On the computational basis of the confirmation bias
Richard D. Lange Ankani Chattoraj
Ralf M. Haefner
University of Rochester
evidence integ.
(weighted sum)
In evidence integration tasks, subjects report one of two categories of a stimulus
after observing a sequence e1 ...en of evidence. Each ei is independent and slightly
predictive of the correct choice (Gold and Shadlen 2007)
1 )P (et |C)
T ime
The probabilistic inference model is empirically
distinguishable from the bounded integration model by
changing the variance of the evidence sequence e2
Time
0.5
1
0.5
Time
Internal variables
context, beliefs
(prior)
pposterior(x|E,I)
Kiani, Hanks, and
Shadlen 2008
External observations
-1
p(C)
p(C)
High evidence
variance
Wyatt et al. 2012
C
+1
e
e
t
t
C is discrete, x and e are continuous.
The neural sampling hypothesis states that the variability of neural responses
reflects the dynamics of MCMC sampling over an internal model of the world
(Fiser et al. 2010)
(s)
(s 1)
⇠ P (x|C
)P (et |x)
C (s) ⇠ P (x(s) |C)Pt
1 (C)
Sampling Race Model samples of C are counted by a downstream area (e.g.
LIP). We assume a single brain area both counts samples and represents the
posterior
ns
Pt (C = +1) /
+ #C1+ + #C2+ + ... + #Ct+
2
Buffer-less marginalization the posterior over C is updated by approximate
marginalization over x, yielding:
1 (C)P (C|x
+ const
Z
P (C, xt |et )dxt
= P (C|xt )p(xt |et )dxt
with
(s)
xt
⇠ p(xt |et )
(s)
)
log Pt
0
1 (C)
low variance
unbiased
Low σe2
High σe2
Ideal Obs.
0.9
1
1
2
σ
2
e
3
4
5
The standard Bounded Integration model accounts for a decreasing PK by analogy to
reaction time tasks (we ignore further information once reasonably certain)
Our Probabilistic Inference models account for a decreasing PK as the influence of
priors (we see what we expect to see, we expect to see what we’ve seen)
These models are distinguishable from the standard bounded integration model by
changing the variance of evidence
Results of existing studies could be interpreted as low- or high-variance evidence
(Reported Evidence Weighting table, lower left), consistent with the PI model
Future psychophysics studies needed that directly manipulate evidence variance
within a single paradigm
The deviations from optimality in our models come from the assumptions that there
are no synchronized internal “buffers” and that neural responses represent
posteriors
1
ns
Higher variance of evidence
Sharper likelihood
Overcomes priors
P (C|et )
P (et |C) /
Pt 1 (C)
S
X
1
(s)
log P (et |C) = log
P (C|xt )
S s=1
0.50
We provide a new account of perceptual confirmation biases based on probabilistic
inference and sampling
e
0
Left: Sampling Race Model
Right: Buffer-less Marginalization
x
Empirically, subjects appear less
biased in high evidence variance
tasks, consistent with PI models (see
Reported Evidence Weighting table)
Summary
x
S
X
1
(s)
⇡
P (C|xt )
S s=1
Flat PK
Both sampling models are less
biased at high evidence variance
0.55
0.40
0
x
x
P (C|et ) =
Brunton et al. 2013
Time
The bounded integration model is
less biased at low evidence variance
0.45
Pt (C) / Pt
Decreasing
PK
Race Model
Buffer-less Marg
Bounded Integrator
Ideal Observer
0.65
9
C
sensory evidence
(likelihood)
Reported Evidence Weighting
Low evidence
variance
5
Probabilistic Inference Models
model
info.
Results from existing studies favor the sampling-based probabilistic inference
model
Nienborg &
Cumming 2009
The asymptotic offset of the PK
indicates whether a less biased model
is more ideal or more random
0.70
Time
C
PK(t)
The decreasing PK arises in the probabilistic inference
models from reusing the shifting posterior as a prior,
causing early evidence to be “double counted”
0.4
Offset
PK is reverse correlation of
choices and evidence
A decreasing PK represents a confirmation bias; it
Ideal
Forgetful
means the subject tends to weight evidence early in the
Random Conf. bias
0.64
trial more than evidence later in the trial.
We contrast the standard Bounded Integration model
with a sampling based probabilistic inference model
(Fiser et al. 2010)
0.2
0.60
Chose A
Chose B
Models
0.1
0.3
0.8
PK
The psychophysical kernel (PK) quantifies
how much weight a subject gives to
evidence as a function of time. It is
computed as reverse correlation between
the subject’s choice and how much signal
was present at each point in time.
Low σe2
High σe2
Ideal Obs.
The initial slope of the PK measures
the models’ biases
0.0
Each trial evidence weighted with a step function.
Sum over trials is a ramp
1 (C)P (et |C)
Log odds of evidence
Pt (C) / Pt
0.8
PK
P (C|e1 , ..., et ) / P (C|e1 , ..., et
P (ei |C)
Weight
or sequentially (“online”):
i=1
e
p(RT )
P (C|e1 , ..., en ) / P (C)
n
Q
neural response
low σ2e
high σ2e
0.1
PK
C
log post. ratio
The confirmation bias is one of the most well-studied and ubiquitous irrational
human behaviors. It is found both cognitively and in low-level perception
Stronger Bias
Standard Model
Differentiating Models
Higher variance of evidence
Reaches bound faster
More evidence ignored
Stronger bias
Slope
Introduction
0.7
Bufferless Marg.
Race Model
0.6
0.5
1
5
Time
9
References
Brunton, B. W., Botvinick, M. M., & Brody, C. D. (2013). Rats and humans can
optimally accumulate evidence for decision-making. Science, 340(6128), 95–8.
Fiser, J., Berkes, P., Orbán, G., & Lengyel, M. (2010). Statistically optimal perception
and learning: from behavior to neural representations. Trends in Cognitive
Sciences, 14(3), 119–30.
Gold, J. I., & Shadlen, M. N. (2007). The neural basis of decision making. Annual
Review of Neuroscience, 30(30), 535–574.
Haefner, R. M., Berkes, P., & Fiser, J. (2014). Perceptual decision-making as
probabilistic inference by neural sampling. arXiv, (1409.0257v1).
Kiani, R., Hanks, T. D., & Shadlen, M. N. (2008). Bounded integration in parietal
cortex underlies decisions even when viewing duration is dictated by the
environment. The Journal of Neuroscience, 8(12), 3017–3029.
Nienborg, H., & Cumming, B. G. (2009). Decision-related activity in sensory neurons
reflects more than a neuron’s causal effect. Nature, 459(7243), 89–92.
Wyart, V., Gardelle, V. De, Scholl, J., & Summerfield, C. (2012). Rhythmic
Fluctuations in Evidence Accumulation during Decision Making in the Human Brain.
Neuron, 76(4), 847–858.