# 02 Signal Detection Theory

W
Shared by:
Categories
-
Stats
views:
60
posted:
1/20/2011
language:
English
pages:
30
Document Sample

```							           2. SIGNAL DETECTION THEORY
J. Elder     PSYC 6256 Principles of Neural Coding
Signal Detection Theory
2                                   Probability & Bayesian Inference

  Provides a method for characterizing human
performance in detecting, discriminating and
estimating signals.
  For noisy signals, provides a method for identifying

the 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

PSYC 6256 Principles of Neural Coding                        J. Elder
Example 1
3                                   Probability & Bayesian Inference

  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 the

light was flashed or not.
  This is a yes-no detection task.

PSYC 6256 Principles of Neural Coding                        J. Elder
Noise
4                                          Probability & Bayesian Inference

    In this example, the information useful for the task is the light
energy of the stimulus.

    By the time the stimulus information is received by decision
centres 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: the
dispersion increases with the mean.

PSYC 6256 Principles of Neural Coding                        J. Elder
Equal-Variance Gaussian Case
5                                    Probability & Bayesian Inference

  It is often possible to approximate this noise as
Gaussian-distributed, with the same variance for
both stimulus conditions.
  Then the noise is independent of the signal state.

PSYC 6256 Principles of Neural Coding                        J. Elder
Discriminability d’
6                                                  Probability & Bayesian Inference

(
⎛ x−µ
)       ⎞
2
1
(         )
p x | S = sH =         exp ⎜ −
⎜   2σ
H
2
⎟
⎟
2πσ     ⎝                   ⎠

⎛ x−µ
(       )       ⎞
2
1
(         )
p x | S = sL =        exp ⎜ −
⎜   2σ 2
L
⎟
⎟
2πσ     ⎝                    ⎠

signal separation µH − µL                                                     µH − µL
d'=                   =
signal dispersion    σ

σ

PSYC 6256 Principles of Neural Coding                                            J. Elder
Criterion Threshold
7                                        Probability & Bayesian Inference

    The internal response is often approximated as a continuous
variable, called the decision variable.
    But to yield an actual decision, this has to be converted to a
binary variable (yes/no).
    A reasonable way to do this is to define a criterion threshold z:
z
x ≥ z → ' yes'
x < z → 'no'

x

x

PSYC 6256 Principles of Neural Coding                                J. Elder
Effect of Shifting the Criterion
8                                 Probability & Bayesian Inference

PSYC 6256 Principles of Neural Coding                        J. Elder
How did we calculate these numbers?
9                                                  Probability & Bayesian Inference

(
⎛ x−µ
)       ⎞
2
1
(         )
p x | S = sH =         exp ⎜ −
⎜   2σ
H
2
⎟
⎟
2πσ     ⎝                   ⎠

⎛ x−µ
(       )       ⎞
2
1
(         )
p x | S = sL =        exp ⎜ −
⎜   2σ 2
L
⎟
⎟
2πσ     ⎝                    ⎠

µH − µL
d ' = zFA − zHIT

σ

PSYC 6256 Principles of Neural Coding                                            J. Elder
What is the right criterion to use?
10                                            Probability & Bayesian Inference

    Suppose the observer wants to maximize the expected number
of times they are right.
    Then the optimal decision rule is to always select the state s
with higher probability for the observed internal response x:
( ) ≥ 1→ ' yes '
p x | sH
p (x | s )
L
The ‘likelihood ratio test’
p (x | s )
H
< 1→ ' no '
p (x | s )
L

    This is the maximum likelihood detector.
    For the equal-variance case, this means that the criterion is the
average of the two signal levels:
z
1
2
(
z = µL + µH        )

PSYC 6256 Principles of Neural Coding                            J. Elder
Optimal Performance
11                                       Probability & Bayesian Inference

    The performance of the maximum likelihood
observer for this yes/no task is given by
⎛ d′ ⎞
p(correct) = p(HIT) = p(CORRECT REJECT) = erfc ⎜ −
⎝ 2 2⎟
⎠

PSYC 6256 Principles of Neural Coding                        J. Elder
Bias
12                                    Probability & Bayesian Inference

  For this optimal decision rule, the different types of
errors are balanced: p(FA) = p(MISS)
  For observers that use a different criterion, the
different types of errors will be unbalanced.
