Transcript P(x | i )

Bayesian Decision Theory
(Sections 2.1-2.2)
• Decision problem posed in probabilistic terms
• Bayesian Decision Theory–Continuous Features
• All the relevant probability values are known
Probability Density
Jain CSE 802, Spring 2013
Course Outline
MODEL INFORMATION
COMPLETE
Bayes Decision
Theory
Parametric
Approach
“Optimal”
Rules
Plug-in
Rules
INCOMPLETE
Supervised
Learning
Nonparametric
Approach
Unsupervised
Learning
Parametric
Approach
Density
Geometric Rules
Estimation
(K-NN, MLP)
Mixture
Resolving
Nonparametric
Approach
Cluster Analysis
(Hard, Fuzzy)
Introduction
• From sea bass vs. salmon example to “abstract”
decision making problem
• State of nature; a priori (prior) probability
• State of nature (which type of fish will be observed next) is
unpredictable, so it is a random variable
• The catch of salmon and sea bass is equiprobable
•
P(1) = P(2) (uniform priors)
•
P(1) + P( 2) = 1 (exclusivity and exhaustivity)
• Prior prob. reflects our prior knowledge about how likely we are to
observe a sea bass or salmon; these probabilities may depend on
time of the year or the fishing area!
• Bayes decision rule with only the prior information
• Decide 1 if P(1) > P(2), otherwise decide 2
• Error rate = Min {P(1) , P(2)}
• Suppose now we have a measurement or feature
•
•
on the state of nature - say the fish lightness value
Use of the class-conditional probability density
P(x | 1) and P(x | 2) describe the difference in
lightness feature between populations of sea bass
and salmon
Amount of overlap between the densities determines the “goodness” of feature
• Maximum likelihood decision rule
• Assign input pattern x to class 1 if
P(x | 1) > P(x | 2), otherwise 2
• How does the feature x influence our attitude
(prior) concerning the true state of nature?
• Bayes decision rule
• Posteriori probability, likelihood, evidence
• P(j , x) = P(j | x)p (x) = p(x | j) P (j)
• Bayes formula
P(j | x) = {p(x | j) . P (j)} / p(x)
j2
where
P ( x )   P ( x |  j )P (  j )
j 1
• Posterior = (Likelihood. Prior) / Evidence
• Evidence P(x) can be viewed as a scale factor that
•
guarantees that the posterior probabilities sum to 1
P(x | j) is called the likelihood of j with respect to x; the
category j for which P(x | j) is large is more likely to be
the true category
•
•
P(1 | x) is the probability of the state of nature being 1
given that feature value x has been observed
Decision based on the posterior probabilities is called the
Optimal Bayes Decision rule
For a given observation (feature value) X:
if P(1 | x) > P(2 | x)
if P(1 | x) < P(2 | x)
decide 1
decide 2
To justify the above rule, calculate the probability of error:
P(error | x) = P(1 | x) if we decide 2
P(error | x) = P(2 | x) if we decide 1
• So, for a given x, we can minimize te rob. Of error,
decide 1 if
P(1 | x) > P(2 | x);
otherwise decide 2
Therefore:
P(error | x) = min [P(1 | x), P(2 | x)]
• Thus, for each observation x, Bayes decision rule
•
minimizes the probability of error
Unconditional error: P(error) obtained by
integration over all x w.r.t. p(x)
• Optimal Bayes decision rule
Decide 1 if P(1 | x) > P(2 | x);
otherwise decide 2
• Special cases:
(i) P(1) = P(2); Decide 1 if
p(x | 1) > p(x | 2), otherwise 2
(ii) p(x | 1) = p(x | 2); Decide 1 if
P(1) > P(2), otherwise 2
Bayesian Decision Theory –
Continuous Features
• Generalization of the preceding formulation
• Use of more than one feature (d features)
• Use of more than two states of nature (c classes)
• Allowing other actions besides deciding on the state of
•
nature
Introduce a loss function which is more general than the
probability of error
• Allowing actions other than classification primarily
allows the possibility of rejection
• Refusing to make a decision when it is difficult to
decide between two classes or in noisy cases!
• The loss function specifies the cost of each action
• Let {1, 2,…, c} be the set of c states of nature
(or “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 true state of nature is j
• General decision rule (x) specifies which action to take for every
possible observation x
j c
Conditional Risk
R(  i | x )    (  i |  j )P (  j | x )
j 1
For a given x, suppose we take the action i ; if the true state is j ,
we will incur the loss (i | j). P(j | x) is the prob. that the true
state is j But, any one of the C states is possible for given x.
Overall risk
R = Expected value of R(i | x) w.r.t. p(x)
Conditional risk
Minimizing R
Minimize R(i | x) for i = 1,…, a
Select the action i for which R(i | x) is minimum
The overall risk R is minimized and the resulting risk
is called the Bayes risk; it is the best performance that
can be achieved!
• Two-category classification
1 : deciding 1
2 : deciding 2
ij = (i | j)
loss incurred for deciding i when the true state of nature is j
Conditional risk:
R(1 | x) = 11P(1 | x) + 12P(2 | x)
R(2 | x) = 21P(1 | x) + 22P(2 | x)
Bayes decision rule is stated as:
if R(1 | x) < R(2 | x)
Take action 1: “decide 1”
This results in the equivalent rule:
decide 1 if:
(21- 11) P(x | 1) P(1) >
(12- 22) P(x | 2) P(2)
and decide 2 otherwise
Likelihood ratio:
The preceding rule is equivalent to the following rule:
P ( x |  1 ) 12   22 P (  2 )
if

