Intro to Pattern Recognition :
Bayesian Decision Theory
2. 1 Introduction
2.2 Bayesian Decision Theory–Continuous
Features
Materials used in this course were taken from the textbook “Pattern Classification” by Duda et al., John Wiley & Sons, 2001
with the permission of the authors and the publisher
Credits and Acknowledgments
Materials used in this course were taken from the textbook “Pattern
Classification” by Duda et al., John Wiley & Sons, 2001 with the permission of
the authors and the publisher; and also from
Other material on the web:












Dr. A. Aydin Atalan, Middle East Technical University, Turkey
Dr. Djamel Bouchaffra, Oakland University
Dr. Adam Krzyzak, Concordia University
Dr. Joseph Picone, Mississippi State University
Dr. Robi Polikar, Rowan University
Dr. Stefan A. Robila, University of New Orleans
Dr. Sargur N. Srihari, State University of New York at Buffalo
David G. Stork, Stanford University
Dr. Godfried Toussaint, McGill University
Dr. Chris Wyatt, Virginia Tech
Dr. Alan L. Yuille, University of California, Los Angeles
Dr. Song-Chun Zhu, University of California, Los Angeles
TYPICAL APPLICATIONS
IMAGE PROCESSING EXAMPLE
• Sorting Fish: incoming fish are
sorted according to species
using optical sensing (sea
bass or salmon?)
• Problem Analysis:
 set up a camera and take

some sample images to
extract features
Consider features such as
length, lightness, width,
number and shape of fins,
position of mouth, etc.
Sensing
Segmentation
Feature Extraction
TYPICAL APPLICATIONS
LENGTH AS A DISCRIMINATOR
• Length is a poor discriminator
TYPICAL APPLICATIONS
ADD ANOTHER FEATURE
• Lightness is a better feature than length because it
reduces the misclassification error.
• Can we combine features in such a way that we improve
performance? (Hint: correlation)
TYPICAL APPLICATIONS
WIDTH AND LIGHTNESS
• Treat features as a N-tuple (two-dimensional vector)
• Create a scatter plot
• Draw a line (regression) separating the two classes
TYPICAL APPLICATIONS
WIDTH AND LIGHTNESS
• Treat features as a N-tuple (two-dimensional vector)
• Create a scatter plot
• Draw a line (regression) separating the two classes
TYPICAL APPLICATIONS
DECISION THEORY
• Can we do better than a linear classifier?
• What is wrong with this decision surface?
(hint: generalization)
TYPICAL APPLICATIONS
GENERALIZATION AND RISK
• Why might a smoother decision surface be a better
choice? (hint: Occam’s Razor).
• This course investigates how to find such “optimal”
decision surfaces and how to provide system designers
with the tools to make intelligent trade-offs.
TYPICAL APPLICATIONS
CORRELATION
• Degrees of difficulty:
• Real data is often much
harder:
2.1 Bayesian Decision Theory
Thomas Bayes
At the time of his death, Rev.
Thomas Bayes (1702 –1761) left
behind two unpublished essays
attempting to determine the
probabilities of causes from
observed effects. Forwarded to
the British Royal Society, the
essays had little impact and were
soon forgotten.
When several years later, the
French mathematician Laplace
independently rediscovered a
very similar concept, the English
scientists quickly reclaimed the
ownership of what is now known
as the “Bayes Theorem”.
BAYESIAN DECISION THEORY
PROBABILISTIC DECISION THEORY
Bayesian decision theory is a fundamental statistical
approach to the problem of pattern classification.
Quantify the tradeoffs between various classification
decisions using probability and the costs that
accompany these decisions.
Assume all relevant probability distributions are known
(later we will learn how to estimate these from data).
Can we exploit prior knowledge in our fish classification
problem:



Are the sequence of fish predictable? (statistics)
Is each class equally probable? (uniform priors)
What is the cost of an error? (risk, optimization)
BAYESIAN DECISION THEORY
PRIOR PROBABILITIES
State of nature is prior information
Model as a random variable, :





