virtual journal club may 4th 2020 · confidence refers to the “sense of knowing” that comes...
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Virtual Journal Club May 4th 2020
Confidence refers to the “sense of knowing” that comes with a decision
Confidence refers to the “sense of knowing” that comes with a decisionConfidence affects the planning of subsequent actions
Bayesian Confidence Hypothesis Confidence is based on the posterior probability of the chosen category : p(correct)
e.g., Peirce and Jastrow (1884)Galvin et al. (1959)Kepecs and Mainen (2012)Mamassian (2016)Kiani and Shadlen (2009)Kepecs, Uchida, Zariwala and Mainen (2009)Pouget, Drugowitsch and Kepecs (2016)Sanders, Hangya and Kepecs (2016)
Bayesian Confidence Hypothesis Confidence is based on the posterior probability of the chosen category : p(correct)
Left or right tilted?
1 2Category
0
0.2
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0.6
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1
Post
erio
r pro
babi
lity
Low confidence
p(L | x) p(R | x)
Left or right tilted?
1 2Category
0
0.2
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0.6
0.8
1
Post
erio
r pro
babi
lity High confidence
p(L | x) p(R | x)
Kiani and Shadlen, 2009 Fetsch et al., 2014 Kepecs et al., 2009
Bayesian Confidence Hypothesis Confidence is based on the posterior probability of the chosen category : p(correct)
Adler and Ma, 2018
People often face more complex decisions involving multiple choices
Distinguish different models of confidence report
People often face more complex decisions involving multiple choices
Distinguish different models of confidence report
timeWhich group does the black dot belong to?
How confident are you in your decision?
somewhatlow
verylow
somewhathigh
veryhigh
AExperiment 1 and 3 Experiment 2B C
Fig. 1 Experimental procedure and stimuli.
DEMO
Astimulus s measurement x posterior probability
decision
confidence c
P( |x) P( |x)
P( |x)
σ
C^
B
P( |x) P( |x)
P( |x)C
conf
iden
ce
high
low
Max model Difference model Entropy model
post
erio
rpr
obab
ility
Fig. 2 Models.
p(C | x) = p(x |C)
p(x | ′C )′C =1
3
∑
p(x |C) = N (mC ,(σ s +σ )I)
Astimulus s measurement x posterior probability
decision
confidence c
P( |x) P( |x)
P( |x)
σ
C^
B
P( |x) P( |x)
P( |x)C
conf
iden
ce
high
low
Max model Difference model Entropy model
post
erio
rpr
obab
ility
Fig. 2 Models.
Categorical decision: Always pick the group with the highest posterior
Confidence report: Three candidate modelsMax, Difference, Entropy
Astimulus s measurement x posterior probability
decision
confidence c
P( |x) P( |x)
P( |x)
σ
C^
B
P( |x) P( |x)
P( |x)C
conf
iden
ce
high
low
Max model Difference model Entropy model
post
erio
rpr
obab
ility
Fig. 2 Models.p(C = 1| x) p C x( )log p C x( )
C∑p(C = 1| x)− p(C = 2 | x)
Max (Maximum a posteriori)
Diff (Difference)
Ent (Negative entropy)p C x( )log p C x( )
C∑
p(C = 1| x)− p(C = 2 | x)
p(C = 1| x)
Astimulus s measurement x posterior probability
decision
confidence c
P( |x) P( |x)
P( |x)
σ
C^
B
P( |x) P( |x)
P( |x)C
conf
iden
ce
high
low
Max model Difference model Entropy model
post
erio
rpr
obab
ility
Max
Diff
Entropy
Confidence
1 2Category
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Post
erio
r pro
babi
lity
p(C = 1| x)
p(C = 1| x)− p(C = 2 | x)
p C x( )log p C x( )C∑
1 2Category
0
0.2
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1
Post
erio
r pro
babi
lity
Confidence
p C = 1 x( )+ p C = 2 x( ) = 1
p(C = 1| x)
p(C = 1| x)− p(C = 2 | x)
p C x( )log p C x( )C∑
Max
Diff
Entropy
Confidence
p(C = 1| x)
p(C = 1| x)− (1− p(C = 1| x))
p C = 1 x( )log p C = 1 x( )+ 1− p C = 1 x( )( )log 1− p C = 1 x( )( )
1 2Category
0
0.2
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1
Post
erio
r pro
babi
lity
p C = 1 x( )+ p C = 2 x( ) = 1
Max
Diff
Entropy
Confidence
Max modelDiff modelEnt model p(C = 1| x)
1 2Category
0
0.2
