomly drawn from a prior distribution that was gaussian with
a mean shift of 1 cm to the right and a standard deviation of 0.5 cm
(Fig. 1b). We refer to this distribution as the true prior. During the
movement, visual feedback of the finger position was only provided
briefly, midway through the movement. We manipulated the
reliability of this visual feedback on each trial. This feedback was
either provided clearly (j0 condition, in which the uncertainty comes
from intrinsic processes only), blurred to increase the uncertainty by
a medium (jM) or large (jL) amount, or was withheld altogether
leading to infinite uncertainty (j1). Visual information about the
position of the finger at the end of the movement was provided only
on clear feedback trials (j0) and subjects were instructed to get as
close to the target as possible on all trials.
Subjects were trained for 1,000 trials on the task, to ensure that
they experienced many samples of the lateral shift drawn from the
underlying gaussian distribution. After this period, when feedback
was withheld (j1), subjects pointed 0.97 ^ 0.06 cm (mean ^
s.e.m. across subjects) to the left of the target showing that they
had learned the average shift of 1 cm experienced over the ensemble
of trials (Fig. 1a, example finger and cursor paths shown in green).
Subsequently, we examined the relationship between imposed
lateral shift and the final location that subjects pointed to. On trials
in which feedback was provided, there was compensation during the
second half of the movement (Fig. 1a, example finger and cursor
paths for a trial with lateral shift of 2 cm shown in blue). The visual
feedback midway through the movement provides information
about the current lateral shift. However, we expect some uncertainty
in the visual estimate of this lateral shift. For example, if the lateral
shift is 2 cm, the distribution of sensed shifts over a large number of
trials would be expected to have a gaussian distribution centred on
2 cm with a standard deviation that increases with the blur (Fig. 1c).
There are several possible computational models that subjects
could use to determine the compensation needed to reach the target
on the basis of the sensed location of the finger midway through the
movement. First (model 1), subjects could compensate fully for the
8
Figure 1 The experiment and models. a, As the finger moves from the starting circle, the
cursor is extinguished and shifted laterally from the true finger location. The hand is never
visible. Halfway to the target, feedback is briefly provided clearly (j0) or with different
degrees of blur (jM and jL ), or withheld (j1). Subjects are required to place the cursor on
the target, thereby compensating for the lateral shift. The finger paths illustrate typical
trajectories at the end of the experiment when the lateral shift was 2 cm (the colours
correspond to two of the feedback conditions). b, The experimentally imposed
prior
distribution of lateral shifts is gaussian with a mean of 1 cm. c, A diagram of the probability
distribution of possible visually experienced shifts under the clear and the two blurred
feedback conditions (colours as in a) for a trial in which the true lateral shift is 2 cm. d, The
estimate of the lateral shift for an optimal observer that combines the prior with the
evidence. e, The average lateral deviation from the target as a function of the true lateral
shift for the models (for jL the green shading shows the variability of the lateral deviation).
Left, the full compensation model; middle, the bayesian probabilistic model; right, the
mapping model (see the text for details).
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visual estimate of the lateral shift. In this model, increasing the
uncertainty of the feedback for a particular lateral shift (by increas-
ing the blur) would affect the variability of the pointing but not the
average location. Crucially, this model does not require subjects to
estimate their visual uncertainty or the prior distribution of shifts.
Second (model 2), subjects could optimally use information about
the prior distribution and the uncertainty of the visual feedback to
estimate the lateral shift. We can see intuitively why model 1 is sub-
optimal. If, on a given trial, the subject sensed a lateral shift of 2 cm,
there are many true lateral shifts that can give rise to such a
perception. For example, the true lateral shift could be 1.8 cm
with a visual error of þ0.2 cm, or it could be a lateral shift of
2.2 cm with a visual error of 20.2 cm. Which of the two possibilities
is more probable? Given gaussian noise on the visual feedback,
visual errors of þ0.2 cm and 20.2 cm are equally probable. How-
ever, a true lateral shift of 1.8 cm is more probable than a shift of
2.2 cm given that the prior distribution has a mean of 1 cm (Fig. 1b).
