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The development of Bayesian integration in sensorimotor estimation

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Examining development is important in addressing questions about whether Bayesian principles are hard coded in the brain. If the brain is inherently Bayesian, then behavior should show the signatures of Bayesian computation from an early stage in life. Children should integrate probabilistic information from prior and likelihood distributions to reach decisions and should be as statistically efficient as adults, when individual reliabilities are taken into account. To test this idea, we examined the integration of prior and likelihood information in a simple position-estimation task comparing children ages 6-11 years and adults. Some combination of prior and likelihood was present in the youngest sample tested (6-8 years old), and in most participants a Bayesian model fit the data better than simple baseline models. However, younger subjects tended to have parameters further from the optimal values, and all groups showed considerable biases. Our findings support some level of Bayesian integration in all age groups, with evidence that children use probabilistic quantities less efficiently than adults do during sensorimotor estimation.
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The development of Bayesian integration in sensorimotor
estimation
Claire Chambers
Department of Bioengineering,
University of Pennsylvania, Philadelphia, PA, USA
Department of Neuroscience,
University of Pennsylvania, Philadelphia, PA, USA $
Taegh Sokhey
Sensory Motor Performance Program,
Shirley Ryan Abilitylab, Chicago, IL, USA
Department of Biological Sciences,
Northwestern University, Evanston, IL, USA $
Deborah Gaebler-Spira
Sensory Motor Performance Program,
Shirley Ryan Abilitylab, Chicago, IL, USA
Department of Physical Medicine and Rehabilitation,
Feinberg School of Medicine, Northwestern University,
Chicago, IL, USA $
Konrad Paul Kording
Department of Bioengineering,
University of Pennsylvania, Philadelphia, PA, USA
Department of Neuroscience,
University of Pennsylvania, Philadelphia, PA, USA $
Examining development is important in addressing
questions about whether Bayesian principles are hard
coded in the brain. If the brain is inherently Bayesian,
then behavior should show the signatures of Bayesian
computation from an early stage in life. Children should
integrate probabilistic information from prior and
likelihood distributions to reach decisions and should be
as statistically efficient as adults, when individual
reliabilities are taken into account. To test this idea, we
examined the integration of prior and likelihood
information in a simple position-estimation task
comparing children ages 6–11 years and adults. Some
combination of prior and likelihood was present in the
youngest sample tested (6–8 years old), and in most
participants a Bayesian model fit the data better than
simple baseline models. However, younger subjects
tended to have parameters further from the optimal
values, and all groups showed considerable biases. Our
findings support some level of Bayesian integration in all
age groups, with evidence that children use probabilistic
quantities less efficiently than adults do during
sensorimotor estimation.
Introduction
The behavior of adults under uncertainty is well
described by Bayesian inference, in that adult humans
weight different sources of information according to
their relative uncertainty. Behavior is consistent with
Bayesian computations in sensorimotor behavior
(Berniker, Voss, & Kording, 2010; Kording & Wolpert,
2004), perception (Knill & Richards, 1996; Mamassian
& Goutcher, 2001), cognition and reasoning tasks
(Battaglia, Hamrick, & Tenenbaum, 2013; Tenenbaum
& Griffiths, 2001), and cue combination across and
within sensory modalities (Ernst & Banks, 2002; Hillis,
Watt, Landy, & Banks, 2004). Adult humans seem to
integrate information in a way that is well predicted by
Bayesian inference (but see Bowers & Davis, 2012;
Jones & Love, 2011; Rahnev & Denison, 2018).
These numerous findings of Bayesian behavior have
led to the theory that the underlying neural computa-
tions are inherently Bayesian. For example, it has been
argued that the activity of neural populations reflects
probabilistic population codes that directly implement
Citation: Chambers, C., Sokhey, T., Gaebler-Spira, D., & Kording, K. P. (2018). The development of Bayesian integration in
sensorimotor estimation. Journal of Vision,18(12):8, 1–16, https://doi.org/10.1167/18.12.8.
Journal of Vision (2018) 18(12):8, 1–16 1
https://doi.org/10.11 67/18 . 1 2 . 8 ISSN 1534-7362 Copyright 2018 The AuthorsReceived January 11, 2018; published November 15, 2018
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Bayesian computations (Beck et al., 2008; Ma, Beck,
Latham, & Pouget, 2006; Ma, Beck, & Pouget, 2008;
Zemel, Dayan, & Pouget, 1998). However, findings of
Bayesian behavior are not sufficient to support this
claim. Bayesian behavior simply represents optimal
behavior under uncertainty, and there are ways of
generating optimal behavior that do not explicitly
implement Bayesian computation (Mandt, Hoffman, &
Blei, 2017; Verstynen & Sabes, 2012; Weisswange,
Rothkopf, Rodemann, & Triesch, 2011). Therefore,
previous research has not fully established whether the
neural code is inherently Bayesian.
If neural circuits are evolved to implement Bayesian
computations, then behavior should always show
Bayesian signatures, including during development.
Therefore, children too should act in accordance with
the rules of Bayesian integration. Specifically, they
should weight information according to its relative
uncertainty in simple tasks.
A good number of articles ask how Bayesian
children are. Work on the development of cue
combination across and within modalities has often
shown that children do not integrate information but
instead process it from each cue separately up to the
age of approximately 9–11 years (Dekker et al., 2015;
Gori, Del Viva, Sandini, & Burr, 2008; Nardini,
Bedford, & Mareschal, 2010; Nardini, Jones, Bedford,
& Braddick, 2008). Work on the integration of value-
based information (reward/penalty) in a sensorimotor
task has shown that children’s behavior is suboptimal
until late in development, at around 11 years (Dekker
& Nardini, 2016). However, other findings suggest that
whether children integrate information or not may be
task dependent. In a hand-localization task, variance
reduction consistent with multisensory integration has
been reported in the responses of young children ages
4–6 years (Nardini, Begus, & Mareschal, 2013). Some
work on looking times is consistent with optimal
integration of information as early as infancy (T´
egl´
as,
Tenenbaum, & Bonatti, 2011). It has been shown that
children as young as 4 years old use probabilistic
information to infer causality when performing actions
(Gopnik & Wellman, 2013; Kushnir & Gopnik, 2007;
Sobel, Tenenbaum, & Gopnik, 2004). Therefore, based
on previous research, it is unclear whether the behavior
of children is consistent with use of Bayesian inference.
Here, we investigate whether integration of infor-
mation to perform sensorimotor estimation is present
in young children or is acquired over the course of
development. In our paradigm, we examined the use of
probabilistic information in a simple sensorimotor
estimation task, previously used in adults to examine
integration of prior and likelihood information under
uncertainty (Acuna, Berniker, Fernandes, & Kording,
2015; Berniker et al., 2010; Kording & Wolpert, 2004;
Vilares, Howard, Fernandes, Gottfried, & Kording,
2012). The results of these studies show that adults
learn experimentally imposed prior distributions and
integrate prior statistics with uncertain sensory infor-
mation efficiently and without instructions to do so.
Our task was designed to specifically probe the
integration of prior and likelihood information during
sensorimotor estimation. We isolated integration from
the many other factors that contribute to movement
execution under naturalistic conditions. In the real
world, priors and likelihoods are complex and multi-
dimensional. In order to discount the influence of the
complexity of the distribution and task, we used simple
unidimensional Gaussian distributions (Berniker et al.,
2010). We also minimized the influence on our results
of having to learn the prior distribution by presenting
samples from the prior distribution on-screen. Move-
ment in the real world has an associated loss function
(Wolpert & Landy, 2012). In this task, we minimized
the effect of motor effort: All possible responses could
be made using minimal movement of a computer
mouse. Our experiment thus isolated Bayesian inte-
gration from other processes.