.
P ( x |  2 )  21  11 P (  1 )
then take action 1 (decide 1); otherwise take action 2
(decide 2)
Note that the posteriori porbabilities are scaled by the loss
differences.
Interpretation of the Bayes decision rule:
“If the likelihood ratio of class 1 and class 2
exceeds a threshold value (that is independent of
the input pattern x), the optimal action is to decide
1”
Maximum likelihood decision rule: the threshold
value is 1; 0-1 loss function and equal class prior
probability
Bayesian Decision Theory
(Sections 2.3-2.5)
• Minimum Error Rate Classification
• Classifiers, Discriminant Functions and Decision Surfaces
• The Normal Density
Minimum Error Rate Classification
• Actions are decisions on classes
If action i is taken and the true state of nature is j then:
the decision is correct if i = j and in error if i  j
• Seek a decision rule that minimizes the probability
of error or the error rate
• Zero-one (0-1) loss function: no loss for correct decision
and a unit loss for any error
0 i  j
 (  i , j )  
1 i  j
i , j  1 ,..., c
The conditional risk can now be simplified as:
j c
R(  i | x )    (  i |  j )P (  j | x )
j 1
  P(  j | x )  1  P(  i | x )
j 1
“The risk corresponding to the 0-1 loss function is the
average probability of error”
• Minimizing the risk requires maximizing the
posterior probability P(i | x) since
R(i | x) = 1 – P(i | x))
• For Minimum error rate
• Decide i if P (i | x) > P(j | x) j  i
• Decision boundaries and decision regions
12   22 P (  2 )
P( x | 1 )
Let
.
   then decide  1 if :
 
21  11 P (  1 )
P( x |  2 )
• If  is the 0-1 loss function then the threshold involves
only the priors:
 0 1

1 0
  
then   
P(  2 )
 a
P( 1 )
0 2 
2 P(  2 )
 then   
if   
 b
P( 1 )
1 0
Classifiers, Discriminant Functions
and Decision Surfaces
• Many different ways to represent pattern
classifiers; one of the most useful is in terms of
discriminant functions
• The multi-category case
• Set of discriminant functions gi(x), i = 1,…,c
• Classifier assigns a feature vector x to class i if:
gi(x) > gj(x) j  i
Network Representation of a Classifier
• Bayes classifier can be represented in this way, but
the choice of discriminant function is not unique
• gi(x) = - R(i | x)
(max. discriminant corresponds to min. risk!)
• For the minimum error rate, we take
gi(x) = P(i | x)
(max. discrimination corresponds to max. posterior!)
gi(x)  P(x | i) P(i)
gi(x) = ln P(x | i) + ln P(i)
(ln: natural logarithm!)
• Effect of any decision rule is to divide the feature
space into c decision regions
if gi(x) > gj(x) j  i then x is in Ri
(Region
Ri means assign x to i)
• The two-category case
• Here a classifier is a “dichotomizer” that has two
discriminant functions g1 and g2
Let g(x)  g1(x) – g2(x)
Decide 1 if g(x) > 0 ; Otherwise decide 2
•
So, a “dichotomizer” computes a single
discriminant function g(x) and classifies x
according to whether g(x) is positive or
not.
• Computation of g(x) = g1(x) – g2(x)
g( x )  P (  1 | x )  P (  2 | x )
P( x | 1 )
P( 1 )
 ln
 ln
P( x |  2 )
P(  2 )
The Normal Density
• Univariate density: N( , 2)
• Normal density is analytically tractable
• Continuous density
• A number of processes are asymptotically Gaussian
• Patterns (e.g., handwritten characters, speech signals ) can be
viewed as randomly corrupted versions of a single typical or
prototype (Central Limit theorem)
P( x ) 
2

