Unformatted text preview: Banburismus and the Brain: Decoding
Review the Relationship between Sensory
Stimuli, Decisions, and Reward
Joshua I. Gold and Michael N. Shadlen
Presenter: He Huang
04/22/2010 Outline
•
•
• Turing’s theoretical framework and “Banburismus”
Application to 2AFC decisions on RT perceptual tasks
Experimental evidence Background: Banburismus at Bletchley Park • Problem: Decrypt the ‘Enigma machine’
Solution: Compare if two messages received are from
•
the same encoding scheme. Background: Banburismus at Bletchley Park
Formalize the problem • Hypothesis testing (given the evidence from the
messages): • h0: the encoding scheme for the two messages was
different • h1: the encoding system was the same Background: Banburismus at Bletchley Park
Formalize the problem
h1: the encoding systems that generates (m1, m2) is different;
h0: the encoding systems that generates (m1,m2) is the same; •
• Pr(mh0): the likelihood of getting a match given h0. Pr(mh1): the likelihood of getting a match given h1; What is the decision rule?
When is the stopping time? of letters they encoded also negative number describing the
weight of ,evidence crossed eithereach other. decision
matched barrier, the There one pair of c
tions w
te givendecide h1 thePr(mh1a was greater than the
)
h1, denoted other ),
on
he fore, themade. evidence formulation for statistical When the
was expected “match rate” fromdecide h . applicaA similar needed to the encoded mesthe war
nt
Second, it pre
elihoodweight of a match given h0, denoted Pr(mh0).
of getting
0
e sages woulddeveloped crossed eitherby Wald shortly after
tions was evidence independently barrier, the decision
be the same as that evidence unencodedPark As Tu
from the proBackground: Banburismus at Bletchleyfollows. By Bay
of
weight of
ccordingly, Turing defined the weight of
s messages, (Wald, is equal to 1/13 in ordinary German
the war which 1947).
advanta
e
be related to s
ded byFormalizeA similaraformulation for statistical applicaawas made.similarly, nonmatch) in favor of h1
match (or, the problem
ed text. In other words, the likelihood of getting a match
As Turing was aware, this formulation had two distinct
instruct
ethe logarithm of the independently by Wald shortly after
ver h0 astions was developed ratio of these likelihoods:
an (m )Evidence 1favoring h1 Pr(mahdefinite stoppingthan the
advantages. First, it had  1), was greater time that Pr(on m)
h1 one
given h , denoted / evidence favoring h0
sthe war (Wald, 1947).
log
ch likelihood of getting a match given hwhen to stop working Pr(Second
instructed the codebreaking team , denoted Pr(mh ).
h0m)
0
0
d
As Turing was aware, thisPr(mh1)
formulation had two distinct
he Accordingly, Turing defined the weight of evidence proon one of evidence
1)
Weight pair of ciphers and turn their attention (to another.
log
follows
n
advantages. First, it had aPr(mh0) stopping time that
definite
Second, it predicted the accuracy of the decision,hwhere be h0m
as
Pr( rela
0).
h vided by a match (or, similarly, athe weight ofto stop working
instructed the codebreaking nonmatch) in favor ofcan
team when evidence 1
ofollows. By Bayes’ theorem,
probabilities a
over h0 of the logarithm of the ratio of attention to another.
Bayes probabilities, and probabilities:
ea logarithmasrule:to several this quantity isthese likelihoods:
on one pair Pr(hm) =otherturn their measured
of ciphers Pr(mh)Pr(h)/Pr(m)
s1
be related
h
pothesis after
Pr(
log
).
Second, it predicted of accuracy of the decision, as ) and Pr(h
0s:
units that depend on the basethethe logarithm (Good,
Pr(h0
Pr(
Pr(m of
Pr( 1m)
Pr(hof
o In Banburismus, logarithms 1) base weighthused
follows.hBy Bayes’ theorem, the 10 were 1) evidence describe the p
can
979).