If we consider all possible shifts and visual errors that can give rise to
a sensed shift of 2 cm, we find that the most probable true shift is less
than 2 cm. The amount by which it is less depends on two factors,
the prior distribution and the degree of uncertainty in the visual
feedback. As we increase the blur, and thus the degree of uncertainty,
the estimate moves away from the visually sensed shift towards the
mean of the prior distribution (Fig. 1d). Without any feedback (j1)
the estimate should be the mean of the prior. Such a strategy can be
derived from bayesian statistics and minimizes the subject’s mean
squared error.
A third computational strategy (model 3) is to learn a mapping
from the visual feedback to an estimate of the lateral shift. By
minimizing the error over repeated trials, subjects could achieve a
combination similar to model 2 but without any explicit represen-
tation of the prior distribution or visual uncertainty. However, to
learn such a mapping requires knowledge of the error at the end of
the movement. In our experiment we only revealed the shifted
position of the finger at the end of the movement on the clear
feedback trials (j0). Therefore, if subjects learn a mapping, they can
only do so for these trials and apply the same mapping to the blurred
conditions (jM, jL). This model therefore predicts that the average
shift of the response towards the mean of the prior should be the
same for all amounts of blur.
By examining the influence of the visual feedback on the final
deviation from the target we can distinguish between these three
models (Fig. 1e). If subjects compensate fully for the visual feedback
(model 1), the average lateral deviation of the cursor from the target
should be zero for all conditions. If subjects combine the prior and
the evidence provided by sensory feedback (model 2), the estimated
lateral shift should move towards the mean of the prior by an
amount that depends on the sensory uncertainty. For a gaussian
distribution of sensory uncertainty, this predicts a linear relation-
ship between lateral deviation and the true lateral shift, which
should intercept the abscissa at the mean of the prior (1 cm) and
with a slope that increases with uncertainty. Finally, the mapping
model (model 3) predicts that subjects should compensate for the
seen position independently of the degree of uncertainty. Thus, all
conditions should exhibit the same slope as the clear feedback
condition (j0) of model 2. An examination of the theoretically
determined mean squared error for the three models shows that it is
minimal for model 2. Even though model 1 is on average on target,
the variability in the response is higher than in model 2 (green
shading in Fig. 1e shows the variability for the jL condition),
leading to a larger mean squared error.
The lateral deviation from the target as a function of the lateral
shift is shown for a representative subject in Fig. 2a. This shows a
slope that increases with increasing uncertainty and is, therefore,
incompatible with models 1 and 3. As predicted by model 2, the
influence of the feedback on the final pointing location decreases
with increasing uncertainty. The slope increases significantly with
uncertainty in the visual feedback over the subjects tested (Fig. 2b).
The bias and the slope should have a fixed relationship if we assume
that subjects do bayesian estimation. We expect no deviation from
the target if the true lateral shift is at the mean of the prior (1 cm).
This predicts that the sum of the slope and offset should be zero, as
observed in Fig. 2c. Subjects thus combine prior knowledge of the
distribution with sensory evidence to generate appropriate com-
pensatory movements.
Assuming that subjects use a bayesian strategy, we can further-
more use the errors that the subjects made during the trials to infer
their degree of uncertainty in the feedback. For the three levels of
imposed uncertainty, j0, jM and jL, we find that subjects’ estimates
of their visual uncertainty are 0.36 ^ 0.04, 0.67 ^ 0.1 and
8
8
Figure 2 Results for a gaussian distribution. Colour codes as in Fig. 1. a, The lateral
deviation of the cursor at the end of the trial as a function of the imposed lateral shift for a
typical subject. Error bars denote s.e.m. The horizontal dotted lines indicate the prediction
from the full compensation model and the dashed line is the fit for a model that ignores
sensory feedback on the current trial and corrects only for the mean over all trials. The
solid line is the bayesian model with the level of uncertainty fitted to the data. b, The
slopes for the linear fits are shown for the full population of subjects. On the basis of the
hypothesis that the slope should increase with increasing visual uncertainty, we
performed a repeated-measures analysis of variance on the slope, with visual uncertainty
as a factor (main effect of visual uncertainty F 3,27 ¼ 82.7; p , 0.001). Planned
comparisons of the slopes between adjacent uncertainty levels were all significant
(asterisk, p , 0.05; three asterisks, p , 0.001). c, The bias against gain for the linear
fits for each subjects and condition. The solid line shows the bayesian solutions. d, The
inferred priors and the true prior (red) for each subject and condition.