In our experiment, visual targets were drawn from a
prior distribution and participants were shown uncer-
tain sensory information about each target. We
measured reliance on sensory information and found
that while young children ages 6–8 years used sensory
information according to its precision, we did not find
evidence that they used the uncertainty of the prior
distribution, as did children ages 9–11 years or adults.
In all age groups, a Bayesian model predicted
estimation behavior better than models which used
only one source of information (prior or likelihood) or
switched between prior and likelihood (Laquitaine &
Gardner, 2018).
Methods
Experimental details
We aimed to examine probabilistic inference during
sensorimotor estimation in a child population. Our task
was designed to examine use of probabilistic informa-
tion during sensorimotor estimation (Acuna et al.,
2015; Berniker et al., 2010; Vilares et al., 2012).
Previous findings indicate that adults weight informa-
tion according to its reliability and learn priors in a
manner which resembles Bayesian integration during
sensorimotor estimation. For the purposes of the
current study, we adapted the experimental protocol
for child participants by using a concept that was
engaging to children, using simplified instructions, and
reducing the number of trials.
Journal of Vision (2018) 18(12):8, 1–16 Chambers, Sokhey, Gaebler-Spira, & Kording 2
Participants were 16 children (eight boys, eight girls)
ages 6–8 years (M¼6.94, SD ¼0.77), 17 children (eight
boys, nine girls) ages 9–11 years (M¼10.06, SD ¼0.75),
and 11 adults (five men, six women) over 18 years old
(M¼27.27, SD ¼5.31). The data of two participants
were excluded due to their looking away from the
screen during the experiment. Participants overlapped
with the sample of a previous study, where some
participants from the current study were included as
age-matched controls and compared with a clinical
population (Chambers, Sokhey, Gaebler-Spira, &
Kording, 2017).
In a quiet room, participants sat at a comfortable
distance (typically between 30 and 60 cm) from a
computer monitor 52 cm wide and 32.5 cm high. Before
starting the experiment, we presented participants with
the instructions that someone behind them was
throwing pieces of candy into a pond, represented by
the screen, and that their aim was to estimate where the
candy target landed and catch as many pieces of candy
as possible over the course of the experiment. We told
participants that one piece of candy was being thrown
into the water per trial. We also told participants that
the candy target was hidden and caused a splash on the
surface of the water. We showed them ‘‘where the
candy usually lands.’’ The participants’ goal was to hit
these targets (‘‘catch the candy’’) by combining the
information from the splash and where the candy
usually lands. The instructions from the beginning of
the experiment are shown in the Appendix (Figures A1
and A2).
Candy targets were drawn from a Gaussian prior
distribution centered at the middle of the screen,
Nl;r2
s

. The prior distribution was fixed for the
duration of a block. On each trial, participants were
presented with an uncertain ‘‘splash’’ stimulus for 1 s
and were told that the splash was caused by a hidden
candy target (Figure 1A). The splash was n¼4 samples
(white dots on a blue background, diameter ¼2%
screen width) from a Gaussian likelihood distribution
that was centered on the target location Ns;r2
l

. The
stimulus-generation process was hierarchical: The
target on each trial was sampled from a prior
distribution that was fixed across trials; the likelihood
distribution was centered on the target; and samples
were then drawn from the likelihood distribution to
form the splash stimulus that was displayed to
participants. Participants provided an estimate of the
candy target’s location on the horizontal axis using a
vertical bar that extended from the top to the bottom of
the screen (2%of screen width). In order to successfully
catch a target, the center of the ‘‘net’’ or vertical bar
had to be within 2%of screen width of the target
center. The net appeared at the same time as the splash
at a random location on screen. Participants had 6 s to
respond. After providing a response, they were shown
the true candy-target location.
The stimuli in this experiment were presented well
above the threshold of perception (in terms of their size
and contrast) and were clearly visible to those with
normal or corrected-to-normal vision. Therefore,
physical measurements of stimulus dimensions like the
exact visual angle were not crucial in determining how
Figure 1. (A) Experimental protocol. Participants were shown a visual cue with experimentally controlled uncertainty or likelihood
which was presented as a ‘‘splash’’ created by a hidden target or ‘‘piece of candy’’ drawn from a prior distribution. Participants were
told that the splash was created by candy falling into a pond. They were prompted to place a vertical bar (‘‘net’’) where the hidden
target fell and were then shown feedback on the target location. (B) Relying on the likelihood. A simple strategy would be to rely
entirely on likelihood information by pointing at its centroid on each trial. This strategy is close to optimal when the likelihood is
precise or narrow. The black bar or net overlaps with the target in the left panel. However, this strategy is less successful when the
likelihood is wider, as samples from the likelihood become a less reliable indicator of target location and the optimal estimate shifts
closer to the prior mean. The net is far from the target in the right panel. The optimal strategy involves weighting prior and likelihood
information according to their relative uncertainties. (C) Experimental design. In order to quantify integration of the prior and
likelihood, we measured reliance on the likelihood under different conditions of prior width and likelihood width. The prior could be
narrow or wide, and the likelihood could be narrow, medium, or wide.
Journal of Vision (2018) 18(12):8, 1–16 Chambers, Sokhey, Gaebler-Spira, & Kording 3
participants weighted information from the prior and
likelihood distributions in this experiment.
One simple strategy for performing sensorimotor
estimation under uncertainty is to consistently judge
target location at the centroid cof the splash. This
consists of full reliance on the likelihood and works
well when the likelihood distribution is narrow, because
the closely spaced points of the splash are an accurate
indicator of target location (Figure 1B, left). However,
full reliance on the likelihood would cause a participant
to miss targets more frequently as the likelihood
distribution widens (Figure 1B, right). When sensory
information is unreliable, rather than relying on the
likelihood completely we maximize performance by
giving more weight to our prior belief on target
location. More generally, the optimal strategy involves
weighting sources of information according to their
relative uncertainties.
Formally, weighting sources of information ac-
cording to their relative precision corresponds to
Bayesian inference. An optimal Bayesian observer
combines noisy sensory information from the likeli-
hood Ns;r2
l=n

with the prior Nl;r2
s

,resultingina
posterior distribution over target location:
Nl
r2
s
þc
r2
l=n

.1
r2
s
þ1
r2
l=n

;1.1
r2
s
þ1
r2
l=n

:
The mean of the posterior is a mean of the prior and
sensory information weighted by their precisions. From
this posterior distribution, an estimate is computed.
Therefore, the optimal reliance on the likelihood is a
function of prior and likelihood uncertainties,
r2
s=r2
sþr2
l=n

. We can manipulate the prior and
likelihood variance and measure their influence on
participants’ reliance on the likelihood to investigate
probabilistic information during sensorimotor estima-
tion.
To investigate how children use probabilistic infor-
mation during sensorimotor estimation, we manipu-
lated the variances of prior distribution and likelihood
distribution (Figure 1C). We used a Gaussian prior
distribution with a mean at the center of the screen and
standard deviation of 0.03 (Narrow Prior) or 0.1 (Wide
Prior) in units of screen width. The likelihood
distribution was centered on the target location and
could have a standard deviation of 0.05 (Narrow
Likelihood), 0.1 (Medium Likelihood), or 0.25 (Wide
Likelihood) in units of screen width. There were six
conditions: Narrow Prior/Narrow Likelihood, Narrow
Prior/Medium Likelihood, Narrow Prior/Wide Likeli-
hood, Wide Prior/Narrow Likelihood, Wide Prior/
Medium Likelihood, and Wide Prior/Wide Likelihood.
The experiment consisted of four blocks, each lasting
120 trials, preceded by a practice block lasting 10 trials.
Trials were blocked by prior condition, with all
likelihood conditions presented in randomized order
within one block. The prior over target location
switched from block to block with a randomly chosen
starting condition for each participant (i.e., Narrow-
Wide-Narrow-Wide or Wide-Narrow-Wide-Narrow).