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0.6
0.8
1
Post
erio
r pro
babi
lity
p C = 1 x( )+ p C = 2 x( ) = 1
Astimulus s measurement x posterior probability
decision
confidence c
P( |x) P( |x)
P( |x)
σ
C^
B
P( |x) P( |x)
P( |x)C
conf
iden
ce
high
low
Max model Difference model Entropy model
post
erio
rpr
obab
ility
Fig. 2 Models. Sources of variability:(1) sensory noise (noisy measurement of the target dot location)(2) inference noise (decision noise)
Astimulus s measurement x posterior probability
decision
confidence c
P( |x) P( |x)
P( |x)
σ
C^
B
P( |x) P( |x)
P( |x)C
conf
iden
ce
high
low
Max model Difference model Entropy model
post
erio
rpr
obab
ility
Astimulus s measurement x posterior probability
decision
confidence c
P( |x) P( |x)
P( |x)
σ
C^
B
P( |x) P( |x)
P( |x)C
conf
iden
ce
high
low
Max model Difference model Entropy model
post
erio
rpr
obab
ility
Astimulus s measurement x posterior probability
decision
confidence c
P( |x) P( |x)
P( |x)
σ
C^
B
P( |x) P( |x)
P( |x)C
conf
iden
ce
high
low
Max model Difference model Entropy model
post
erio
rpr
obab
ility
Astimulus s measurement x posterior probability
decision
confidence c
P( |x) P( |x)
P( |x)
σ
C^
B
P( |x) P( |x)
P( |x)C
conf
iden
ce
high
low
Max model Difference model Entropy model
post
erio
rpr
obab
ility
low decision noise high decision noise
q ~ Dirichlet(p;α )
The expected value of q is p. The concentration parameter 𝛼, a scalar that determines the magnitude of the decision noise
Max modelDiff modelEnt model{ } Very
unconfidentSomewhat
unconfidentSomewhatconfident
Veryconfident
b1 b2 b3
c*1 2 3Category
0
0.2
0.4
0.6
0.8
1
Post
erio
r pro
babi
lity
b1,b2 ,b3Criteria:Noise:
Lapse rate:σ ,αλ
Free parameters
1 2 3 4Confidence
1
2
3
Decision
p(di ,ci | si ,b1,b2 ,b3,σ ,α ,λ)joint probability for trial i
b1,b2 ,b3Criteria:Noise:
Lapse rate:σ ,αλ
Free parameters
Maximum likelihood estimationLM (θ ) = p(di ,ci | si ,b1,b2 ,b3,σ ,α ,λ)
i=1
n
∏
timeWhich group does the black dot belong to?
How confident are you in your decision?
somewhatlow
verylow
somewhathigh
veryhigh
AExperiment 1 and 3 Experiment 2B C
Fig. 1 Experimental procedure and stimuli.
Diffe
renc
eM
axEn
tropy
A
B
1
2
3
4
1
2
3
4
1
2
3
4
Target location (deg)
Mea
n co
nfide
nce
repo
rtDataModel fit
-2 0 2 -2 0 2 -2 0 2 -2 0 2
Fig. 3. Experiment 1.
Diffe
renc
eM
axEn
tropy
A
B
1
2
3
4
1
2
3
4
1
2
3
4
Target location (deg)
Mea
n co
nfide
nce
repo
rt
DataModel fit
-2 0 2 -2 0 2 -2 0 2 -2 0 2
Diffe
renc
eM
axEn
tropy
A
B
1
2
3
4
1
2
3
4
1
2
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4
Target location (deg)
Mea
n co
nfide
nce
repo
rtDataModel fit
-2 0 2 -2 0 2 -2 0 2 -2 0 2
Fig. 3. Experiment 1.
0
0.5
1
0
0.5
1
0
0.5
1
Prop
ortio
n ch
oice
Diffe
renc
eM
axEn
tropy
A
B
-2 0 2 -2 0 2 -2 0 2 -2 0 2Target location (deg)
Data
Model fit
Supplementary Figure 2.
A B Experiment 2Experiment 1
0
500
1000
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2000
2500
3000
AIC
relat
ive to
th
e Di
ffere
nce
mod
el
0
500
1000
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3000
AIC
relat
ive to
th
e Di
ffere
nce
mod
el
Diff
Max
Ent
Model
Diff
Max
Ent
Model
Fig. 4. Experiment 1. Model comparisons using ΔAIC:AIC of each model compared with the Difference model
sensory noise and
decision noise
decision noise sensory noiseA
B
C
Model
Diff-Sen-Dir
Max-Sen-Dir
Ent-Sen-Dir0
500100015002000250030003500400045005000
Expe
rimen
t 1
AIC
relat
ive to
th
e Di
ffere
nce-
Sen-
Dir m
odel
Diff-DirMax-Dir
Ent-DirDiff-S
enMax-Sen
Ent-Sen
Diff-Sen-Dir
Max-Sen-Dir
Ent-Sen-Dir0
500100015002000250030003500400045005000
Expe
rimen
t 2
AIC
relat
ive to
th
e Di
ffere
nce-
Sen-
Dir m
odel
Diff-DirMax-Dir
Ent-DirDiff-S
enMax-Sen
Ent-Sen
Diff-Sen-Dir
Max-Sen-Dir
Ent-Sen-Dir0
500100015002000250030003500400045005000
Expe
rimen
t 3
AIC
relat
ive to
th
e Di
ffere
nce-
Sen-
Dir m
odel
Diff-DirMax-Dir
Ent-DirDiff-S
enMax-Sen
Ent-Sen
Supplementary Figure 4.
