Bayesian Perception
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![Page 1: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/1.jpg)
Bayesian Perception
![Page 2: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/2.jpg)
General Idea
Ernst and Banks, Nature, 2002
![Page 3: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/3.jpg)
General Idea
• Bayesian formulation:
, || ,
,
| |
,
| |
P t v w P wP w t v
P t v
P t w P v w P w
P t v
P t w P v w P w
ConditionalIndependence assumption
ˆ arg max | ,w
w P w t v
![Page 4: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/4.jpg)
General Idea
Ernst and Banks, Nature, 2002
w
v=w+n
t=w+n
+
+
noise
noise
w?
Generative model: , |P t v w
![Page 5: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/5.jpg)
General Idea
Width
Prob
abil
ity
VisualP(v|w)
TouchP(t|w)
BimodalP(w|t,v)= P(v|w) P(t|w)
ˆ arg max | ,w
w P w t v
![Page 6: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/6.jpg)
General Idea
2
2
2
2
2 2
2 2
2 22 2
2 2
2 2 2 2 2
2 2
2 22
2 2
2 2 2 2
2 2
2 2
| exp2
log |2
log | |2 2
2
2
2
2
2 /
v
v
v t
t v
v t
t v t v
v t
t v
t v
v t t v
t v
t v
v wP v w
v wP v w
v w t wP v w P t w
v w t w
w v t w C
v tw w C
v tw
2
2 2 2 22 /v t t v
C
![Page 7: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/7.jpg)
General Idea
Mean and variance
2 2
2 2 2 2t v
t v t v
w v t
22 2
2 2
2 2 2 2log | ,
2 /
t v
t v
v t t v
v tw
P w v t
![Page 8: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/8.jpg)
General Idea
Width
Prob
abil
ity
VisualP(v|w)
TouchP(t|w)
tv
2 2
2 2 2 2t v
t v t v
w v t
![Page 9: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/9.jpg)
General Idea
Mean and variance
2 2
2 2 2 2t v
t v t v
w v t
22 2
2 2
2 2 2 2log | ,
2 /
t v
t v
v t t v
v tw
P w v t
2 2
2
2 2v t
w
t v
![Page 10: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/10.jpg)
Optimal Variance
Variance
2 2 2
1 1 1
w v t
w v tI I I
22 2
2 2
2 2 2 2log | ,
2 /
t v
t v
v t t v
v tw
P w v t
Fisher information sums for independent signals
![Page 11: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/11.jpg)
General Idea
0 67 133 2000
0.05
0.1
0.15
0.2
Th
resh
old
(S
TD
)
Visual noise level (%)
Measured bimodal STD
Predicted by the Bayesian model
Unimodal visual STD
Unimodal Tactile STD
Ernst and Banks, Nature, 2002 Note: unimodal estimates may not be optimal but the multimodal estimate is optimal
![Page 12: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/12.jpg)
Adaptive Cue Integration
• Note: the reliability of the cue change on every trial
• This implies that the weights of the linear combination have to be changed on every trial!
• Or do they?
2 2
2 2 2 2t v
t v t v
w v t
![Page 13: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/13.jpg)
General Idea
• Perception is a statistical inference
• The brain stores knowledge about P(I,V) where I is the set of natural images, and V are the perceptual variables (color, motion, object identity)
• Given an image, the brain computes P(V|I)
, ||
P P PP
P P
I V I V VV I
I I
![Page 14: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/14.jpg)
General Idea
• Decisions are made by collapsing the distribution onto a single value:
• or
ˆ |P dV V I V V
ˆ arg max |PV
V V I
![Page 15: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/15.jpg)
Key Ideas
• The nervous systems represents probability distributions. i.e., it represents the uncertainty inherent to all stimuli.
• The nervous system stores generative models, or forward models, of the world (P(I|V)), and prior knowlege about the state of the world (P(V))
• Biological neural networks can perform complex statistical inferences.
