02 Signal Detection Theory.pdf

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J. Elder PSYC 6256 Principles of Neural Coding

2. SIGNAL DETECTION THEORY


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Probability & Bayesian Inference

J. ElderPSYC 6256 Principles of Neural Coding

2

Signal Detection Theory

Provides a method for characterizing humanperformance in detecting, discriminating and

estimating signals.

For noisy signals, provides a method for identifyingthe optimal detector (the ideal observer) and for

expressing human performance relative to this.

Origins in radar detection theory Developed through the 1950s and on by Peterson,

Birdsall, Fox, Tanner, Green & Swets


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Example 1

The observer sits in a dark room On every trial, a dim light will be flashed with 50%

probability.

The observer indicates whether she believes thelight was flashed or not.

This is a yes-no detection task.


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Noise

In this example, the information useful for the task is the lightenergy of the stimulus.

By the time the stimulus information is received by decisioncentres in the brain, it will be corrupted by many sources of

noise:

photon noise isomerization noise

neural noise

Many of these noise sources are Poisson in nature: thedispersion increases with the mean.


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Equal-Variance Gaussian Case

It is often possible to approximate this noise asGaussian-distributed, with the same variance for

both stimulus conditions.

Then the noise is independent of the signal state.


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Discriminability d

p x|S = sH( ) =

1

2exp

x H( )

2

22

p x|S = sL( ) =

1

2exp

x L( )

2

22

H

L

d' =

signal separation

signal dispersion=

H

L


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Criterion Threshold

The internal response is often approximated as a continuousvariable, called the decision variable.

But to yield an actual decision, this has to be converted to abinary variable (yes/no).

A reasonable way to do this is to define a criterion thresholdz:x z 'yes'

x< z 'no'

x

z

x


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Effect of Shifting the Criterion


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How did we calculate these numbers?

p x|S = sH( ) =

1

2exp

x H( )

2

22

p x|S = sL( ) =

1

2exp

x L( )

2

22

H

L

d' = z

FA z

HIT


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What is the right criterion to use?

Suppose the observer wants to maximize the expected numberof times they are right.

Then the optimal decision rule is to always select the state swith higher probability for the observed internal responsex:

This is the maximum likelihood detector. For the equal-variance case, this means that the criterion is the

average of the two signal levels:

p x|sH( )p x| sL( )1 'yes '

p x|sH( )

p x| sL( )


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Optimal Performance

The performance of the maximum likelihoodobserver for this yes/no task is given by

p(correct) =p(HIT) =p(CORRECT REJECT) = erfc d

2 2


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Bias

For this optimal decision rule, the different types oferrors are balanced: p(FA) = p(MISS)

For observers that use a different criterion, thedifferent types of errors will be unbalanced.

Such observers have lower p(correct) and are saidto be biased.

z


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ROC Curves

Suppose the experiment is repeated many timesunder different instructions.

The first time, the observer is instructed to beextremely stringent in their criterion, only reportingyes when they are 100% sure the light was flashed.

On subsequent repetitions, the observer is instructedto gradually relax their criterion.


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ROC Curves

As the criterion threshold is swept from right to left, p(HIT)increases, but p(FA) also increases.

The resulting plot of p(HIT) vs p(FA) is called a receiver-operating characteristic (ROC).

d = 0

Increasing d


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ROC Curves

Note that d remains fixed as the criterion is varied! Thus d is criterion-invariant, and is thus a pure

reflection of the signal-to-noise ratio.


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Example 2: Motion Direction Discrimination

Random dot kinematogram Signal dots are either all moving up or all moving down Noise dots are moving in random directions

Britten et al (1992)


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100% Coherence


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30% Coherence


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5% Coherence


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0% Coherence


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The Medial Temporal Area (V5)

www.thebrain.mcgill.ca


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Experimental Details

Signal direction always in preferred or anti-preferred direction for cell.

What kind of task is this?

Note that now there isexternal

noise as well asinternal noise.

To calculate neural discrimination performance,assumed neuron paired with identical neuron, tuned

to opposite direction of motion.


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Behaviour Neuron

Anti-Preferred

Direction

Preferred

Direction


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False Alarm Rate

HitRate


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Priors

Note that if the probabilities of the two signalstates are not equal, the maximum likelihood

observer will be suboptimal.

In this case we must make use of the posterior ratio.p s

H|x( )

p sL|x( )

1 'yes '

p sH

|x( )p s

L|x( )


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MAP Inference

Using Bayes rule, we obtain:

Thus we simply scale the likelihoods by the priors.

p sH

| x( )p s

L| x( )

=

p x| sH( )p sH( )

p x| sL( )p sL( )


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Loss and Risk

Maximizing p(correct) is not always the best thing todo.

How would you adjust your criterion if you wereA venture capitalist trying to detect the next Google?A pilot looking for obstacles on a runway?


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Loss Function

In general, different types of correct decision or action willyield different payoffs, and different types of errors will yield

different costs.

These differences can be accounted for through a loss function:Let a(x) represent the action of the observer, given internal response x.

Then L s,a(x)( ) represents the cost of taking action a, given world state s.


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The Ideal Observer

TheIdeal Observeruses the decision rule thatminimizes the Expected Loss, aka the Risk R(a|x):

R(a |x) = L s,a(x)( )p(s,x)s

= L s,a(x)( )p(x| s)s

p(s)


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Example 3: Slant Estimation

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