letters to nature
NATURE | VOL 427 | 15 JANUARY 2004 | www.nature.com/nature 245© 2004 Nature Publishing Group
0.8 ^ 0.1 cm (means ^ s.e.m. across subjects), respectively. We
have also developed a novel technique that uses these estimates to
infer the priors used by the subjects. Figure 2d shows the priors
inferred for each subject and condition. This shows that the true
prior (red line) was reliably learned by each subject.
To examine whether subjects can learn complex distributions, a
new group of subjects were exposed to a bimodal distribution
(Fig. 3a) consisting of a mixture of two gaussians separated by
4 cm. Here, the bayesian model predicts a nonlinear relationship
between true shift and lateral deviation, with the precise shape
depending on the uncertainty of the visual feedback. Figure 3b
shows a single subject’s lateral deviation together with the fit of a
bayesian model (solid line) in which we fit two parameters: the
separation of the two gaussians and the variance of the visual
uncertainty. The nonlinear properties are reflected in the empirical
data and are consistent over the subjects (Fig. 3c) with a fitted
separation of 4.8 ^ 0.8 cm (mean ^ s.e.m. across subjects), close to
the true value of 4 cm, suggesting that subjects represent the
bimodal prior. Taken together, our results demonstrate that subjects
implicitly use bayesian statistics.
Many technically challenging problems have been addressed
successfully within the bayesian framework7,8. It has been proposed
that the architecture of the nervous system is well suited for bayesian
inference9–13 and that some visual illusions can be understood
within the bayesian framework14. However, most models of the
sensorimotor system consider a cascade of mappings from sensory
inputs to the motor output15–17. These models consider input–
output relationships and do not explicitly take into account the
probabilistic nature of either the sensors or the task. Recent models
of motor control have begun to emphasize probabilistic proper-
ties18–24. Unlike the visual system, which loses much of its plasticity
once it has passed its critical period, the motor system retains much
of its plasticity throughout adult life. We could therefore impose a
novel prior on the subjects and measure its influence on sensor-
imotor processing. To show quantitatively that the system performs
optimally would require a direct measure of sensory uncertainty
before it is integrated with the prior. However, such a measure
cannot easily be obtained as even a naive subject would integrate
feedback with their natural, but unknown, prior. However, by
imposing experimentally controlled priors we have shown that
our results qualitatively match a bayesian integration process. A
bayesian view of sensorimotor learning is consistent with neuro-
physiological studies showing that the brain represents the degree of
uncertainty when estimating rewards25–27 and with psychophysical
studies addressing the timing of movements28,29. Although we have
shown only the use of a prior in learning hand trajectories during a
visuomotor displacement, we expect that such a bayesian process
might be fundamental to all aspects of sensorimotor control and
learning. For example, representing the distribution of dynamics of
objects, such as their mass, would facilitate our interactions with
them. Similarly, although the possible configurations of the human
body are immense, they are not all equally likely and knowledge of
their distribution could be used to refine estimates of our current
state. Taking into account a priori knowledge might be key to
winning a tennis match. Tennis professionals spend a great deal
of time studying their opponent before playing an important
match, ensuring that they start the match with correct a priori
knowledge. A
Methods
Experimental details
Six male and four female subjects participated in this study after giving informed consent.
Subjects made reaching movements on a table during which an Optotrak 3020 tracking
system (Northern Digital) measured the position of their right index finger. A projection–
mirror system prevented direct view of their arm and allowed us to generate a cursor
representing their finger position that could be displayed in the plane of the movement
(for details of the set-up see ref. 30). Subjects saw a blue sphere representing the starting
location, a green sphere representing the target and a white sphere representing the
position of their finger (Fig. 1a). Subjects were requested to point accurately to the target.