![Page 16: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/16.jpg)
Motion Perception
![Page 17: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/17.jpg)
The Aperture Problem
![Page 18: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/18.jpg)
The Aperture Problem
![Page 19: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/19.jpg)
The Aperture Problem
![Page 20: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/20.jpg)
The Aperture Problem
Horizontal velocity (deg/s)V
erti
cal v
eloc
ity
(deg
/s)
![Page 21: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/21.jpg)
The Aperture Problem
Horizontal velocity (deg/s)V
erti
cal v
eloc
ity
(deg
/s)
![Page 22: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/22.jpg)
The Aperture Problem
![Page 23: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/23.jpg)
The Aperture Problem
Horizontal velocity (deg/s)V
erti
cal v
eloc
ity
(deg
/s)
![Page 24: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/24.jpg)
The Aperture Problem
Horizontal velocity (deg/s)V
erti
cal v
eloc
ity
(deg
/s)
![Page 25: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/25.jpg)
The Aperture Problem
Horizontal velocity (deg/s)V
erti
cal v
eloc
ity
(deg
/s)
![Page 26: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/26.jpg)
Standard Models of Motion Perception
• IOC: interception of constraints
• VA: Vector average
• Feature tracking
![Page 27: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/27.jpg)
Standard Models of Motion Perception
Horizontal velocity (deg/s)V
erti
cal v
eloc
ity
(deg
/s)
IOCVA
![Page 28: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/28.jpg)
Standard Models of Motion Perception
Horizontal velocity (deg/s)V
erti
cal v
eloc
ity
(deg
/s)
IOCVA
![Page 29: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/29.jpg)
Standard Models of Motion Perception
Horizontal velocity (deg/s)V
erti
cal v
eloc
ity
(deg
/s)
VA
IOC
![Page 30: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/30.jpg)
Standard Models of Motion Perception
Horizontal velocity (deg/s)V
erti
cal v
eloc
ity
(deg
/s)
IOCVA
![Page 31: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/31.jpg)
Standard Models of Motion Perception
• Problem: perceived motion is close to either IOC or VA depending on stimulus duration, eccentricity, contrast and other factors.
![Page 32: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/32.jpg)
Standard Models of Motion Perception
• Example: Rhombus
Horizontal velocity (deg/s)
Ver
tica
l vel
ocit
y (d
eg/s
) IOCVA
Horizontal velocity (deg/s)
Ver
tica
l vel
ocit
y (d
eg/s
) IOCVA
Percept: VAPercept: IOC
![Page 33: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/33.jpg)
Moving Rhombus
![Page 34: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/34.jpg)
Bayesian Model of Motion Perception
• Perceived motion correspond to the MAP estimate
* arg max |
|| |
, |i ii
P I
P I PP I P I P
P I
P P I x y
vv v
v vv v v
v v
![Page 35: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/35.jpg)
Prior
• Human observers favor slow motions
-50 0 50
-50
0
50
Horizontal Velocity
Ver
tica
l Ve
loci
ty
2 2exp / 2 pP v v
![Page 36: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/36.jpg)
Likelihood
• Weiss and Adelson
-50 0 50
-50
0
50
Horizontal Velocity
Ver
tica
l Ve
loci
ty
, , |i i iP I x y t v
![Page 37: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/37.jpg)
Likelihood
, , , ,
, , , ,
x y
x y
I x y t I x t y t t t
I x y t I x t y t t t
v v
v v
, , , ,
, , , ,
i x i y i i x x y y t
i i i x i y x x y y t
x x y y t
I x t y t t t I x y t I t I t I t
I x y t I x t y t t t I t I t I t
I I I t
v v v v
v v v v
v v
2
2
, , , ,, , | exp
2
i i x y
i i
I x y t I x t y t t tP I x y t
v vv
2
2, , | exp
2x x y y t
i i
I I IP I x y t
v vv
![Page 38: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/38.jpg)
Likelihood
2
2
2
2,
, , | , , |
1exp
2
1exp ,
2
i ii
x x y y ti
x x y y t
x y
P I x y t P I x y t
I I I
w x y I I I dxdy
v v
v v
v v
Binary maskPresumably, this is set by segmentation cues
![Page 39: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/39.jpg)
Posterior
2 2 22 2
,
log | , , log , , | log
1 1
2 2x x y y t x yx y p
P I x y t P I x y t P
I I I
v v v
v v v v
![Page 40: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/40.jpg)
Bayesian Model of Motion Perception
• Perceived motion corresponds to the MAP estimate
*
22
2
*
22
2
arg max | , | , |ii
i
x x yp x t
y tx y y
p
P I P I P P I x y
I I II I
I II I I
vv v v v v
v
Only one free parameter
![Page 41: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/41.jpg)
Likelihood
2 2 22 2
, ,
, ,
2 2
,,
22
2, ,
2 22
2, ,
1 1( )
2 2
2 2( ) 1 1
2 2 22
1
x x y y t x yx y x yp
x x y y t x xx y x y
p yx x y y t yx yx y
x y x tx y x yp
y x yx y x y p
L I I I
I I I IL
I I I I
I I I I I
I I I
v v v v v
v v vv
v vv v
v,
,
0
xx y
t yx y
I I
![Page 42: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/42.jpg)
Motion through an Aperture
• Humans perceive the slowest motion.