Weight oflog
evidence lweight of evidence, (1)
og
(2)
log
Pr(m actuPr(h0m)
be related to (the Banburismus processh
other
h1 were called “bans” severalPr(h0) probabilities:0)
nd
where
evidence is sa
(1)
s: computed weights of evidence in units of 1/10th of
lly
probab
prior probabili
wherer(Pr(mm) and Pr(h11)m) are called the posterior
P h1 h0of probabilities, this quantity is measured
)
Pr(h
As log
logarithm
ban—a adeciban—that the codebreakers considered
log
weight of evidence, hy(2) pothesi
assumption m
probabilities) and describe) the probability of each
Pr(h0smalleston the baseevidence that is (Good,
m
Pr(h0of of the logarithm
be in units that depend the evidence has been sampled, and
Pr(h a
ed “about the after all weight
evidence is0)eq
pothesis
1)
1979). In Banburismus, Pr(h m) of called were used
irectly perceptible htom) ) andlogarithmsare base 10 the posterior
human called the prior probabilities and
intuition”; Good, 1979).
describ
d,
posterior prob
where and Pr(h1 are
Pr(h0) Pr( 0
1 robabilities and describe the probability of each hyh
instructed the codebreaking team when to stop working
othesison oneall the evidence has been sampled, and
after pair of ciphers and turn their attention to another.
e
r(
Background: Banburismus of the and
). h0) and Pr(h1) it predictedthe prior probabilitiesdecision, as
Second, are called the accuracy at Bletchley Park
escribefollows. By Bayes’ problem weight of evidence can
the probability of theorem, the
each hypothesis before any
Formalize the
vidence isrelated to severalexample, there is an equal
be sampled. If, for other probabilities:
1
rior probability of either hypothesis (which was the
:
Pr(h1 in
Pr(h1)
ssumption madem) Banburismus), then theof evidence, (2)
weight of
log
weight
log
Pr(h0 to
Pr(h0)
vidence is equalm) the logarithm of the ratio of the
)
osterior probabilities. In this case, for a given weight
ifwhere knowledge is equal 1for the two hypothesis
prior Pr(h0m) and Pr(h m) are called the posterior
f evidence (e.g., the stopping point in Banburismus):
probabilities and describe the probability of each hypothesis m) all the evidence has been sampled, and
after
Pr(h1
eight of evidence B
(3)
log h0) and Pr(h1)ware called the prior probabilities and
,
Pr( Pr(h m)
0
describe the probability of each hypothesis before any
here Bevidence is sampled. If, for example, there is an equal
is a constant that represents the barrier height
h1 is1 true two of How much the hypotheses
f favor prior .probabilitymutually exclusive evidence favors theover h0
either hypothesis (which was h1
of h For
1
Pr( 1m) in Banburismus), then 1
r(h0m)assumptionhmadeor, equivalently, Pr(h0) the weight of
s(h )), and assuming the to the logarithm of the ratio of the
evidence is equal weight of evidence is exr1 a constant that represents the barrier height
f h1. For two mutually exclusive hypotheses
1 Br(h1m) or, equivalently, Pr(h0)
P ackground: Banburismus1at Bletchley
d assuming the weight problemdecision rule
of evidence is exFormalize the
bans, this rearranges to:
Pr(h1m) 1 1
.
B
10 Park P(h1m)
(4) indicates that the posterior probability of h1
B
nly on the value of the barrier, B, and not on
h1 is true
How much the evidence favors h1 over h0
lar samples of evidence, m, encountered. In
Decision rule: setting B to achieve an expected level of performance (confidence that h1
s, as ilong as the weight of evidence reaches
s true).
e probability that h1 is correct is a fixed value.