 = 1: the event that the next fish is a sea bass
category 1: sea bass; category 2: salmon
P(1) = probability of category 1
P(2) = probability of category 2
P(1) + P( 2) = 1


•
Exclusivity: 1 and 2 share no basic events
Exhaustivity: the union of all outcomes is the sample space
(either 1 or 2 must occur)
If all incorrect classifications have an equal cost:
 Decide
1 if P(1) > P(2); otherwise, decide 2
BAYESIAN DECISION THEORY
CLASS-CONDITIONAL PROBABILITIES
A decision rule with only prior information always
produces the same result and ignores measurements.
If P(1) >> P( 2), we will be correct most of the time.
Probability of error: P(E) = min(P(1),P( 2)).
• Given a feature, x (lightness),
which is a continuous random
variable, p(x|2) is the classconditional probability density
function:
• p(x|1) and p(x|2) describe the
difference in lightness between
populations of sea and salmon.
BAYESIAN DECISION THEORY
PROBABILITY FUNCTIONS
A probability density function is denoted in lowercase and represents
a function of a continuous variable.
px(x|), often abbreviated as p(x), denotes a probability density
function for the random variable X. Note that px(x|) and py(y|) can
be two different functions.
P(x|) denotes a probability mass function, and must obey the
following constraints:
P( x )  0
 P(x)  1
x X
• Probability mass functions are typically used for discrete
random variables while densities describe continuous
random variables (latter must be integrated).
BAYESIAN DECISION THEORY
BAYES FORMULA
Suppose we know both P(j) and p(x|j), and we can
measure x. How does this influence our decision?
The joint probability that of finding a pattern that is in
category j and that this pattern has a feature value of x is:

p (  j , x )  P  j x  p  x   p x 
j
P  j 
• Rearranging terms, we arrive at Bayes formula:
P  j x  

p x
j
 P  j 
p x 
where in the case of two categories:
2

p x    p x 
j 1
j
 P  j 
BAYESIAN DECISION THEORY
POSTERIOR PROBABILITIES
Bayes formula:
P  j x  

p x
j
 P  j 
p x 
can be expressed in words as:
posterior

likelihood
 prior
evidence
By measuring x, we can convert the prior probability,
P(j), into a posterior probability, P(j|x).
Evidence can be viewed as a scale factor and is often
ignored in optimization applications (e.g., speech
recognition).
BAYESIAN DECISION THEORY
POSTERIOR PROBABILITIES
Two-class fish sorting problem (P(1) = 2/3, P(2) = 1/3):
For every value of x, the posteriors sum to 1.0.
At x=14, the probability it is in category 2 is 0.08, and for
category 1 is 0.92.
BAYESIAN DECISION THEORY
BAYES DECISION RULE
Decision rule:

For an observation x, decide 1 if P(1|x) > P(2|x); otherwise,
decide 2
Probability of error:
P( 2 x )
P  error | x   
 P ( 1 x )
x  1
x  2
The average probability of error is given by:




P ( error )   P ( error , x )dx   P ( error | x ) p ( x )dx
P ( error | x )  min[ P (  1 x ), P (  2 x )]
If for every x we ensure that P(error|x) is as small as
possible, then the integral is as small as possible. Thus,
Bayes decision rule for minimizes P(error).
Bayes Decision Rule
BAYESIAN DECISION THEORY
EVIDENCE
The evidence, p(x), is a scale factor that assures
conditional probabilities sum to 1:
P(1|x)+P(2|x)=1
We can eliminate the scale factor (which appears
on both sides of the equation):
 Decide
1 if p(x|1)P(1) > p(x|2)P(2)
Special cases:
if p(x| 1)=p(x| 2): x gives us no useful information
if P(1) = P(2): decision is based entirely on the
likelihood, p(x|j).
CONTINUOUS FEATURES
GENERALIZATION OF TWO-CLASS PROBLEM
Generalization of the preceding ideas:





Use of more than one feature
(e.g., length and lightness)
Use more than two states of nature
(e.g., N-way classification)
Allowing actions other than a decision to decide on the
state of nature (e.g., rejection: refusing to take an
action when alternatives are close or confidence is low)
Introduce a loss of function which is more general than
the probability of error
(e.g., errors are not equally costly)
Let us replace the scalar x by the vector x in a
d-dimensional Euclidean space, Rd, called
the feature space.
CONTINUOUS FEATURES
LOSS FUNCTION 1
Let {1, 2,…, c} be the set of “c” categories
Let {1, 2,…, a} be the set of “a” possible actions
Let (i|j) be the loss incurred for taking action i when
the state of nature is j
Examples
Ex 1: Fish classification
X= is the image of fish
 x =(brightness, length, fin #,
etc.)
  is our belief what the fish
type is
C
 = {“sea bass”, “salmon”,
“trout”, etc}
  is a decision for the fish
type, in this case

 = {“sea bass”, “salmon”,
“trout”, “manual expection
needed”, etc}

Ex 2: Medical diagnosis
X= all the available medical
tests, imaging scans that a
doctor can order for a patient
 x =(blood pressure, glucose
level, cough, x-ray, etc.)
 is an illness type
={“Flu”, “cold”, “TB”,
C
 “pneumonia”, “lung cancer”,
etc}

is a decision for treatment,

 = {“Tylenol”, “Hospitalize”,
“more tests needed”, etc}


CONTINUOUS FEATURES
LOSS FUNCTION
(i|j) be the loss incurred for taking action i when the
state of nature is j
The posterior, P(j|x), can be computed from Bayes
formula:
p( x |  j )P (  j )
P( j x ) 
p( x )
where the evidence is:
c
p ( x )   p ( x |  j )P (  j )
j 1
• The expected loss from taking action i is:
c
R ( i | x ) 
  (
j 1
i
|  j ) P ( j | x )
CONTINUOUS FEATURES
BAYES RISK
An expected loss is called a risk.
R(i|x) is called the conditional risk.
A general decision rule is a function (x) that tells us
which action to take for every possible observation.
The overall risk is given by:
R   R (  ( x ) | x ) p ( x )d x
If we choose (x) so that R(i(x)) is as small as possible
for every x, the overall risk will be minimized.
Compute the conditional risk for every  and select the
action that minimizes R(i|x). This is denoted R*, and is
referred to as the Bayes risk.
The Bayes risk is the best performance that can be
achieved.
CONTINUOUS FEATURES
TWO-CATEGORY CLASSIFICATION
Let 1 correspond to 1, 2 to 2, and ij = (i|j)
The conditional risk is given by:
R(1|x) = 11P(1|x) + 12P(2|x)
R(2|x) = 21P(1|x) + 22P(2|x)
Our decision rule is:
choose 1 if: R(1|x) < R(2|x);
otherwise decide 2
This results in the equivalent rule:
choose 1 if: (21- 11) P(x|1) > (12- 22) P(x|2);
otherwise decide 2
If the loss incurred for making an error is greater than that
incurred for being correct, the factors (21- 11) and
(12- 22) are positive, and the ratio of these factors simply
scales the posteriors.
CONTINUOUS FEATURES
LIKELIHOOD
By employing Bayes formula, we can replace the
posteriors by the prior probabilities and conditional
densities:
choose 1 if:
(21- 11) p(x|1) P(1) > (12- 22) p(x|2) P(2);
otherwise decide 2
If 21- 11 is positive, our rule becomes:
choose  1 if :
p( x | 1 )
p( x |  2 )

 12   22 P (  2 )
 21   11 P (  1 )
If the loss factors are identical, and the prior probabilities
are equal, this reduces to a standard likelihood ratio:
choose  1 if :
p( x | 1 )
p( x |  2 )
1
 0 1
 