• More generally: we tend to perceive the most likely interpretation of an image
![Page 43: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/43.jpg)
-50 0 50
-50
0
50
Horizontal Velocity
Ver
tica
l Ve
loci
ty
-50 0 50
-50
0
50
Horizontal Velocity
Ver
tica
l Ve
loci
ty
-50 0 50
-50
0
50
Horizontal Velocity
Ver
tica
l Ve
loci
ty
Motion through an Aperture
ML
MAP
Prior Posterior
Likelihood
![Page 44: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/44.jpg)
Motion and Constrast
• Humans tend to underestimate velocity in low contrast situations
![Page 45: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/45.jpg)
-50 0 50
-50
0
50
Horizontal Velocity
Ver
tica
l Ve
loci
ty
-50 0 50
-50
0
50
Horizontal Velocity
Ver
tica
l Ve
loci
ty
-50 0 50
-50
0
50
Horizontal Velocity
Ver
tica
l Ve
loci
ty
Motion and Contrast
ML
MAP
Prior Posterior
HighContrast
Likelihood
![Page 46: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/46.jpg)
-50 0 50
-50
0
50
Horizontal Velocity
Ver
tica
l Ve
loci
ty
-50 0 50
-50
0
50
Horizontal Velocity
Ver
tica
l Ve
loci
ty
-50 0 50
-50
0
50
Horizontal Velocity
Ver
tica
l Ve
loci
ty
Motion and Contrast
ML
MAP
Prior Posterior
LowContrast
Likelihood
![Page 47: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/47.jpg)
Motion and Contrast
• Driving in the fog: in low contrast situations, the prior dominates
![Page 48: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/48.jpg)
-50 0 50
-50
0
50
Horizontal Velocity
Ver
tica
l Ve
loci
ty
-50 0 50
-50
0
50
Horizontal Velocity
Ver
tica
l Ve
loci
ty
-50 0 50
-50
0
50
Horizontal Velocity
Ver
tica
l Ve
loci
ty
-50 0 50
-50
0
50
Horizontal Velocity
Ver
tica
l Ve
loci
ty
Moving Rhombus
IOC
MAP
Prior Posterior
HighContrast
Likelihood
![Page 49: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/49.jpg)
Moving Rhombus
-50 0 50
-50
0
50
Horizontal Velocity
Ver
tica
l Ve
loci
ty
-50 0 50
-50
0
50
Horizontal Velocity
Ver
tica
l Ve
loci
ty
-50 0 50
-50
0
50
Horizontal Velocity
Ver
tica
l Ve
loci
ty
-50 0 50
-50
0
50
Horizontal Velocity
Ver
tica
l Ve
loci
ty
IOC
MAP
Prior Posterior
LowContrast
Likelihood
![Page 50: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/50.jpg)
Moving Rhombus
![Page 51: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/51.jpg)
Moving Rhombus
• Example: Rhombus
Horizontal velocity (deg/s)
Ver
tica
l vel
ocit
y (d
eg/s
) IOCVA
Horizontal velocity (deg/s)
Ver
tica
l vel
ocit
y (d
eg/s
) IOCVA
Percept: VAPercept: IOC
![Page 52: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/52.jpg)
Barberpole Illusion
![Page 53: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/53.jpg)
Plaid Motion: Type I and II
![Page 54: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/54.jpg)
Plaids and Contrast
Lower contrast
![Page 55: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/55.jpg)
Plaids and Time
• Viewing time reduces uncertainty
![Page 56: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/56.jpg)
Ellipses
• Fat vs narrow ellipses
![Page 57: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/57.jpg)
Ellipses
• Fat vs narrow ellipses
• All motions agree
![Page 58: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/58.jpg)
Ellipses
![Page 59: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/59.jpg)
Ellipses
• Adding unambiguous motion
![Page 60: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/60.jpg)
Ellipses
• Adding unambiguous motion
![Page 61: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/61.jpg)
Other Prior
• Prior on direction of lightning
![Page 62: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/62.jpg)
Generalization
• All computation are subject to uncertainty (ill-posed)
• This includes syntax processing, language acquisition… etc.
• Solution: compute with probability distributions
![Page 63: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/63.jpg)
Binary Decision Making
Shadlen et al.
![Page 64: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/64.jpg)
Race Model
• Standard theory: some signal is accumulated (or integrated) to a bound. Also known as race models.
• The signal to be integrated could be the response of sensory neurons.