SpeedAccuracy trade off: more evidence needs to be accumulated in order to increase
Banburismus, setting thethe correctof the barrier
height answer.
the the probability of getting
mined the weight of evidence to accumulate
mmitting to a decision was equivalent to setpected level of performance. For example, Implementation: Banburismus
Performed by Neurons (in 2 AFC choice) • How does the activity of sensory neurons compute
the “weight of evidence”? • How does the brain make use of the sensory
evidence? • What is the decision rule based on this evidence? 992;(and assume opponency, giving an ,analogous expres ,
Salzman
Pr(x yh1)
2
2
in these areas h0):
2
√1
sion under
that the brain Banburismus Performed by Neurons
Q
where
e2
is was most
Sensory xSignalsTuring’s Weight of Evidence
,
(5)
Pr( ,yh1)
2
22
2
Neuron
(y
2 (x
1 2 (x √1 1)
0)
1)(y
302
Q
2
2
1
Using the difference (xy) in spike rates from two
where • (5) ) 0 , neurons (one 2 the common variance, other favors Down
favors Up motion, the and are the means of
2
is
cisions about (x
2
1
)
(y
)
2 (x
)(y
)0
1
1
0
1
0
motion) x and y, respectively, under h1, and , is the covariance
Qlike Tu 2
tities
2 1
between x and y. Solving for the weight of evidence (in
ically related
h1: Up motion ( x is larger)
2
is the common variance, 1 and 0 are the versus h0 yields:
units of natural bans) for h1 means of
1966). Briefly,
x and rate
y, Down motion (y h1, and
s case, ah2: respectively, under is larger) is the covariance
Pr(x,yh1(in
)
between x
Solving for the weight log
thought of as and y. weight of evidence of evidence
units of natural bans) for h1 versus h0 yields:Pr(x,yh0)
us conditions
value but can
( 1 constant 1 Relationship between Neural Responses an
Figure 1. Theoretical
Pr(x,yh1)
0)
·
· hypothetical distributions of respon
y) (6)
The panels on the left depict (x
nt orweight of evidence log
trialto2
example, the upper curve could represent the responses of a dire
(1
)
Pr(x,yh0)
then represent the responses of the opposing “antineuron” that pr
on numerous
responses are normally distributed with mean values of
and
the exponential and Poisson, lead to similar results (Gold and S
)
( 1numerous 1
ese probabiliNote that
factors could cause responses plotted as a function of their diff
nonzero values
0
computed from these
·
· (x y) (6)
2
easurements
of , including intrinsic factors like attention and arousal
(1
) •
• 0 1 measured
pothesis after all the by the neuron and antineuron up to that
tions of the total number of spikes generated evidence has been sampled, and
hm (Good, barrier at 0B, andprocess are called the decision is rendered for h1.
e reaches theReview Pr(h ) the Pr(h1) is stopped and a prior probabilities and
were used 303 reaches B (not shown), a decision for h0hypothesiserror.
Banburismus Performed by before any
ight of evidence
describe the probability of each is issued inNeurons
cess actuevidence is sampled. If, for example, there is an equal
Sensory SignalsTuring’s Weight of Evidence
th
f 1/10 of
prior probability of either hypothesis (which was the
onsidered
eant that
1968; Link and Heath, 1975; Luce, 1986;the weight of
assumption made in Banburismus), then Ratcliff and
ce that is
, implied
Rouder, 1998;equal to the logarithm of the ratio of the
evidence is Stone, 1960; Usher and McClelland, 2001;
od, Howposterior probabilities. In this case, for a given weight
er. 1979).
Vickers, 1979).
df the co“natural
of evidence (e.g., is analogous to in Banburismus):
This problem the stopping point onedimensional
ponses and Turing’s Weight of Evidence
ted quanlar responses from a pair of sensory neurons when h a pairtrue, barriers (Link, 1992; Ratcliff
Brownian motion to or h is of as indicated. For
of tradePr(h1mmotion. The lower curve would
)
obabilities
es of a directionselective neuron that prefers upward The weight of evidence
articular
and Rouder, 1998).
means of the normal B
distribu(3)
log
ron” that prefers downward motion. Here wePr(h the simplifying assumption that the
make m)
surprisal”
r can be equal variances ( represent the drift rates, i (where the subscript i
tions ). Note that other0 forms of the distribution, including
f
and
and
bans but
old and Shadlen, 2001). The panelthe particular Turing’s weight of evidenceThe psychometon the
f reward
reflects B is a right shows motion strength).the barrier height
where
constant that represents
of their difference; note the linear relationship.