We introduced a number of modifications to engage
child participants in the task. Participants were shown
how much candy they had won on-screen and won a
‘‘bonus’’ piece of candy for every 10 pieces of candy
they caught. Sounds were presented to signal success-
fully catching a target and missed responses when they
did not respond within the 6-s time window. Step-by-
step instructions were shown to participants on-screen
before the experiment, to ensure that all participants
received the same instructions. Before the experiment,
we told all participants that their payment was
proportional to the number of pieces of candy they
caught. Their target score was displayed on-screen.
Then at the end of the experiment we paid a fixed
amount, which was different for the parents of child
participants ($25) and for adults ($10). These modifi-
cations ensured that child participants were not
discouraged during the experiment.
Ethical approval was provided by the Northwestern
University Institutional Review Board
(#20142500001072). This study was performed in
accordance with the Declaration of Helsinki. Partici-
pants signed a consent form before participation. For
child participants, a parent provided consent for their
child to take part and completed the Developmental
Coordination Disorder questionnaire (Wilson et al.,
2009), a modified Vanderbilt questionnaire to assess for
attention deficit–hyperactivity disorder (Wolraich et
al., 2003), and the Behavior Assessment System for
Children parent rating scales (Reynolds & Kamphaus,
2004). After the participant completed the game, we
administered the child Mini-Mental State Evaluation to
obtain an approximate assessment of cognitive ability
(Ouvrier, Goldsmith, Ouvrier, & Williams, 1993). No
data were excluded on the basis of neuropsychological
test results.
Data analysis
We were interested in the integration of probabilistic
information from a prior distribution and sensory
information from the likelihood during sensorimotor
estimation. To investigate this, we examined whether
samples from the likelihood distribution,
X¼x1;x2;x3;x4
fg
, were combined with information
about the prior distribution Nl;r2
s

in producing an
estimate of target location. We quantified this for each
condition using the extent to which participants relied
on the likelihood, given by the linear relationship
between the centroid of the splash c¼PN
ixi=nand
their estimate on each trial. We performed a linear
Journal of Vision (2018) 18(12):8, 1–16 Chambers, Sokhey, Gaebler-Spira, & Kording 4
regression with the estimate (placement of the net) as
the dependent variable and the likelihood centroid cas
the independent variable. The slope provides an
estimate of the reliance on the likelihood, which we
term the estimation slope. If participants relied only on
the likelihood to generate their estimate, then they
should point close to the centroid of the splash con all
trials, leading to an estimation slope 1. If instead
participants ignore the likelihood and use only their
representation of the prior, then their estimates should
not depend on c, leading to an estimation slope 0.
The intercept/(1 estimation slope) computed from the
fitted function reflects the subjective prior mean used to
provide estimates. Therefore, from participants’ esti-
mates we obtain a measure of their reliance on the
likelihood and the subjective prior mean.
For a Bayesian observer, sources of information are
weighted according to their relative reliabilities, leading
to an estimation slope of r2
s=ðr2
sþrlþDl
ðÞ
2=n). To
account for the fact that participants may not have
learned the experimentally imposed likelihood variance
r2
l, we added an extra parameter Dl, a constant source
of variance added to the likelihood variance. We use
this equation to model the estimation slopes and to
infer probabilistic variables used by participants (r2
s,
Dl).
We also investigated a simple switching model as an
alternative to full Bayesian integration, where partici-
pants randomly switched between using the prior and
likelihood across trials. The proportion of trials where
participants used the likelihood to form their estimate,
p(likelihood), was a free parameter. In this model,
participants ignored the uncertainty of prior and
likelihood information and used a fixed p(likelihood)
for all conditions.
We quantified task performance using the propor-
tion of correct responses, p(correct), and the root mean
square error (RMSE) with respect to the regression line
in each condition. We note that the RMSE contains
contributions from both additional noise Dladded to
the likelihood and motor errors.
Motor errors, which occur after the combination of
task-relevant information, influence estimates. Linear
regression provides an unbiased estimate of the
estimation slope for realistic amounts of motor noise
added to estimates with standard deviations of up to
approximately 10%–15%of screen width. Larger
amounts of motor noise lead to many estimates at
screen limits and estimation slopes that are closer to
zero than the true values predicted by prior and
likelihood parameters. Here, our analysis of estimation
slopes assumes that sensorimotor estimates are not
corrupted by large amounts of motor noise. However,
such large motor errors are unlikely given that the
RMSE of the most variable subject was less than 15%
of screen width (see Figure 2A, right panel).
We examined whether children’s estimation behavior
resembled Bayesian inference using model selection and
by estimating the parameters of the best-fitting model.
We first compared the performance of a Bayesian
model with three alternative models: a model where
participants alternated between the likelihood and
prior (Switch model), a model where they fully relied on
the prior (Prior-only model), and a model where they
fully relied on the likelihood (Likelihood-only model).
We computed estimation slopes from each participant’s
data, then fitted the estimation slopes to the Bayesian
model and the Switch model. The Bayesian model
contained two prior variance parameters and the
variance parameter added to the likelihood (r2
sNP
,
r2
sWP
, and Dl). We minimized the mean squared error
(MSE) of the objective function r2
s=ðr2
sþrlþDl
ðÞ
2=nÞ
relative to the data, using bounds of 0.02 and 0.4 for
the prior variance parameters. For the Switch model,
we minimized the MSE between estimation slopes and
the p(likelihood) parameter, using bounds of 0.1 and
0.9. We ran the optimizer five times with randomly
selected initial parameters. We performed leave-one-
out cross validation on the data of each participant by
fitting model parameters to five conditions and
computing the MSE for the left-out condition. For the
Prior-only model, the MSE was computed by com-
paring estimation slopes to 0; for the Likelihood-only
model, it was computed by comparing estimation
slopes to 1. We selected the model with the lowest MSE
summed across left-out conditions. We estimated the
parameters of the Bayesian and Switch models by
fitting the models to 1,000 data sets resampled with
replacement from each participant’s data. We used
estimation slopes measured from participants’ data for
model selection and parameter estimation.
For validation purposes, we performed model
selection on 1,000 data sets simulated from each model.
Each simulated participant generated estimates using a
maximum a posteriori decision rule with added motor
noise (SD ¼5%in screen units). If estimates fell outside
screen limits, they were set to the screen limit. Prior-
only and Likelihood-only participants were generated
by excluding the likelihood and prior terms from the
posterior distribution, respectively. Switch subjects
alternated between Prior-only and Likelihood-only
strategies. In order to illustrate that we can infer model
parameters with reasonable accuracy, we performed
parameter estimation for 1,000 simulated Bayesian
subjects whose subjective prior variance was at
intermediate values between the theoretical variance
parameters. We simulated cases where subjective prior
variances were undifferentiated (r2
sNP¼0.06, r2
sWP¼
0.06), where they were partly differentiated (r2
sNP¼
0.048, r2
sNP¼0.083), and where subjects used the
experimentally imposed prior (r2
sNP¼0.03; r2
sWP¼0.1).
We simulated conditions where different amounts of
Journal of Vision (2018) 18(12):8, 1–16 Chambers, Sokhey, Gaebler-Spira, & Kording 5
variance were added to the experimentally imposed
likelihood (Dl¼0, 0.05, 0.1). We also inferred the
p(likelihood) parameter of the Switch model (0.2, 0.4,
0.6, 0.8) from the data of 1,000 simulated subjects per
condition. Simulations allowed us to ensure that our
model-selection and parameter-estimation procedures
produced unbiased results.
Results
We wanted to investigate the development of
Bayesian integration. To do so, we examined whether
children ages 6–11 years and adults could learn to use
uncertainty of different sources of information (prior
and likelihood) during sensorimotor estimation (Figure
1). We first quantified participants’ task performance.