1
1 x 
exp   
 ,
2 
 2    
where:
 = mean (or expected value) of x
2 = variance (or expected squared deviation) of x
• Multivariate density: N( , )
• Multivariate normal density in d dimensions:
P( x ) 
1
( 2 )
d/2

1/ 2
 1

t
1
exp  ( x   )  ( x   )
 2

where:
x = (x1, x2, …, xd)t (t stands for the transpose of a vector)
 = (1, 2, …, d)t mean vector
 = d*d covariance matrix
•
•
•
|| and -1 are determinant and inverse of , respectively
The covariance matrix is always symmetric and positive semidefinite; we
assume  is positive definite so the determinant of  is strictly positive
Multivariate normal density is completely specified by [d + d(d+1)/2]
parameters
If variables x1 and x2 are statistically independent then the covariance
of x1 and x2 is zero.
Multivariate Normal density
Samples drawn from a normal population tend to fall in a single
cloud or cluster; cluster center is determined by the mean vector
and shape by the covariance matrix
The loci of points of constant density are hyperellipsoids whose
principal axes are the eigenvectors of 
r 2  ( x   )t 1 ( x   )
Transformation of Normal Variables
Linear combinations of jointly normally distributed random variables are
normally distributed
Coordinate transformation can convert an arbitrary multivariate normal
distribution into a spherical one
Bayesian Decision Theory
(Sections 2-6 to 2-9)
• Discriminant Functions for the Normal Density
• Bayes Decision Theory – Discrete Features
Discriminant Functions for the
Normal Density
• The minimum error-rate classification can be
achieved by the discriminant function
gi(x) = ln P(x | i) + ln P(i)
• In case of multivariate normal densities
1
1
d
1
t
g i ( x )   ( x   i )  ( x   i )  ln 2  ln  i  ln P (  i )
2
2
2
i
• Case i = 2.I
(I is the identity matrix)
Features are statistically independent and each
feature has the same variance
g i ( x )  w x  w i 0 (linear discriminant function)
t
i
where :
i
1
t
wi  2 ; wi 0  

i  i  ln P (  i )
2

2
(  i 0 is called the threshold for the ith category! )
• A classifier that uses linear discriminant functions is called
“a linear machine”
• The decision surfaces for a linear machine are pieces of
hyperplanes defined by the linear equations:
gi(x) = gj(x)
• The hyperplane separating Ri and Rj
1
2
x0  (  i   j ) 
2
i   j
2
P(  i )
ln
( i   j )
P(  j )
is orthogonal to the line linking the means!
1
if P (  i )  P (  j ) then x0  (  i   j )
2
• Case 2: i =  (covariance matrices of all classes
are identical but otherwise arbitrary!)
• Hyperplane separating Ri and Rj