sensory noise and
decision noise
decision noise sensory noiseA
B
C
Model
Diff-Sen-Dir
Max-Sen-Dir
Ent-Sen-Dir0
500
1000
1500
2000
2500
Expe
rimen
t 1
AIC
relat
ive to
th
e Di
ffere
nce-
Sen-
Dir m
odel
Diff-DirMax-Dir
Ent-DirDiff-S
enMax-Sen
Ent-Sen
Diff-Sen-Dir
Max-Sen-Dir
Ent-Sen-Dir0
500
1000
1500
2000
2500
Expe
rimen
t 2
AIC
relat
ive to
th
e Di
ffere
nce-
Sen-
Dir m
odel
Diff-DirMax-Dir
Ent-DirDiff-S
enMax-Sen
Ent-Sen
Diff-Sen-Dir
Max-Sen-Dir
Ent-Sen-Dir0
500
1000
1500
2000
2500
Expe
rimen
t 3
AIC
relat
ive to
th
e Di
ffere
nce-
Sen-
Dir m
odel
Diff-DirMax-Dir
Ent-DirDiff-S
enMax-Sen
Ent-Sen
Supplementary Figure 6. Fitting confidence reports only
timeWhich group does the black dot belong to?
How confident are you in your decision?
somewhatlow
verylow
somewhathigh
veryhigh
AExperiment 1 and 3 Experiment 2B C
Supplementary Figure 1.
Supplementary Figure 2.
Target location (x)
Targ
et lo
catio
n (y
)
Fig. 5 Experiment 2.
A
Diffe
renc
eM
axEn
tropy
Target location (deg)
Targ
et lo
catio
n (d
eg)
1 2 3 4confidence
B
Fig. 5 Experiment 2.
A B Experiment 2Experiment 1
0
500
1000
1500
2000
2500
3000
AIC
relat
ive to
th
e Di
ffere
nce
mod
el
0
500
1000
1500
2000
2500
3000
AIC
relat
ive to
th
e Di
ffere
nce
mod
el
Diff
Max
Ent
Model
Diff
Max
Ent
Model
Fig. 4. Experiment 1. Model comparisons using ΔAIC:AIC of each model compared with the Difference model
1. Feedback2. Sampling from the true stimulus distribution
timeWhich group does the black dot belong to?
How confident are you in your decision?
somewhatlow
verylow
somewhathigh
veryhigh
AExperiment 1 and 3 Experiment 2B C
Experiment 3
1
2
3
4
1
2
3
4
1
2
3
4
-5 0 5Target location (deg)
Mea
n co
nfide
nce
repo
rt
-5 0 5 -5 0 5 -5 0 5
Diffe
renc
eM
axEn
tropy
A
B
C
DataModel fit
ModelDiff-S
en-Dir
Max-Sen-Dir
Ent-Sen-Dir
0
500
1000
1500
2000
2500
AIC
relat
ive to
th
e Di
ffere
nce-
Sen-
Dir m
odel
Supplementary Figure 11.
If you are interested in:-Model recovery analysis (Supplementary Figure 3)-Model comparisons using BIC index (Supplementary Figure 5)-Fitting confidence reports alone (Supplementary Figure 6 and 7)-Fitting category decisions alone (Supplementary Figure 9)-21 additional models (various implementations of variabilities; heuristic models; Supplementary Figure 10)-Ratio model (Supplementary Figure 12)
Difference model outperforms Max and Entropy modelPosteriors of unchosen options matter
If you are interested in:-Model recovery analysis (Supplementary Figure 3)-Model comparisons using BIC index (Supplementary Figure 5)-Fitting confidence reports alone (Supplementary Figure 6 and 7)-Fitting category decisions alone (Supplementary Figure 9)-21 additional models (various implementations of variabilities; heuristic models; Supplementary Figure 10)-Ratio model (Supplementary Figure 12)
Difference model outperforms Max and Entropy modelPosteriors of unchosen options matter
THANK YOU! Time to discuss