We then examined how their estimates depended on the
prior and the likelihood. We then performed model
selection on the data of each participant to assess
whether their behavior was more consistent with
Bayesian inference, full reliance on the prior, full
reliance on the likelihood, or alternation between prior
and likelihood. Finally, we devised estimated parame-
ters of the Bayesian model for each age group.
It was first important to establish that the all age
groups understood and carried out the task. We
Figure 2. Task performance and estimation data. (A) Left: The proportion of candy targets caught as a function of age group (median,
error bars ¼95% confidence intervals [CIs]). Right: The root mean square error relative to the regression line in the Wide Prior/
Narrow Likelihood condition gives an indication of how noisy participants were and is shown as a function of age group (median,
error bars ¼95% CIs). (B) Estimation data overlaid with linear fit for a representative participant age 11 years. The net position as a
function of the centroid of the likelihood is shown for each trial (points). The fitted (blue) and optimal (red) functions are displayed.
Note that optimal values here and in (C–D) and Figure 3B assume that participants use the experimentally imposed likelihoods and
priors. Each panel displays estimation data for one condition, as defined by prior and likelihood width. (C) The median bootstrapped
intercept of individual participants is shown as a function of age group (error bars ¼95% CI). The optimal intercept at zero is shown
(red). (D) The median bootstrapped estimation slope of individual participants is shown as a function of age group (error bars ¼95%
CI). The optimal estimation slope values are shown (red).
Journal of Vision (2018) 18(12):8, 1–16 Chambers, Sokhey, Gaebler-Spira, & Kording 6
therefore examined the proportion p(correct) of candy
caught (Figure 2A, left panel). Performance increased
significantly with age (one-way analysis of variance
[ANOVA]), F(2, 41) ¼27.44, p,0.0001, with
significant differences between age groups—6–8 years
versus 9–11 years: t(31) ¼3.86, p,0.001; 6–8 years
versus 18þyears: t(25) ¼7.72, p,0.0005; 9–11 years
versus 18þyears: t(26) ¼3.83, p,0.001 (corrected a¼
0.0167). We compared the proportion of candy targets
caught, p(correct), to chance level for each age group
(Figure 2A). The performance of all age groups far
exceeded chance level, as tested with one-sample t
tests—6–8 years: t(15) ¼33.77, p,0.0001; 9–11 years:
t(16) ¼32.26, p,0.0001; 18þyears: t(10) ¼29.31, p,
0.0001 (corrected a¼0.0167). In addition, we examined
the RMSE of estimates relative to the regression line in
the Narrow Likelihood/Wide Prior condition, which
gave an indication of the variability of sensorimotor
estimates (Figure 2A, right panel). We found a
significant effect of age group on RMSE (one-way
ANOVA), F(2, 41) ¼10.76, p,0.0005, and significant
differences between age groups—6–8 years versus 9–11
years: t(31) ¼2.80, p,0.01; 6–8 years versus 18þyears:
t(25) ¼4.31, p,0.001; 9–11 years versus 18þyears:
t(26) ¼2.19, p¼0.0378 (corrected a¼0.0167). The
RMSE (in units of screen width) was within an
acceptably low range for all age groups—6–8 years: M
(SD)¼0.07 (0.03); 9–11 years: 0.04 (0.02); 18þyears:
0.03 (0.01). This shows that all age groups understood
the candy-catching task and carried out the task above
chance level. Although we do observe differences
between age groups, these differences between cannot
be attributed to a lack of understanding of the task.
Having found differences in performance between
groups, we examined how weighting of prior and
likelihood information changes over the course of
development. In order to quantify the nature of
integration between the likelihood and the prior, we
used the parameters of the linear fit between estimates
and the likelihood centroid. The mean of the prior used
by participants is calculated using the parameters of the
linear fit: intercept/(1 estimation slope) (Berniker et
al., 2010). Full reliance on the likelihood indicates a
close relationship between estimates and the likelihood
and results in estimation slope 1. Full reliance on the
prior indicates a lack of relationship between estimates
and the likelihood and results in estimation slope 0.
We display the fitted estimation slope, intercept, and
estimation data for an 11-year old participant (Figure
2B), and the reliability of the slope and intercept
estimates for each individual participant (Figure 2C
and 2D). The linear fit explains a reasonable amount of
variance in raw estimates in each age group—6–8 years:
mean R
2
(SD)¼0.32 (0.22); 9–11 years: 0.54 (0.20); 18þ
years: 0.70 (0.12). This procedure allowed us to
quantify the nature of prior and likelihood integration
for each participant.
We were interested in whether sensorimotor inte-
gration of prior and likelihood improved during
development. To examine this, we first tested whether
participants’ estimates were consistent with simple
prior statistics. We analyzed the prior mean used by
participants (Figure 3A). We note that this information
was provided to participants on-screen, since they were
shown samples from the prior distribution. However,
since we were examining a child population, it was
necessary to make sure that they used visually available
information. We examined the influence of the prior
width, likelihood width, and age group on the prior
mean (Intercept/[1 estimation slope]). The analysis of
the prior mean did not reveal a significant influence of
prior width or age group (repeated-measures AN-
OVA)—prior width: F(1, 42) ¼0.37, p¼0.55; age
group: F(1, 42) ¼2.53, p¼0.12—nor their interaction,
F(1, 42) ¼0.76, p¼0.39. However, there was a
significant effect of likelihood width, F(2, 84), 3.94, p¼
0.03, and a significant Likelihood width 3Age
interaction, F(2, 84) ¼3.69, p¼0.04). Post hoc ttests
comparing levels of the likelihood-width variable and
the interaction between likelihood width and age group
did not show significant differences between conditions
(see Appendix, Tables A1 and A2). Therefore, age
group and stimulus factors did not play a strong role in
influencing the prior mean used by participants.
We next examined whether the weighting of prior
and likelihood information changed during develop-
ment by analyzing the estimation slope (Figure 3B). We
examined the influence of the prior width, likelihood
width, and age group on the estimation slope. A
repeated-measures ANOVA applied to the estimation
slope revealed significant main effects of prior width,
F(1, 42) ¼5.66, p,0.05, and likelihood width, F(2, 84)
¼53.78, p,0.0001, as well as a nonsignificant main
effect of age group, F(1, 42) ¼0.42, p¼0.52. Therefore,
there is evidence that the sample as a whole integrated
both the prior and the likelihood into their judgments.
In our analysis of estimation slopes, we then turned
to statistical interactions which reveal age-specific
effects. There was a significant Age group 3Prior width
interaction, F(1, 42) ¼26.34, p,0.0001, and no
significant Age group 3Likelihood width interaction,
F(2, 84) ¼3.39, p¼0.06. The use of likelihood may not
change considerably over the course of development.
However, there is evidence for an influence of age
group on use of prior statistics. We therefore examined
the influence of the prior width on the estimation slope
for different age groups, using paired ttests—6–8 years:
t(30) ¼0.54, p¼0.59; 9–11 years: t(32) ¼3.76, p,
0.001; 18þyears: t(20) ¼7.08, p,0.0001 (corrected a¼
0.0167). At a group level, 6- to 8-year-olds do not
distinguish between Narrow and Wide Prior condi-
Journal of Vision (2018) 18(12):8, 1–16 Chambers, Sokhey, Gaebler-Spira, & Kording 7
tions, with this difference becoming significant at 9–11
years. Therefore, young children ages 6–8 years show
the ability to incorporate likelihood into their judg-
ments as adults do, but there is no evidence that they
make use of the prior width until 9–11 years.
Children ages 6–8 years did not appear to use the
prior distribution when making sensorimotor estimates.