  Such observers have lower p(correct) and are said

to be biased.

z

PSYC 6256 Principles of Neural Coding                        J. Elder
ROC Curves
13                                    Probability & Bayesian Inference

  Suppose the experiment is repeated many times
under different instructions.
  The first time, the observer is instructed to be

extremely stringent in their criterion, only reporting
‘yes’ when they are 100% sure the light was flashed.
  On subsequent repetitions, the observer is instructed

PSYC 6256 Principles of Neural Coding                        J. Elder
ROC Curves
14                                       Probability & Bayesian Inference

    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).

Increasing d ′

d′ = 0

PSYC 6256 Principles of Neural Coding                        J. Elder
ROC Curves
15                                    Probability & Bayesian Inference

  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.

PSYC 6256 Principles of Neural Coding                        J. Elder
Example 2: Motion Direction Discrimination
16                                     Probability & Bayesian Inference

  Random dot kinematogram                     Britten et al (1992)

  Signal dots are either all moving up or all moving down

  Noise dots are moving in random directions

PSYC 6256 Principles of Neural Coding                        J. Elder
100% Coherence
17                                Probability & Bayesian Inference

PSYC 6256 Principles of Neural Coding                        J. Elder
30% Coherence
18                                Probability & Bayesian Inference

PSYC 6256 Principles of Neural Coding                        J. Elder
5% Coherence
19                                Probability & Bayesian Inference

PSYC 6256 Principles of Neural Coding                        J. Elder
0% Coherence
20                                Probability & Bayesian Inference

PSYC 6256 Principles of Neural Coding                        J. Elder
The Medial Temporal Area (V5)
21                                 Probability & Bayesian Inference

PSYC 6256 Principles of Neural Coding               www.thebrain.mcgill.ca   J. Elder
Experimental Details
22                                   Probability & Bayesian Inference

  Signal direction always in preferred or anti-
preferred direction for cell.
  What kind of task is this?

  Note that now there is external noise as well as

internal noise.
  To calculate neural discrimination performance,

assumed neuron paired with identical neuron, tuned
to opposite direction of motion.

PSYC 6256 Principles of Neural Coding                        J. Elder
Anti-Preferred            Preferred
Behaviour   Direction        Neuron   Direction
Hit Rate

False Alarm Rate
Priors
25                                          Probability & Bayesian Inference

  Note that if the probabilities of the two signal
states are not equal, the maximum likelihood
observer will be suboptimal.
  In this case we must make use of the posterior ratio.

( ) ≥ 1→ ' yes '
p sH | x
p (s | x )
L
Maximum a posteriori (MAP) rule
p (s | x )
H
< 1→ ' no '
p (s | x )
L

PSYC 6256 Principles of Neural Coding                        J. Elder
MAP Inference
26                                          Probability & Bayesian Inference

    Using Bayes’ rule, we obtain:
( ) = p ( x | s ) p (s )
p sH | x            H      H

p (s | x ) p ( x | s ) p (s )
L               L      L

    Thus we simply scale the likelihoods by the priors.

PSYC 6256 Principles of Neural Coding                        J. Elder
Loss and Risk
27                                         Probability & Bayesian Inference

  Maximizing p(correct) is not always the best thing to
do.

  A venture capitalist trying to detect the next Google?
  A pilot looking for obstacles on a runway?

PSYC 6256 Principles of Neural Coding                        J. Elder
Loss Function
28                                        Probability & Bayesian Inference

    In general, different types of correct decision or action will
yield 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.

PSYC 6256 Principles of Neural Coding                                  J. Elder
The Ideal Observer
29                                        Probability & Bayesian Inference

    The Ideal Observer uses the decision rule that
minimizes the Expected Loss, aka the Risk R(a|x):
(         )                    (
R(a | x) = ∑ L s,a(x) p(s, x) = ∑ L s,a(x) p(x | s)p(s)   )
s                             s

PSYC 6256 Principles of Neural Coding                        J. Elder
Example 3: Slant Estimation
30                                 Probability & Bayesian Inference

PSYC 6256 Principles of Neural Coding                        J. Elder

```
Related docs
Other docs by dfsdf224s
Pottery Decorating Workshops