m of base
ric function describing the probability of correctly reachexclusive hypotheses
in favor of h1. For two mutually accumulate/integrate over
Decision Model
tiontime Figure 2. ing eitherh the is“up” r(h difference between equivalently, given) antineuron is:
or “down” barrier on a Pr(h trial
The weight (Pr(h favor of
evidence in m) over h
the accumulated m) or, responses of a neuron that prefers h and an
1depicted P computed under thetcondition) thatyh( is)true. The thin, wavy line depicts a simulated
1
0
0
ntlities has that prefers weight of7).evidence atare
motion
(
d (7)
total h (see Equation The curves
time1t k
ihich the trajectory that represents how the weight of evidence might grow on a single trial as a function of time. The dashed line depicts the expectation
0
Pr(h1)), and assuming the weight of evidence is exn motion
(mean value) of this trajectory at each time point. Note that changing the constant of proportionality used to relate the accumulated difference
100 years to the weight of evidence (see text for details) simply scales the ordinate. The two insets illustrate the correspondence between the weight
to make ofwhere kthe underlying rule:antineuron this ifatk does 1 viewed as hypothetical (normal, equivariate) probability(8)
stimulus
is a constant. Note that two time points, equal the
not
Decision in bans, responsesrearranges to: by the neuron and antineuron up to that
,
pi
evidencepressed neuron and
and
2
hesis x
i
value of that density functions.that h proportionality in Equation 6, thenBgenerated
constantThese is true. If the weightthe distributions of thethe 1number Be process isvalueand a decision is rendered for h .
of functions describe of evidence reaches total at of, spikes the stopped
sensory time point, given
barrier
the
llthat “two This is the is computed h is true. If the weight of evidence reaches to (not shown), a decision for h is issued in error.
value of
that expected outcome when is merely proportional B the weight
1
rtex. We of evidence. Regardless of the value of k, however, the
(Recall) Equation 4, 1is the logistic function (here using
between
.
(4)
Pr(h m)
t, that is
neither
which,speed. Higher barriers meant that of1968; 1 and10 B1975; Luce, 1986; Ratcliff and
like
le (x
tween accuracy and
Link
ence
algorithm for updating the weight
evidenceHeath,the
is
ght, when more evidence was e ). Unlike in turn, implied Rouder, 1998; Stone, 1960; Usher and McClelland, 2001;
base accumulated, which, in Equation 4, however, the probability
0 0 1 2 1 1 0 0 1 1 1 1 1 0 time measured
pothesis after all the by the neuron and antineuron up to that
tions of the total number of spikes generated evidence has been sampled, and
hm (Good, barrier at 0B, andprocess are called the decision is rendered for h1.
e reaches theReview Pr(h ) the Pr(h1) is stopped and a prior probabilities and
were used 303 reaches B (not shown), a decision for h0hypothesiserror.
Banburismus Performed by before any
ight of evidence
describe the probability of each is issued inNeurons
cess actuevidence is sampled. If, for example, there is an equal
Sensory SignalsTuring’s Weight of Evidence
th
f 1/10 of
prior probability of either hypothesis (which was the
onsidered
eant that
1968; Link and Heath, 1975; Luce, 1986;the weight of
assumption made in Banburismus), then Ratcliff and
ce that is
, implied
Rouder, 1998;equal to the logarithm of the ratio of the
evidence is Stone, 1960; Usher and McClelland, 2001;
od, Howposterior probabilities. In this case, for a given weight
er. 1979).
Vickers, 1979).