It could have been the case that they learned to do so
over the course of the experiment. We therefore
examined how the influence of the prior width on the
estimation slope changed during the experiment. We
computed slopes for bins of 40 consecutive trials and
averaged estimation slopes across likelihood conditions
(Figure 3C–3E). As shown by a repeated-measures
ANOVA, we did not find significant main effects of age
group, F(1, 41) ¼0.01, p¼0.94; prior width, F(1, 41) ¼
1.96, p¼0.17; or trial bin, F(2, 82) ¼0.60, p¼0.55; but
we did find a significant main effect of experimental
block, F(1, 41) ¼7.79, p,0.01. As before, we found a
significant Prior width 3Age group interaction, F(1,
41) ¼14.58, p,0.001. However, interactions between
age group, prior width, trial bin and experimental block
were not significant. The significant effect of experi-
mental block revealed a subtle tendency to rely more on
the likelihood in Block 1—M(SD)¼0.55 (0.25)—than
in Block 2—0.46 (0.22). However, we did not find
evidence for a change over the course of the experiment
in how participants distinguished between conditions
based on uncertainty. Therefore, we did not find
evidence that young children learned the prior variance
over the course of the experiment.
We formally tested whether estimation behavior is
consistent with Bayesian inference by comparing the
performance of a model which integrated prior and
likelihood information (Bayesian model) with baseline
models that used only prior information (Prior-only
model), used only likelihood information (Likelihood-
only model), and alternated between prior and
likelihood across trials (Switch model). We first show
that we can reliably infer the correct model from
simulated estimation data (Figure 4A). When applied
to estimation data, on average, we found a lower MSE
for the Bayesian model compared to the baseline
models (Figure 4B). Most individual participants’ data
were best fit by the Bayesian model: 11 out of 16
participants ages 6–8 years, 15 out 17 ages 9–11 years,
and 11 out of 11 adults (Figure 4C). On average, the
Bayesian model accounts for a reasonable amount of
variance of estimation slopes—6–8 years: mean R
2
(SD)
¼0.74 (0.26); 9–11 years: 0.94 (0.07); 18þyears: 0.96
Figure 3. Prior mean, estimation slope, and estimation slope as a function of trial bin. (A) Median prior mean as a function of prior
width (NP ¼narrow prior, WP ¼wide prior), likelihood width (NL ¼narrow likelihood, ML ¼medium likelihood, WL ¼wide likelihood),
and age group (error bars ¼95% confidence interval). The optimal value is shown (red). (B) Median estimation slope as a function of
prior width, likelihood width, and age group (error bars¼95% confidence interval). Optimal values are shown (red). (C–E) Estimation
slopes were computed for separate blocks and bins of 40 consecutive trials, then averaged across likelihood conditions. Median
estimation slopes (error bars ¼95% confidence) are shown for the three age groups: (C) 6–8 years, (D) 9–11 years, and (E) 18þyears.
Journal of Vision (2018) 18(12):8, 1–16 Chambers, Sokhey, Gaebler-Spira, & Kording 8
(0.02). Our findings suggest that some combination of
prior and likelihood was present in children as young as
6 years.
Having found that the Bayesian model provides a
better fit for the majority of participants across age
groups, we estimated the parameters of the Bayesian
model. We first show that our parameter-estimation
procedure leads to unbiased estimates using simulated
data (Figure 5A–5C). We simulated optimal partici-
pants with different amounts of noise added to the
likelihood (Dl¼0.01, 0.05, 0.1), who either used a prior
whose standard deviation was the mean of the
experimentally imposed prior standard deviations
(rsNP
,rsWP¼0.065), used partly differentiated priors
(rsNP¼0.048, rsWP ¼0.083), or used the experimen-
tally imposed priors (rsNP¼0.03, rsWP ¼0.1). The
parameters inferred from the model agreed with the
simulated parameters (Figure 5A–5C). We then esti-
mated the prior width and noise added to the likelihood
(rsNP
,rsWP
, and Dl) from the data of the 37 out of 44
participants whose data were better fit by the Bayesian
model. We inferred the prior variance from data sets
sampled with replacement 1,000 times from each
participant’s data. While parameters estimated from
the model are corrupted by noise, there is a trend for
prior variance parameters to be unaffected by the task
(experimental prior width) in young children (Figure
5D) and to shift in the direction of experimentally
imposed prior variances during development (Figure
5D–5F). We observe that, in young children ages 6–8
years, sensorimotor estimates are unaffected by prior
width both here and in our analysis of estimation
slopes. While the priors used by all age groups were
suboptimal, this bias appears to decrease with age.
The behavior of a small number of younger
participants was better fit by the switching model
(Figure 5H and 5I). We can accurately infer the
proportion p(likelihood) of trials where participants use
the likelihood from simulated data (Figure 5H). The
p(likelihood) is variable across participants who use
this strategy (Figure 5I). This simple strategy provides
an alternative to Bayesian integration but is not
prevalent in our sample.
Discussion
We investigated the development of Bayesian inte-
gration during sensorimotor estimation. Statistically
efficient estimation required integrating information
from a prior distribution with sensory information
from a likelihood distribution. We found that partic-
ipants’ estimates reflect the experimentally imposed
prior mean, which was expected, since samples of the
prior were displayed on screen. Our analysis of
estimation slopes showed that all age groups relied on
the likelihood according to its uncertainty. While the
older child group (9–11 years) and adults distinguished
between conditions based on prior width, we did not
find evidence for this in the youngest group (6–8 years).
We did not find learning related to the prior over the
course of the experiment, but we did find slightly less
reliance on the likelihood in later blocks. In all age
groups, a Bayesian model that integrated likelihood
and prior information performed better than simple
models that used only one or the other source of
information and a model that alternated between them.
This finding is consistent with the significant main
effect of the likelihood width on estimation slopes. We
also found that younger children exhibit larger
deviations from optimality in fitted parameters.
Figure 4. Model selection. (A) Confusion matrix showing the proportion of cases where each model was selected, computed from the
data of 1,000 participants per simulated model. We can infer the correct model from simulated data with reasonable accuracy. (B)
Median mean squared error for each model as a function of age group (error bars ¼95% confidence interval). (C) The number of
participants for whom each model was selected. The Bayesian model provides an improved fit for most participants (11 out of 16 ages
6–8 years, 15 out of 17 ages 9–11 years, 11 out of 11 adults).
Journal of Vision (2018) 18(12):8, 1–16 Chambers, Sokhey, Gaebler-Spira, & Kording 9
We examined the possibility that instead of inte-
grating information from the prior and likelihood,
subjects relied on only one source of information on
each trial and randomly switched between the two
sources across trials. The form of switching that we
investigated was simple, in that the probability that
subjects relied on the likelihood (the parameter of this
model) was fixed across conditions. However, other
forms of switching behavior that resemble Bayesian
integration are possible and have been explored in
previous work (Landy & Kojima, 2001; Laquitaine &
Gardner, 2018). For example, participants may weight
the prior and likelihood proportionally to their
uncertainties, which would lead to the same set of
estimation slopes in our experiment but differences in
the variance of estimates. Our experiment did not allow
us to accurately distinguish between Bayesian integra-
tion and this more complex variant of switching
behavior. Switching rather than full integration during
sensorimotor estimation remains a possible explanation
for sensorimotor estimation in children to be addressed
in future work.