When the finger left the start position, the cursor representing the finger was extinguished
and displaced to the right by an amount that was drawn each trial from a gaussian
distribution with mean of 1 cm and standard deviation of 0.5 cm. Midway through the
movement (10 cm), feedback of the cursor centred at the displaced finger position was
flashed for 100 ms. On each trial one of four types of feedback (j0, jM, jL, j1) was
displayed; the selection of the feedback was random, with the relative frequencies of the
four types being (3, 1, 1, 1) respectively. The j0 feedback was a small white sphere. The jM
feedback was 25 small translucent spheres, distributed as a two-dimensional gaussian with
a standard deviation of 1 cm, giving a cloud-type impression. The jL feedback was
analogous but had a standard deviation of 2 cm. No feedback was provided in the j1 case.
After another 10 cm of movement the trial was finished; feedback of the final cursor
location was provided only in the j0 condition. The experiment consisted of 2,000 trials
for each subject. On post-experimental questioning, all subjects reported being unaware of
the displacement of the visual feedback. Only the last 1,000 trials were used for analysis.
Subjects were instructed to take into account what they saw at the midpoint and to get as
close to the target as possible; we took the lateral deviation of the finger from the target as a
measure of subjects’ estimate of the lateral shift. By averaging over trials we could obtain
this estimate uncorrupted by any motor output noise, which we assumed to have mean of
zero.
Bayesian estimation
We wish to estimate the lateral shift x true of the current trial given a sensed shift x sensed (also
known as the evidence) and the prior distribution of lateral shifts p(x true). From Bayes rule
we can obtain the posterior distribution, that is the probability of each possible lateral shift
taking into account both the prior and the evidence,
pðxtruejxsensedÞ ¼ pðxsensedjxtrueÞ
pðxtrueÞ
pðxsensedÞ
where pðxsensedjxtrueÞ is the likelihood of perceiving x sensed when the lateral shift really is
x true. We assume that visual estimation is unbiased and corrupted by gaussian noise so that
pðxtruejxsensedÞ ¼
1
j
sensed
ffiffiffiffiffiffi
2p
p e2ðxtrue2xsensed Þ
2=2j2
sensed
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
pðxsensed jxtrueÞ
1
jprior
ffiffiffiffiffiffi
2p
p e2ðxtrue21cmÞ2=2j2
prior
.
pðxsensedÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
pðxtrue Þ=pðxsensed Þ
For the optimal estimate we can find the maximum by differentiation, which represents
the most probable lateral shift. For gaussian distributions such an estimate also has the
Figure 3 Results for a mixture of gaussian distributions. a, The experimentally imposed
prior distribution of lateral shifts is a mixture of two gaussians. b, The lateral deviation of
the cursor at the end of the trial as a function of the true lateral shift for a typical
subject. Error bars denote s.e.m. The horizontal dotted lines indicate the prediction from
the full compensation model, the dashed line is the fit for a bayesian model with a single
gaussian prior, and the solid line is the fit for a bayesian model with a prior that is a mixture
of two gaussians. c, The lateral deviation across subjects (mean ^ s.e.m. across
subjects) is shown with a linear regression fit, demonstrating the nonlinearity of the data.
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NATURE | VOL 427 | 15 JANUARY 2004 | www.nature.com/nature246 © 2004 Nature Publishing Group
smallest mean squared error. This estimate is a weighted sum of the mean of the prior and
the sensed feedback position:
xestimated ¼
j2
sensed
j
2
sensed þ j2
prior
½1cm
þ
j2
prior
j2
sensed þ j2
prior
xsensed
Given that we know j2
prior; we can estimate the uncertainty in the feedback j sensed by
linear regression from Fig. 2a.
Resulting mean squared error
The mean squared error (MSE) is determined by integrating the squared error over all
possible sensed feedbacks and actual lateral shifts
MSE ¼
ð1
21
ð1
21
ðxestimated 2 xtrueÞ
2pðxsensedjxtrueÞpðxtrueÞdxsensed
dxtrue
For model 1, xestimated ¼ x sensed, and this gives MSE ¼ j2
sensed :
Using the result for xestimated from above for model 2 gives
MSE ¼ j2
sensedj
2
prior=ðj
2
sensed þ j2
priorÞ; which is always lower than the MSE for model 1. If
the variance of the prior is equal to the variance of the feedback, the MSE for model 2 is half
that of model 1.