df the co“natural
of evidence (e.g., is analogous to in Banburismus):
This problem the stopping point onedimensional
ted quanlar tradeBrownian motion to a pair of barriers (Link, 1992; Ratcliff
Pr(h1m)
obabilities
articular
and Rouder, 1998). The weight of evidence B
means of the normal distribu(3)
log
Pr(h0mdrift rates, (where the subscript i
)
surprisal”
r can be
tions represent the
i
bans but
f reward
reflects B is particular motion strength).the barrier height
the a constant that represents The psychometwhere
m of base
ric function describing the probability of correctly reachin favor of h1. For two mutually exclusive hypotheses
Decision Model
tiontime Figure 2. ing eitherh the is“up” r(h difference between equivalently, given) antineuron is:
or “down” barrier on a Pr(h trial
The weight (Pr(h favor of
evidence in m) over h
the accumulated m) or, responses of a neuron that prefers h and an
1
1
0 7). The curves depicted P computed under the condition that h is true. The thin, wavy line depicts a simulated
1
0
ihich the trajectory that represents how the weight of evidence are grow on a single trial as a function of time. The dashed line depicts the expectation
lities has that prefers h (see Equation
and assuming the of proportionality of relate the accumulated is exPr(h1)), at each time point. Notemightchanging the constantweight used to evidence difference
(mean value) of this trajectory
100 years to the weight of evidence (see text for details) simply that the ordinate. The two insets illustrate the correspondence between the weight
1
to make of evidencepressedrule:antineuronscales i rearranges as hypothetical (normal, equivariate) probability(8)
Decision neuron and the distributions of theattotal number of spikesBgenerated by the neuron and antineuron up to that
,
p two time points, viewed to:
and the underlying in bans, this
responses
2i
hesis that density functions. These functions describe
1 at Bethe process is stopped and a decision is rendered for h .
sensory time point, given that h is true. If the weight of evidence reaches the barrier ,
that “two This is the expected outcome when h is true. If the weight of evidence reaches B (not shown), a decision for h is issued in error.
1
rtex. We
(Recall) Equation 4, 1is the logistic function (here using
.
(4)
Pr(h m)
t, that is
neither
which,speed. Higher barriers meant that 1968; 1 and10 B1975; Luce, 1986; Ratcliff and
like
le
tween accuracy and
Link
Heath,
ght, when more evidence was e ). Unlike in turn, implied Rouder, 1998; Stone, 1960; Usher and McClelland, 2001;
base accumulated, which, in Equation 4, however, the probability
1 0 0 1 1 1 1 1 0 Pr(x,yh1) 994). Thus, neural activity in these areas
Banburismus Performed by Neurons
ncoded, noisy evidence that the brain
Neuron
where
304
SpeedAccuracy Tradeoff
o decide which hypothesis was most
Neuron
304 1 (x ) 1 2 2 (y √1 2 ) 0 2 2 P(Correct)
QPredicted Effect of Barrier Height
Accuracy
y Signals to Turing’s
Figure 3.
2
2
1
on Speed, Accuracy, and Reward Rate in a
Figure 3. Predicted Effect of
DirectionDiscrimination Task Barrier Height
nce
on Speed, Accuracy, and Reward Rate in
The calculations assume that an experiment a
2
DirectionDiscrimination (see
uses six levels ofthe Task text for dedifficulty
is assume common variance, 1 and
consider categorical decisions about
The (A) Accuracy.
that an experiment
tails). calculations The probability of a coruses six levels of difficulty all text for derect choice averaged across(see six motion
x Accuracy. The function of barrier
gnals by formulating quantities like Tutails). (A)and y, respectively, under h1, and
strengths is plotted as a probability of a correct choice averaged ensures six deciheight. A higher barrieracross all that motion
strengthsbased on aas a xweight of evi .
vidence that are monotonically related
sions between function of barrier Solving for the weig
are is plotted larger and y
height. A higher barrier ensures on decidence and thus are more accurate thataversions are based of larger weight of evion a natural bans) for h
age. The dashed vertical line is the barrier
units rate maximum on averratio (Green and Swets, 1966). Briefly,
1 versus
Reward are more accurate rate of
= Accuracy/ Response Timeh
dence and thus
associated
Reward rate
Response Time height The dashedwith the line is the barrier
age.
reward. (B) Reward vertical reward rate derate. The
a sensory neuron—in this case, a rate
height associated with the maximum rate of
pends on the decision time plus fixed times
Pr(x,yh1)
reward. (B) Reward plus a “time out” for dein and between trials rate. The reward rate an
kes per second—can be thought of as
pends weight time occurs times
error. on the
maximum reward rate evidence
log
B=A b...
View
Full Document
 Spring '08
 LEWIS
 Banburismus

Click to edit the document details