ln P (  i ) / P (  j )
1
x0  (  i   j ) 
.(  i   j )
t
1
2
( i   j )  ( i   j )
• The hyperplane separating Ri and Rj is generally
not orthogonal to the line between the means!
• To classify a feature vector x, measure the
squared Mahalanobis distance from x to each of
the c means; assign x to the category of the
nearest mean
Discriminant Functions for 1D Gaussian
• Case 3: i = arbitrary
• The covariance matrices are different for each category
g i ( x )  x tWi x  w it x  w i 0
where :
1 1
Wi    i
2
w i   i 1  i
1 t 1
1
w i 0    i  i  i  ln  i  ln P (  i )
2
2
In the 2-category case, the decision surfaces are
hyperquadrics that can assume any of the general forms:
hyperplanes, pairs of hyperplanes, hyperspheres,
hyperellipsoids, hyperparaboloids, hyperhyperboloids)
Discriminant Functions for the Normal Density
Discriminant Functions for the Normal Density
Discriminant Functions for the Normal Density
Decision Regions for Two-Dimensional Gaussian Data
x2  3.514  1.125 x1  0.1875 x12
Error Probabilities and Integrals
• 2-class problem
• There are two types of errors
• Multi-class problem
– Simpler to computer the prob. of being correct (more
ways to be wrong than to be right)
Error Probabilities and Integrals
Bayes optimal decision boundary in 1-D case
Error Bounds for Normal Densities
•
•
The exact calculation of the error for the
general Guassian case (case 3) is extremely
difficult
However, in the 2-category case the general
error can be approximated analytically to give
us an upper bound on the error
Error Rate of Linear Discriminant Function (LDF)
• Assume a 2-class problem
p  x   ~ N (  , ), p  x   ~ N (  , )
1
1
2
2
1
gi ( x)  log  P( x i )    ( x   i )t  1 ( x   i )  log  P(i ) 
2
• Due to the symmetry of the problem (identical
), the two types of errors are identical
• Decide x  if g ( x)  g ( x) or
1
1
2
1
1
t 1
 ( x   1 )  ( x   1 )  log  P(1 )    ( x   2 )t  1 ( x   2 )  log  P(2 ) 
2
2
or


1 t 1
t
(  2   1 )  x   1  1   2 1  2  log  P(1 ) / P(2 ) 
2
t
1
Error Rate of LDF
• Let h( x)  (   )  x  12         
• Compute expected values & variances of
when x  1 & x  2
t
2
1
1
t
1
1
t
1
1
2
1  E  h( x) x  1   (  2   1 )t  1 E  x 1  
1
  (  2   1 )t  1 (  2   1 )
2
 
where
2

h( x)
1 t 1
 1  1   t2  1  2
2

1
(  2  1 )t 1 (  2  1 )
= squared Mahalanobis distance between
1 &  2
Error Rate of LDF
• Similarly
1
2
2   (  2   1 )t  1 (  2   1 )
 
 12  E  h( x)  1  1   E (  2   1 )t  1 ( x   1 ) x  1 
2


 (  2   1 )t  1 (  2   1 )
 2
 22  2
p  h( x) x  1  ~ N ( , 2 )
p  h( x) x  2  ~ N ( , 2 )
Error Rate of LDF

1  P  g1 ( x)  g 2 ( x) x  1    P  h( x) 1  dh
t



n t
2
1
 2
1
e 2 d
2
1 1
  erf
2 2
 t 


 4 
h( x ) ~
1
 ()
1
e 2
2  2
Error Rate of LDF
 P (1 ) 
t  log 

P
(

)

2 
2
erf (r ) 

1 1
 2   erf
2 2
r
e
 x2
dx
0
 t

 4



Total probability of error
Pe  P (1 )1  P (2 ) 2
Error Rate of LDF
1
P 1   P 2  
 t 0
2
   1 1
1 1
1   2   erf 
   erf
2 2
 4  2 2
 (  1   2 )t  1 (  1   2 ) 




2 2


(i) No Class Separation
(  1   2 )t  1 (  1   2 )  0

1   2 
1
2
(ii) Perfect Class Separation
(  1   2 )t  1 (  1   2 )  0
∞

1   2  0
(erf  1)
Mahalanobis distance is a good measure of separation between classes
Chernoff Bound
• To derive a bound for the error, we need the
following inequality
Assume conditional prob. are normal
where
Chernoff Bound
Chernoff bound for P(error) is found by determining the
value of  that minimizes exp(-k())
Error Bounds for Normal Densities
• Bhattacharyya Bound
• Assume  = 1/2
• computationally simpler
• slightly less tight bound
• Now, Eq. (73) has the form
When the two covariance matrices are equal, k(1/2) is te
same as the Mahalanobis distance between the two means
Error Bounds for Gaussian Distributions
Chernoff Bound
P(error )  P  (1 ) P1  (1 )  p  ( x | 1 ) p1  ( x | 2 )dx
p