Inferring the used prior
An obvious choice of xestimated is the maximum of the posterior
pðxtruejxsensedÞ ¼
1
jsensed
ffiffiffiffiffiffi
2p
p e2ðxtrue2xsensed Þ
2=2j2
sensed pðxtrueÞ=pðxsensedÞ
The derivative of this posterior with respect to x true must vanish at xestimated. This
allows us to estimate the prior used by each subject. Differentiating and setting to zero we
get
dpðxtrueÞ
dxtrue
1
pðxtrueÞ
����
xestimated
¼
ðxestimated 2 xsensedÞ
j2
sensed
We assume that x sensed has a narrow peak around x true and thus approximate it by x true.
We insert the j sensed obtained above, affecting the scaling of the integral but not its form.
The average of x sensed across many trials is the imposed shift x true. The right-hand side is
therefore measured in the experiment and the left-hand side approximates the derivative
of log(p(x true)). Since p(x true) must approach zero for both very small and very large x true,
we subtract the mean of the right-hand side before integrating numerically to obtain
log(p(x true)), which we can then transform to estimate the prior p(x true).
Bimodal distribution
Six new subjects participated in a similar experiment in which the lateral shift was
bimodally distributed as a mixture of two gaussians:
pðxtrueÞ ¼
1
2
ffiffiffiffiffiffi
2p
p
jprior
e2ðx2xsep=2Þ2=j2
prior þ e2ðxþxsep=2Þ2=j2
prior
� �
where x sep ¼ 4 cm and jprior ¼ 0.5 cm. Because we expected this prior to be more difficult
to learn, each subject performed 4,000 trials split between two consecutive days. In
addition, to speed up learning, feedback midway through the movement was always
blurred (25 spheres distributed as a two-dimensional gaussian with a standard deviation of
4 cm), and feedback at the end of the movement was provided on every trial. Fitting the
bayesian model (using the correct form of the prior and true jprior) to minimize the MSE
between actual and predicted lateral deviations of the last 1,000 trials was used to infer the
subject’s internal estimates of both x sep and j sensed. Some aspects of the nonlinear
relationship between lateral shift and lateral deviation (Fig. 3a) can be understood
intuitively. When the sensed shift is zero, the actual shift is equally likely to be to the right
or the left and, on average, there should be no deviation from the target. If the sensed shift
is slightly to the right, such as at 0.25 cm, then the actual shift is more likely to come from
the right-hand gaussian than the left, and subjects should point to the right of the target.
However, if the sensed shift is far to the right, such as at 3 cm, then because the bulk of the
prior lies to the left, subjects should point to the left of the target.
Received 30 June; accepted 10 October 2003; doi:10.1038/nature02169.
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Acknowledgements We thank Z. Ghahramani for discussions, and J. Ingram for technical
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Competing interests statement The authors declare that they have no competing financial
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Correspondence and requests for materials should be addressed to K.P.K. (e-
mail: konrad@koerding.de).
……………………………………………………..
Functional genomic hypothesis
generation and experimentation
by a robot scientist
Ross D. King1, Kenneth E. Whelan1, Ffion M. Jones1, Philip G. K. Reiser1,
Christopher H. Bryant2, Stephen H. Muggleton3, Douglas B. Kell4
& Stephen G. Oliver5
1Department of Computer Science, University of Wales, Aberystwyth SY23 3DB,
UK
2School of Computing, The Robert Gordon University, Aberdeen AB10 1FR, UK
3Department of Computing, Imperial College, London SW7 2AZ, UK
4Department of Chemistry, UMIST, P.O. Box 88, Manchester M60 1QD, UK
5School of Biological Sciences, University of Manchester, 2.205 Stopford Building,
Manchester M13 9PT, UK
………………………………………………………………………………………………………………………………………………………..
The question of whether it is possible to automate the scientific
process is of both great theoretical interest1,2 and increasing
practical importance because, in many scientific areas, data are
being generated much faster than they can be effectively ana-
lysed. We describe a physically implemented robotic system that
applies techniques from artificial intelligence3–8 to carry out
cycles of scientific experimentation. The system automatically
letters to nature
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