Dynamic estimation of prior probabilities in an orientation-discrimination task
1 Department
Elyse H. Norton 1 & Michael S. Landy 1,2
of Psychology & 2 Center for Neural Science, New York University
Introduction
Signal detection theory: Unequal prior probabilities
Experimental tasks
shift in
Overt-criterion task
decision criterion1
Start
Previous studies: Vary category probability between blocks,
Rotate
Stimulus & Feedback
Points: 1
Points: 1 Points: 0 Points: 0
“Rotate the line to
indicate your
criterion”
300 ms
300 ms
RLP: Reinforcement-learning on probability
Estimates probability as an exponentially weighted average of previously
seen categories
Estimates probability by a proportion of the error when receiving negative
feedback
EMA + bias
Weighted average of EMA estimate and a prior of 0.5
Omniscient criterion
n
rio
RLP + bias
Weighted average of RLP estimate and a prior of 0.5
Updated every 80-120 trials
Ranged from [.2, .8]
.5
P(B) = 1-P(A)
.25
.25
0
0
200
400
600
800
0
200
400
600
800
cr
nt
ie
−45
-30
−30
-70
−70
Slope
sc
ni
s
pen
om
er-c
Und
Fixed criterion
O
m
-45
Under-compensating
.4.4
Overt
−90
200
400
600
800
200
400
600
800
-150
-150
−150
−150
−90
-90
−30
30
-30
30
00
30
2
3
2 3
4
80
5
4 5
Omniscient criterion (deg)
*Estimated over a 25-trial moving
30
1
1
6
7
8
6 7
9
10
11
Observer
11
80
.8.8
window
20
20
-10
−10
Overt-criterion task
Covert-criterion task
.6.6
.4.4
6
2.5
5
2
600
Fixed criterion
12
8 9 10 11 Avg
DICFC-DICAlternative
S6
−120
0
x102
x102
Omniscient criterion
-120
Covert
.2.2
−95
250
500
4
+/- 2 SE
200
400
1.5
150
3
300
1
2
100
200
.5
1
100
0
-1
−100
95% C.I.
50
0
0
0
-.5
−50
1
2
3
4
5
RLC EMA RLP EMA+ RLP+
bias bias
1
2
RLC EMA
3
4
5
RLP EMA+ RLP+
bias bias
-40
−40
.2.2
-50
S10
−50
0
200
400
600
0
200
400
600
-100
800
-100
800
Trial
Trial
−100
−100
−40
20
80
-40
20
80
Conclusions
00
1
1
2
3
2 3
4
5
4 5
6
7
8
6 7
9
10
11
12
8 9 10 11 Avg
A1: Observers dynamically update criterion as
probability changes, but under-compensate (i.e., they
exhibit conservatism).
Observer
Omniscient criterion (deg)
History of previously seen categories:
Omniscient criterion:
Model fits
.6.6
-90
-95
Estimated* vs. omniscient criterion (covert):
Estimated criterion (deg)
.75
.8.8
g
atin
Better fit
1
.75
P(A)
30
Trial
Random stepwise changes of P(A):
0
Points: 1 Points: 0
2
11
Slope
60
30
−20
Orientation
.5
1
ite
30
Observed criterion (deg)
Probability density
μA ~ [-90, 90]
μB = μA + Δθ
σA = σB
c - neutral criterion
-20
0
1
Updates criterion by a proportion of the error when receiving negative
feedback
EMA: Exponentially weighted moving-average
300 ms
“To which category Time
does the ellipse
belong?
Observed vs. omniscient criterion (overt):
Category
B
0
Feedback
Results
Δθ
c
−30
RLC: Reinforcement-learning on criterion
Stimulus
Response
500 ms
Time
Stimuli
Categories of ellipses:
Category
A
Fixes neutral criterion
+
Set
Q1: Can observers track sudden changes in
category probability?
Q2: How is prior probability estimated?
FC: Fixed Criterion
Fixation
assume a fixed criterion, and explicitly state probability (e.g. Refs 1-2)
−60
Models
Covert-criterion task
5
-40
.4
5
0.4
−65
μA
-90
0
Knows the category
distributions, prior probability,
and sensory uncertainty on
each trial
4
4
3
3
2
2
1
1
0
0
-1
Linear regression
Predicting the current
criterion setting from the
previous 14 categories
−1
−90
0
200
200
400
400
Trial
600
600
800
800
Regression weight
μB
-65
“Omniscient observer”
Regression weight
Orientation (deg)
−40
A2: Prior probability is estimated as an exponentially
Logistic regression weighted average of previously experienced categories
with bias towards a prior of 0.5.
Predicting the current
.3
0.3
.2
0.2
decision from the
previous 14 categories
.1
0.1
0
-.1
1Green,
−0.1
0
5
0
5
10
10
Trial lag
15
15
References / Acknowledgements
0
0
5
0
5
Trial lag
10
15
10
15
D. & Swets, J. (1966). Signal detection theory and psychophysics. New York: Wiley.
2Ackermann,
J. F. & Landy, M. S. (2015). Suboptimal decision criteria are predicted by subjectively weighted
probabilities and rewards. Attention, Perception & Psychophysics, 77, 638-658.
NIH EY08266
[email protected]