0   1
( x | 1 ) p1  ( x | 2 )dx  e  k (  )
k ( ) 
 (1  )
2
1
t
(  2  1 ) [ 1  (1   )  2 ] (  2  1 ) 
1 (1  ) 2
1
ln
2
|1| |2 |1
Best Chernoff error bound is 0.008190
Bhattacharya Bound (β=1/2)
P(error )  P(1 ) P(2 )  P( x | 1 ) P( x | 2 )dx  P (1 ) P (2 )e  k (1/2)
k (1 / 2)  1 / 8(  2  1 )
t
2
 1 

2 
1
2–category, 2D data
1 2
1
2
(  2  1 )  ln
2
|1 ||2 |
Bhattacharya error bound is 0.008191
True error using numerical integration = 0.0021
Neyman-Pearson Rule
“Classification, Estimation and Pattern recognition” by Young and Calvert
Neyman-Pearson Rule
Neyman-Pearson Rule
Neyman-Pearson Rule
Neyman-Pearson Rule
Neyman-Pearson Rule
Signal Detection Theory
We are interested in detecting a single weak pulse,
e.g. radar reflection; the internal signal (x) in detector
has mean m1 (m2) when pulse is absent (present)
p( x | 1 ) ~ N ( 1 ,  2 )
p ( x | 2 ) ~ N (  2 ,  2 )
The detector uses a threshold
x* to determine the presence of pulse
Discriminability: ease of determining
whether the pulse is present or not
d'
| 1  2 |

For given threshold, define hit,
false alarm, miss and correct
rejection
P( x  x*| x  2 ) :
P( x  x*| x  1 ) :
P( x  x*| x  2 ) :
P( x  x*| x  1 ) :
hit
false alarm
miss
correct rejection
Receiver Operating Characteristic
(ROC)
• Experimentally compute hit and false alarm rates for
•
•
fixed x*
Changing x* will change the hit and false alarm rates
A plot of hit and false alarm rates is called the ROC
curve
Performance
shown at different
operating points
Operating Characteristic
• In practice, distributions may not be Gaussian
•
and will be multidimensional; ROC curve can still
be plotted
Vary a single control parameter for the decision
rule and plot the resulting hit and false alarm rates
Bayes Decision Theory – Discrete Features
•
•
Components of x are binary or integer valued; x can
take only one of m discrete values
v1, v2, …,vm
Case of independent binary features for 2-category
problem
Let x = [x1, x2, …, xd ]t where each xi is either 0 or 1, with
probabilities:
pi = P(xi = 1 | 1)
qi = P(xi = 1 | 2)
• The discriminant function in this case is:
d
g ( x )   w i x i  w0
i 1
where :
pi ( 1  q i )
w i  ln
q i ( 1  pi )
i  1 ,..., d
and :
1  pi
P( 1 )
w0   ln
 ln
1  qi
P(  2 )
i 1
d
decide  1 if g(x)  0 and  2 if g(x)  0
Bayesian Decision for Three-dimensional
Binary Data
• Consider a 2-class problem with three independent binary
features; class priors are equal and pi = 0.8 and qi = 0.5, i =
1,2,3
• wi = 1.3863
• w0 = 1.2
• Decision surface g(x) = 0 is shown below
Decision boundary for 3D binary features. Left figure shows the case when pi=.8 and
qi=.5. Right figure shows case when p3=q3 (Feature 3 is not providing any
discriminatory information) so decision surface is parallel to x3 axis
Handling Missing Features
• Suppose it is not possible to measure a certain
feature for a given pattern
• Possible solutions:
• Reject the pattern
• Approximate the missing feature
•
•
Mean of all the available values for the missing feature
Marginalize over the distribution of the missing feature
Handling Missing Features
Other Topics
• Compound Bayes Decision Theory & Context
– Consecutive states of nature might not be statistically independent; in
sorting two types of fish, arrival of next fish may not be independent of
the previous fish
– Can we exploit such statistical dependence to gain improved
performance (use of context)
– Compound decision vs. sequential compound decision problems
– Markov dependence
• Sequential Decision Making
– Feature measurement process is sequential (as in medical diagnosis)
– Feature measurement cost
– Minimize the no. of features to be measured while achieving a
sufficient accuracy; minimize a combination of feature measurement
cost & classification accuracy
Context in Text Recognition