Our finding that younger children exhibit larger
deviations from optimality than adults suggests that
efficient statistical inference is at least partly learned
during development, but leaves open questions on the
Figure 5. Estimates of model parameters. (A–C) Estimates of model parameters from 1,000 simulated Bayesian participants. (A)
Estimates of the Narrow Prior standard deviation (rsNP
). (B) Estimates of the Wide Prior standard deviation (rsWP
). (C) Estimates of
the standard deviation added to the likelihood (Dl). Simulated participants used the same prior in both conditions (rsNP
,rsWP ¼
0.065), used partly differentiated priors (rsNP¼0.048, rsWP¼0.083), or used the experimentally imposed prior (rsNP¼0.03, rsWP¼
0.1). We also varied the amount of noise added to the likelihood (Dl¼0.01, 0.05, 0.1). The median (error bars ¼95% confidence
interval) is shown in all panels. (D–F) Wide Prior variance as a function of the Narrow Prior variance inferred from the estimation data
of participants whose data was best fit by the Bayesian model: (D) ages 6–8 years, (E) ages 9–11 years, and (F) adults. Blue and green
lines show the experimentally imposed prior variance in the Narrow and Wide Prior conditions, respectively. (G) Shows the variance
added to the likelihood inferred from the estimation data. (H–I) Switch model. (H) Shows the p(likelihood) inferred from the Switch
models for 1,000 simulated subjects per p(likelihood) condition. (I) Shows the p(likelihood) for participants whose data was best fit by
the Switch model.
Journal of Vision (2018) 18(12):8, 1–16 Chambers, Sokhey, Gaebler-Spira, & Kording 10
sources of this suboptimality. Inaccurate likelihoods,
priors, cost functions, decisions rules, and approxima-
tions to Bayesian inference can all lead to subopti-
mality (Rahnev & Denison, 2018). It will be interesting
to tease apart their contributions in future work.
In a real-world context, priors must be learned from
our interactions with the world (Berniker et al., 2010). In
future investigations, it may be interesting to examine
learning of prior statistics without providing visual cues
on screen. Although this was outside the scope of the
current work, our findings provide certain predictions on
the changes that might occur during development. We
found that young children can weight information by its
relative uncertainty and that their judgments reflect the
mean of the prior but not its variance. This could be
further tested by varying the parameters of an unseen
prior. It may be the case that children can learn simple
statistics of distributions such as the mean but have
difficulty with higher order statistics, and do not show
signs of representing full distributions as adults do until
late in development (Acerbi, Vijayakumar, & Wolpert,
2014; Kording & Wolpert, 2004).
Previous work on Bayesian cue combination has
found that children’s behavior suggests late integration
in childhood, often finding that children’s behavior up
to the ages of 9–12 years is more consistent with
reliance on separate sources of available information
and switching between sources of information (Adams,
2016; Gori et al., 2008; Nardini et al., 2008). However,
in our study, most of the children in the youngest group
tested (6–8 years) integrated prior and likelihood
information as shown by the improved fit of a Bayesian
model to the data compared with baseline models that
use one source of information only or alternate
between sources of information.
Why do we observe early signs of integration, which
are typically not observed in the case of cue combina-
tion? It may be that we learn to integrate information
early to perform functions that are crucial to survival
such as movement, as in the present work, or basic
inferences on the behavior of visual objects (T´
egl´
aset
al., 2011). It could be that we integrate at an earlier
stage of development when the nature of the informa-
tion to be integrated is the same, like in the current
work. Efficient cue combination, on the other hand,
requires bringing together signals that are processed
separately, whether this be across or within the senses
(Gori et al., 2008; Nardini et al., 2010), and it could be
that maturation of the circuits which process these cues
is necessary before information can be combined
(Dekker et al., 2015).
Apparent inefficiencies in the behavior of young
children might be in some sense efficient given child-
ren’s goals and constraints. For example, in the case of
cue combination it may be beneficial to avoid fusion
during development in order to process cues separately,
so that the senses may calibrate themselves (Gori et al.,
2008; Nardini et al., 2008). It will be important for
future research to define the factors that lead to
integration of information under certain conditions and
tasks and not under others (Ernst, 2008). ‘‘Bayesian
brain’’ theories must be developed further to account
for how Bayesian inference emerges in the developing
brain, and in particular must account for develop-
mental trajectories of different behaviors.
If Bayesian computation is at the core of the neural
code (Beck et al., 2008; Zemel et al., 1998), behavior
should show the signatures of Bayesian inference under
all conditions, including during development. Our
findings support some level of Bayesian integration at
all age groups, but they also show that children do not
use available probabilistic information to the same
extent as adults. It may be that during development the
brain learns to approximate Bayesian principles by
means other than explicitly implementing Bayesian
computations in neural circuits (Mandt et al., 2017;
Weisswange et al., 2011). Our findings fit with ideas
suggested by Piaget (1954) on the role of constructivism
in child development: that abilities are acquired
through experience by building on more basic forms of
knowledge. In this sense, learning statistics may be seen
as a very basic form of knowledge. While we may be
born with a general learning architecture, it seems that
statistics should be seen not as core knowledge (Spelke
& Kinzler, 2007) but as an acquired skill.
Keywords: development,Bayesian integration,
sensorimotor
Acknowledgments
We would like to acknowledge Joshua Glaser for
useful comments on this manuscript. This work was
funded by NIH grant 5R01NS063399-08 awarded to
KPK.
Commercial relationships: none.
Corresponding author: Claire Chambers.
Email: clairenc@seas.upenn.edu.
Address: Department of Bioengineering and
Department of Neuroscience, University of
Pennsylvania, Philadelphia, PA, USA.
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Journal of Vision (2018) 18(12):8, 1–16 Chambers, Sokhey, Gaebler-Spira, & Kording 13
Appendix: Instructions during
experiment, analysis of prior mean
Figure A1. Instructions Part 1. Screens 1 and 2. We presented participants with the instructions that someone behind them was
throwing candy into a pond, represented by the screen. Screens 3 to 6. We showed participants 200 samples from the prior
distribution (‘‘Where the candy lands’’). The first 10 samples (two shown here) were shown ‘‘falling’’ into the pond one by one.
Journal of Vision (2018) 18(12):8, 1–16 Chambers, Sokhey, Gaebler-Spira, & Kording 14
Figure A2. Instructions Part 2. Screens 7 to 10. We showed participants an example trial, where they were given noisy feedback on
target location in the form of n¼4 samples from the likelihood (Splash). We showed participants a vertical bar and asked them to use
it to catch the candy by moving the bar from left to right. After they provided their estimate, we showed them the candy’s true
location. Screens 11 and 12. We gave participants information on ‘‘bonuses’’ and the duration of the experiment.
Journal of Vision (2018) 18(12):8, 1–16 Chambers, Sokhey, Gaebler-Spira, & Kording 15
Comparison Age group t(df)p
Narrow versus medium
likelihood
6–8 years 1.64 (30) 0.1123
9–11 years 0.60 (32) 0.5537
18þyears 0.96 (86) 0.3395
Medium versus wide
likelihood
6–8 years 1.23 (30) 0.2296
9–11 years 1.07 (32) 0.2921
18þyears 1.50 (86) 0.1365
Narrow versus wide
likelihood
6–8 years 1.01 (30) 0.3197
9–11 years 1.77 (32) 0.0862
18þyears 1.56 (86) 0.8767
Table A1. Paired ttests comparing prior mean across likelihood
width and age conditions (corrected a¼0.0056).
Comparison t(df)p
Narrow versus medium likelihood 0.96 (86) 0.3395
Medium versus wide likelihood 1.50 (86) 0.1365
Narrow versus wide likelihood 0.16 (86) 0.8767
Table A2. Paired ttests comparing prior mean across likelihood
width (corrected a¼0.0167).
Journal of Vision (2018) 18(12):8, 1–16 Chambers, Sokhey, Gaebler-Spira, & Kording 16
... If adaptive combination is true, it is also a breakthrough finding for the study of developing Bayes-like reasoning in perception and memory. Almost all previous studies to look at Bayesian cue combination in children under 10 years old have returned negative results (Adams, 2016;Burr & Gori, 2011;Chambers et al., 2018;Dekker et al., 2015;Gori et al., 2012;Jovanovic & Drewing, 2014;Nardini et al., 2010Nardini et al., , 2013Petrini et al., 2014), including one that looked at combination of cues for spatial recall (Nardini et al., 2008). For example, when judging a horizontal location with a spatialized audio cue and a brief visual cue, children under 10 years old fail to integrate the two efficiently; the precision of their judgements is not any better than with the visual cue alone (Gori et al., 2012). ...