![Page 65: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/65.jpg)
Bayesian Strategy
• The ‘diffusion to bound’ model of Shadlen et al.
![Page 66: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/66.jpg)
High motion strength
High m
otion stre
ngth
Low motion strength
Time
~1 secStimulus
onStimulus
off
Spikes/s
Time
~1 secStimulus
onStimulus
off
Spikes/s
Low motion strength
A Neural Integrator for Decisions?
MT: Sensory Evidence
Motion energy
“step”
LIP: Decision Formation
Accumulation of evidence
“ramp”
Threshold
![Page 67: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/67.jpg)
Diffusion to bound model
Positive bound
Negative bound
![Page 68: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/68.jpg)
Proposed by Wald, 1947 and Turing (WW II, classified); Stone, 1960; then Laming, Link, Ratcliff, Smith, . . .
Diffusion to bound model
Positive bound or Criterion to answer “1”
Negative bound or Criterion to answer “2”
![Page 69: Bayesian Perception](https://reader036.fdocuments.in/reader036/viewer/2022062422/568131a8550346895d9817b4/html5/thumbnails/69.jpg)
Momentary evidencee.g.,
∆Spike rate:MTRight– MTLeft
Accumulated evidencefor Rightward
andagainst Leftward
Criterion to answer “Right”
Criterion to answer “Left”
Diffusion to bound model
Shadlen & Gold (2004)Palmer et al (2005)
kC
C is motion strength (coherence)Seems arbitrary but why not?
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MT responses
60
40
20
0
Firi
ng r
ate
Direction (deg)
Height scales with coherence
MT MTRight Leftr r
Right Left
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Diffusion to bound model
• Performance reaction time trade-off
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Best fitting chronometric function“Diffusion to bound”
t(C) B
kCtanh(BkC) tnd
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Predicted psychometric function “Diffusion to bound”
P 1
1 e 2k C B
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Average LIP activity in RT motion task
Roitman & Shadlen, 2002 J. Neurosci.
choose Tin
choose Tout
Note the clear asymmetry
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Bayesian Strategy
• The Bayesian strategy in this case consists in computing the posterior distribution given all activity patterns from MT up to the current time,
MT:1MT
:1 MT:1
MT
MT MT MT1:1 1:1
MT MT1:1
MT MT1:1
||
|
| , |
| |
| |
t
t
t
t
t t t
t t
t t
p s p sp s
p
p s p s
p s p s p s
p s p s p s
p s p s
rr
r
r
r r r
r r
r r
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Bayesian Strategy
• Race models and Bayesian approach
MT MT MT:1 1:1
MT MT MT:1 1:1
MT
1
| | |
log | log | log |
log |
t t t
t t t
t
p s p s p s
p s p s p s
p s
r r r
r r r
r
Temporal sum
Unless is related to …
But not over rMT, or MT MTRight Leftr r
MTlog |p srMT MTRight Leftr r
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Bayesian Strategy
• Are neurons computing log likelihood?
• The difference of activity between two neurons with preferred directions 180 deg away is proportional to a log likelihood ratio.
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Bayesian Strategy
• Log likelihood ratio:
MT MT:1
MT MT1:1
| |log log
| |
tt
t
p s R p s R
p s L p s L
r r
r r
LR MT MTR, L,L r r
MT:1 MT MT
R, L,MT1:1
|log
|
tt
t
p s R
p s L
r
r rr
MT
LR
MT
|log
|
p s RL
p s L
r
r
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Bayesian Strategy
• Is the log likelihood ration proportional to ?
2MT MTR L R LMT MT
R L 2
2MT MTR L R LMT MT
R L 2
R R| exp
2
L L| exp
2
p R
p L
r rr r
r rr r
MT MTR, L, r r
Coherence level
MT MT2 2R L MT MT MT MT
R L R L R L R LMT MTR L
|log R R L L
|
p R
p L
r rr r r r
r r
MT MTR L R LR R r r
MT MTR LC r r
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Bayesian Strategy
• Note that if you know , you still don’t know the log likelihood ration unless you’re given the coherence level.
• Therefore, the animal can’t know its confidence level (the log likelihood ratio) unless it estimates C…
• Another important point: if we stop the race at a fixed level of we stop at different levels of log likelihood ratio depending on the coherence. This is why performance gets better when coherence increases, even though we always stop at the same activity threshold.
MT MTR, L, r r
MT MTR, L, r r
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Decision Making
• Does that mean the animal does not know how much to trust its own decision?
• Does that mean the brain does not encode uncertainty or probability distribution?
• Seems unlikely…
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• To be continued…