1 0 
1 0 
 

0
1


1 1
 

1 1
0
 
1
0
 
0
1

1
0

1
2.3 Minimum Error Rate
Classification
Minimum Error Rate
MINIMUM ERROR RATE
Consider a symmetrical or zero-one loss function:
0
(  i  j )  
1
i j
i j
i , j  1 ,2 ,..., c
The conditional risk is:
c
R (  i x )   R (  i  j ) P (  j x)
j 1
c
  P (  j x)
j i
 1  P (  i x)
The conditional risk is the average probability of error.
To minimize error, maximize P(i|x) — also known as
maximum a posteriori decoding (MAP).
Minimum Error Rate
LIKELIHOOD RATIO
Minimum error rate classification:
choose i if: P(i| x) > P(j| x) for all ji
Example
3. It is known that 1% of population suffers from a
particular disease. A blood test has a 97% chance to
identify the disease for a diseased individual, by also
has a 6% chance of falsely indicating that a healthy
person has a disease.
a. What is the probability that a random person has a
positive blood test.
b. If a blood test is positive, what’s the probability that
the person has the disease?
c. If a blood test is negative, what’s the probability that
the person does not have the disease?
S is a boolean RV indicating whether a person has
a disease. P(S) = 0.01; P(S’) = 0.99.
T is a boolean RV indicating the test result ( T = true
indicates that test is positive.)


P(T|S) = 0.97; P(T’|S) = 0.03;
P(T|A’) = 0.06; P(T’|S’) = 0.94;
(a) P(T) = P(S) P(T|S) + P(S’)P(T|S’) = 0.01*0.97
+0.99 * 0.06 = 0.0691
(b) P(S|T)=P(T|S)*P(S)/P(T) = 0.97* 0.01/0.0691 =
0.1403
(c) P(S’|T’) = P(T’|S’)P(S’)/P(T’)= P(T’|S’)P(S’)/(1P(T))= 0.94*0.99/(1-.0691)=0.9997
A physician can do two possible actions after seeing
patient’s test results:


A1 - Decide the patient is sick
A2 - Decide the patient is healthy
The costs of those actions are:


If the patient is healthy, but the doctor decides he/she is sick
- $20,000.
If the patient is sick, but the doctor decides he/she is healthy
- $100.000
When the test is positive:


R(A1|T) = R(A1|S)P(S|T) + R(A1|S’) P(S’|T) = R(A1|S’)
*P(S’|T) = 20.000* P(S’|T) = 20.000*0.8597 = $17194.00
R(A2|T) = R(A2|S)P(S|T) + R(A2|S’) P(S’|T) =
R(A2|S)P(S|T) = 100000* 0.1403 = $14030.00
A physician can do three possible actions
after seeing patient’s test results:



Decide the patient is sick
Decide the patient is healthy
Send the patient for another test
The costs of those actions are:



If the patient is healthy, but the doctor decides
he/she is sick - $20,000.
If the patient is sick, but the doctor decides he/she
is healthy - $100.000
Sending the patient for another test costs $15,000
When the test is positive:



R(A1|T) = R(A1|S)P(S|T) + R(A1|S’) P(S’|T) =
R(A1|S’) *P(S’|T) = 20.000* P(S’|T) =
20.000*0.8597 = $17194.00
R(A2|T) = R(A2|S)P(S|T) + R(A2|S’) P(S’|T) =
R(A2|S)P(S|T) = 100000* 0.1403 = $14030.00
R(A3|T) = $15000.00
When the test is negative:



R(A1|T’) = R(A1|S)P(S|T’) + R(A1|S’) P(S’|T’) =
R(A1|S’) P(S’|T’) = 20,000* 0.9997 = $19994.00
R(A2|T’) = R(A2|S)P(S|T’) + R(A2|S’) P(S’|T’) =
R(A1|S) P(S|T’)= 100,000*0.0003 = $30.00
R(A3|T’) = 15000.00
Example
For sea bass population, the lightness x is a normal
random variable distributes according to N(4,1);
for salmon population x is distributed according to
N(10,1);
Select the optimal decision where:
a. The two fish are equiprobable
b. P(sea bass) = 2X P(salmon)
c. The cost of classifying a fish as a salmon when it
truly is seabass is 2$, and t The cost of
classifying a fish as a seabass when it is truly a
salmon is 1$.
2
Exercise
Consider a 2-class problem with P(C1) = 2/3, P(C2)=1/3; a scalar feature x
and three possible actions a1, a2, a3 defined as:
a1: choose C1
a1
a2
a3
a2: choose C2
a3: do not classify
Let the loss matrix (ai | Cj) be:
0
1
1/4
C1
C2
1
0
1/4
And let P(x | C1) = (2-x)/2, P(x | C2) =1/2, 0  x  2
Questions:
1) Which action to decide for a pattern x; 0  x  2
2) What is the proportion of patterns for which action a3 is performed (i.e.,
“do not classify”)
3) Compute the total minimum risk
4) If you decide to take action a1 for all x, then how much the total risk will be
reduced.
Solution:
P(x) = (5-2x)/6
P(C1 | x) = (4-2x)/(5-2x); 0  x  2
P(C2 | x) = 1/ (5-2x)
This leads to conditional risks:
r1(x) = r(a1 | x) = 0.P(C1 | x) + 1. P(C2 | x) = 1/(5-2x)
r2(x) = r(a2 | x) = 1.P(C1 | x) + 0. P(C2 | x) = (4-2x)/(5-2x)
r3(x) = r(a3 | x) = 1/4.P(C1 | x) + 1/4. P(C2 | x) = 1/4
Bayes decision rule assigns to each x the action with the minimum
conditional risk. The conditional risks are sketched in the following
figure and the optimal decision rule is therefore:
r1
r2
r3
0.5
11/6
2
x
If 0  x 0.5 then “action a1” = “choose C1”
If 0.5  x  11/6 then “action a3” = “do not classify”
If 11/6  x  2 then “action a2” = “choose C2”
In this particular case the action “do not classify” is optimal
whenIever x is between ½ and 11/6
2) 11 / 6
11 / 6
5  2x
dx  60 %
 P ( x ) dx  
6
1/ 2
1/ 2
Therefore, “do not classify” action has been performed for 60% of the
input patterns.
3)
Total minimum risk
2
 min
r1 ( x ), r2 ( x ), r3 ( x )P ( x ) dx
0

1/ 2
 r1 ( x )P ( x ) dx 
0

1
12
11 / 6
 r3 ( x )P ( x ) dx 
1/ 2

4
27

1
216
2
 r2 ( x )P ( x ) dx
11 / 6
 0 . 236
4) If instead of using Bayes classifier we choose to take a1 for all x, then
the total risk is:
2
1
r
(
x
)
P
(
x
)
dx

 0 . 236
 1
3
0
GAUSSIAN CLASSIFIERS
Case 1: i = 2I
Features are statistically independent, and all features
have the same variance: Distributions are spherical in d
dimensions.
 2

0
i  
 0

 0
i
1
0

2
0
...
...
...
0
...
 (1 / 
0 

0 

... 

2
 
2
)I
 i   I is independen
2
2d
t of i
6
GAUSSIAN CLASSIFIERS
THRESHOLD DECODING
This has a simple geometric interpretation:
x  μi
2
 x μ
2
j
2
 2  ln
P(  j )
P( i )
The decision region when the priors are equal
and the support regions are spherical is simply
halfway between the means (Euclidean distance).
GAUSSIAN CLASSIFIERS
Note how priors shift the boundary away from the more likely mean !!!
6
GAUSSIAN CLASSIFIERS
3-D case
6
GAUSSIAN CLASSIFIERS
Case 2: i = 
• Covariance matrices are arbitrary, but equal to each other
for all classes. Features then form hyper-ellipsoidal
clusters of equal size and shape.
• Discriminant function is linear:
T
gi (x )  ω x   i0
i
1
1
T
1
ω i  Σ μ i ,  i 0   μ Σ μ i  ln P ( i )
2 i
6
6
6
Case 3: i = arbitrary
The covariance matrices are different for each category
All bets off !In two class case, the decision boundaries form
hyperquadratics.
(Hyperquadrics are: hyperplanes, pairs of hyperplanes,
hyperspheres, hyperellipsoids, hyperparaboloids,
hyperhyperboloids)
6
6
GAUSSIAN CLASSIFIERS
ARBITRARY COVARIANCES