... Adaptive selection is a non-Bayesian process of selecting the best single cue and using it in isolation. For children under 10 years, this is in line with previous research regarding the use of multiple cues (Adams, 2016;Burr & Gori, 2011;Chambers et al., 2018;Dekker et al., 2015;Gori et al., 2012;Jovanovic & Drewing, 2014;Nardini et al., 2008Nardini et al., , 2010Nardini et al., , 2013Petrini et al., 2014). Reanalysis of previous data agrees as well. ...
... This further applies in a navigation task where the two cues are vestibular and proprioceptive (Frissen et 1424(Frissen et al., 2011. Adults also rapidly learn egocentric (sensorimotor) prior distributions and use them in a Bayesian fashion as well (Bejjanki et al., 2016;Berniker et al., 2010;Chambers et al., 2018;Körding & Wolpert, 2004;Kwon & Knill, 2013;Narain et al., 2013;Sato & Kording, 2014;Tassinari et al., 2006). In practice, this means that they learn where targets tend to be and bias their responses toward the places they tend to be most often. ...
Article
After becoming disoriented, an organism must use the local environment to reorient and recover vectors to important locations. A new theory, adaptive combination, suggests that the information from different spatial cues is combined with Bayesian efficiency during reorientation. To test this further, we modified the standard reorientation paradigm to be more amenable to Bayesian cue combination analyses while still requiring reorientation in an allocentric (i.e., world-based, not egocentric) frame. Twelve adults and 20 children at ages 5 to 7 years old were asked to recall locations in a virtual environment after a disorientation. Results were not consistent with adaptive combination. Instead, they are consistent with the use of the most useful (nearest) single landmark in isolation. We term this adaptive selection. Experiment 2 suggests that adults also use the adaptive selection method when they are not disoriented but are still required to use a local allocentric frame. This suggests that the process of recalling a location in the allocentric frame is typically guided by the single most useful landmark rather than a Bayesian combination of landmarks. These results illustrate that there can be important limits to Bayesian theories of the cognition, particularly for complex tasks such as allocentric recall. (PsycInfo Database Record (c) 2021 APA, all rights reserved).
... Recent research has questioned whether the use of BES is inherent in nature or the consequence of experience (Chambers, Sokhey, Gaebler-Spira, & Kording, 2018). To address this issue, Chambers et al. (2018) examined the development of BI in children between the ages of 6-11 compared to young adults. ...
... Recent research has questioned whether the use of BES is inherent in nature or the consequence of experience (Chambers, Sokhey, Gaebler-Spira, & Kording, 2018). To address this issue, Chambers et al. (2018) examined the development of BI in children between the ages of 6-11 compared to young adults. The results indicated that children used some level of BES; however, their use of probabilistic information was not as efficient as adults. ...
... Recent developmental work has indicated some level of BI in young children (Chambers et al., 2018). Improvement across development is consistent with the notion that motor experience can influence the brain's use of probabilistic information to perform SE effectively. ...
Article
An experiment was designed to determine the effects of sensory uncertainty on sensorimotor estimation in elite athletes compared to non-athletes. Nineteen elite athletes and 16 non-athletes were required to estimate when and where a cursor arrived at a target location. The cursor position was displayed through its entire trajectory in the certain condition while only briefly in the uncertain condition. Accuracy and variability in time and spatial domains were calculated. A Bayesian analysis using subsets of subjects' total spatial variance was also performed. The results indicated that athletes and non-athletes used estimation strategies consistent with Bayesian integration. The results also showed a decrease in variability for spatial performance for both groups during the uncertain condition compared to the certain condition, especially when the cursor location was further away from the prior mean. This decrease in variability was significantly greater for non-athletes. By concentrating performance around the end-point mean location, an increase in spatial error occurred. More spatial and timing errors were observed in non-athletes than athletes, indicating athletes were more certain about likelihood information or their interpretation of likelihood information than non-athletes. These results suggest that athletic experience may facilitate the use of probabilistic information for optimal sensorimotor estimations.
... Of the two, allocentric information is more difficult and noisy but more durable and flexible [48][49][50][51] ; purely egocentric information immediately requires updating or discarding whenever the viewpoint changes, but allocentric information can be used flexibly from any viewpoint. There have already been multiple studies suggesting that prior information in an egocentric frame can be used effectively after some exposure to a novel environment [6][7][8][9][10][11][12][13][14] . It is not yet known if the same is true for allocentric information. ...
... To our knowledge, no previous projects have been designed to test if people can use an allocentric prior. However, there have been several reports of successfully using egocentric spatial priors [6][7][8][9][10][11][12][13][14] . In this kind of task, the participant is in front of a screen or other display device and has to point to a target. ...
... We should note carefully that this does not mean that adults do not use spatial priors. Egocentric prior use has been demonstrated several times in adults [6][7][8][9][10][11][12][13][14] . Earlier, we used an example of learning that your car tends to be parked near the security hut. ...
Article
Full-text available
Prior information represents the long-term statistical structure of an environment. For example, colds develop more often than throat cancer, making the former a more likely diagnosis for a sore throat. There is ample evidence for effective use of prior information during a variety of perceptual tasks, including the ability to recall locations using an egocentric (self-based) frame. However, it is not yet known if people can use prior information effectively when using an allocentric (world-based) frame. Forty-eight adults were shown sixty sets of three target locations in a sparse virtual environment with three beacons. The targets were drawn from one of four prior distributions. They were then asked to point to the targets after a delay and a change in perspective. While searches were biased towards the beacons, we did not find any evidence that participants successfully exploited the prior distributions of targets. These results suggest that allocentric reasoning does not conform to normative Bayesian models: we saw no evidence for use of priors in our cognitively-complex (allocentric) task, unlike in previous, simpler (egocentric) recall tasks. It is possible that this reflects the high biological cost of processing precise allocentric information.
... This posits that the brain decodes hidden environmental regularities by means of statistical inference, and as such computes the probability distributions of prior knowledge and estimates the likelihood of sensory information [2]. Multisensory processing is consistent with Bayesian with principles that describe perception rooted in MSI and sensorimotor behavior [37]. ...
... Furthermore, children as young as 6 years of age showed some level of optimal statistical inference in a simple position-estimation Bayesian task, where data was best fitted by Bayesian model fit. These findings enabled the authors to conclude that some level of Bayesian integration is already present in early childhood, which is gradually refined parallel to acquiring experience, as the brain learns to approximate [37]. Alternatively, cross-sensory calibration may be the antecedent of MSI, allowing the more accurate sense to teach the others. ...
Article
Full-text available
At birth, the vestibular system is fully mature, whilst higher order sensory processing is yet to develop in the full-term neonate. The current paper lays out a theoretical framework to account for the role vestibular stimulation may have driving multisensory and sensorimotor integration. Accordingly, vestibular stimulation, by activating the parieto-insular vestibular cortex, and/or the posterior parietal cortex may provide the cortical input for multisensory neurons in the superior colliculus that is needed for multisensory processing. Furthermore, we propose that motor development, by inducing change of reference frames, may shape the receptive field of multisensory neurons. This, by leading to lack of spatial contingency between formally contingent stimuli, may cause degradation of prior motor responses. Additionally, we offer a testable hypothesis explaining the beneficial effect of sensory integration therapies regarding attentional processes. Key concepts of a sensorimotor integration therapy (e.g., targeted sensorimotor therapy (TSMT)) are also put into a neurological context. TSMT utilizes specific tools and instruments. It is administered in 8-weeks long successive treatment regimens, each gradually increasing vestibular and postural stimulation, so sensory-motor integration is facilitated, and muscle strength is increased. Empirically TSMT is indicated for various diseases. Theoretical foundations of this sensorimotor therapy are discussed.
... This posits that the brain decodes hidden environmental regularities by means of statistical inference, and as such computes the probability distributions of prior knowledge and estimates the likelihood of sensory information [5]. Multisensory processing is consistent Bayesian with principles that describe perception rooted in MSI, and sensorimotor behavior [46]. ...
... Furthermore, children as young as 6 years of age showed some level of optimal statistical inference in a simple position-estimation Bayesian task, where data was best fitted by Bayesian model fit. These findings enabled the authors to conclude that some level of Bayesian integration is already present in early childhood, which is gradually refined parallel to acquiring experience, as the brain learns to approximate [46]. Alternatively, cross-sensory calibration may be the antecedent of MSI, allowing the more accurate sense to teach the others. ...
Preprint
Full-text available
Background: At birth the vestibular system is fully mature, and primitive reflexes are functional whilst higher order sensory processing is yet to develop in the full-term neonate. Sequential motor development driven by primitive survival reflexes sets the appropriate framework for development of sensory processing including multisensory processing and sensorimotor integration. Results and conclusions: The current paper lays out a putative theoretical framework to account for the role vestibular stimulation may have driving multisensory and sensorimotor integration. Accordingly, vestibular stimulation, by activating the parieto-insular vestibular cortex, and/or the posterior parietal cortex may provide the cortical input for multisensory neurons in the superior colliculus that is needed for multisensory processing. Furthermore, we propose that primitive survival reflex-driven motor development, by inducing change of reference frames, may shape the receptive field of multisensory neurons. This, by leading to lack of spatial contingency between formally contingent stimuli, may cause degradation of prior motor responses, hence lead to integration of reflexes. Integration of primitive survival reflexes is mandatory prerequisite for cortically controlled motor responses to emerge. Additionally, we offer a testable hypothesis explaining the beneficial effect of sensory integration therapies regarding attentional processes. Key concepts of a sensorimotor integration therapy (e.g. targeted sensorimotor therapy (TSMT)) are also put into a neurological context. TSMT utilizes specific tools and instruments. It is administered in 8-weeks long successive treatment regimes, each gradually increasing vestibular and postural stimulation, so sensory-motor integration is facilitated, primitive reflexes are inhibited, and muscle strength is increased. Empirically TSMT is indicated for various diseases. Theoretical foundations of this sensorimotor therapy are discussed.
... Likewise, if the likelihood information we are receiving seems to be more unreliable, we may tend to rely more on prior knowledge to guide our current decision-making behavior. Previous literature has generally found that those considered to be neuro-typical can make decisions in this Bayesian-optimal manner, by appropriately utilizing the information's relative levels of uncertainty, as predicted by Bayesian Decision Theory (Chambers et al., 2018;Geisler & Kersten, 2002;Genewein et al., 2015;Knill, 2007;Körding & Wolpert, 2004Mathys et al., 2011;Weiss et al., 2002;Wolpert, 2007;Yuille & Bülthoff, 1994). In addition, there is evidence in support of the independent encoding and distinct representation of these two types of information in the brain (Beierholm et al., 2009;Friston et al., 2002;Vilares et al., 2012). ...
Preprint
Existing research presents a working understanding of Borderline Personality Disorder (BPD) patients’ symptomatology, traits, and behavior in everyday life, but how they combine and utilize prior and likelihood (current sensory) information when making decisions remains unclear. Bayesian Decision Theory suggests that optimal decision-making behavior should combine and weigh both pieces of information according to their relative uncertainties, such that people rely more on the information with less uncertainty when making a decision. Though this optimal behavior has been observed in neuro-typical populations, prior literature suggests that certain neuro-atypical populations can deviate. Some characteristics of BPD patients, such as impulsive behavior and drastic changes in the overall perception of themselves and others, suggest that they may be over-relying on likelihood information and not sufficiently taking prior information into account. From a Bayesian perspective, this can be interpreted as having ‘weak’ priors which may lead to suboptimal decision-making. Here, we investigated this hypothesis by having BPD patients (n = 23) and healthy controls (n = 18) perform a coin-catching sensorimotor task with varying levels of prior and likelihood information uncertainty. Our results indicate that, contrary to our prediction, BPD patients were able to use prior information, and that their use of prior and likelihood information follows qualitatively Bayesian behavior. We found that BPD patients, at least in a lower-level and non-affective sensorimotor task, are still able to use both prior and likelihood information and react appropriately to the respective uncertainties. This could be suggesting that any potential deficits in the use of prior information may not be widespread or only be apparent in affectively-charged interpersonal contexts.
... The Bayesian model of multisensory integration suggests that adults fuse redundant sensory inputs in a statistically optimal way by weighting the sources according to their uncertainty 16,17 . The ability to combine different cues to obtain more precise estimates of one's surroundings appears late in childhood development 18,19 , that is, after the individual modalities have matured 20,21 , unless additional feedback on the reliability of each cue is provided 22 . Younger children will thus favor the information provided by the modality with the highest context-dependent reliability 19,23 . ...
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Full-text available
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... Everything we have discussed could be framed in terms of a cue-prior combination task rather than a cue-cue combination task by letting all parameters that represent one of the cues in each of our equations represent the prior knowledge instead. For example, many of the studies focusing on integration of sensory and prior information require observers to estimate the location of a hidden target, on a continuous scale, using an uncertain sensory cue and prior knowledge of the target distribution (e.g., Berniker et al., 2010;Tassinari et al., 2006;Vilares et al., 2012;Chambers et al., 2018;Kiryakova et al., 2020;Bejjanki et al., 2016). In some cases, when the target distribution is centered on the middle of the stimulus-response range, it would be difficult to parse how much of the central bias is due to a reliance on prior knowledge or a central tendency bias. ...
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Observers in perceptual tasks are often reported to combine multiple sensory cues in a weighted average that improves precision—in some studies, approaching statistically optimal (Bayesian) weighting, but in others departing from optimality, or not benefitting from combined cues at all. To correctly conclude which combination rules observers use, it is crucial to have accurate measures of their sensory precision and cue weighting. Here, we present a new approach for accurately recovering these parameters in perceptual tasks with continuous responses. Continuous responses have many advantages, but are susceptible to a central tendency bias, where responses are biased towards the central stimulus value. We show that such biases lead to inaccuracies in estimating both precision gains and cue weightings, two key measures used to assess sensory cue combination. We introduce a method that estimates sensory precision by regressing continuous responses on targets and dividing the variance of the residuals by the squared slope of the regression line, “correcting-out” the error introduced by the central bias and increasing statistical power. We also suggest a complementary analysis that recovers the sensory cue weights. Using both simulations and empirical data, we show that the proposed methods can accurately estimate sensory precision and cue weightings in the presence of central tendency biases. We conclude that central tendency biases should be (and can easily be) accounted for to consistently capture Bayesian cue combination in continuous response data.
Preprint
The acquisition of postural control is an elaborate process, which relies on the balanced integration of multisensory inputs. Current models suggest that young children rely on an ‘en-block’ control of their upper body before sequentially acquiring a segmental control around the age of 7, and that they resort to the former strategy under challenging conditions. While recent works suggest that a virtual sensory environment alters visuomotor integration in healthy adults, little is known about the effects on younger individuals. Here we show that this coordination pattern is disrupted by an immersive virtual reality framework where a steering role is assigned to the trunk, which causes 6- to 8-year-olds to employ an ill-adapted segmental strategy. These results provide an alternate trajectory of motor development and emphasize the immaturity of postural control at these ages.
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