Transcript Basic Probability + Introduction to Pattern Classification

1
Basic Probability
Pattern Classification, Chapter 1
2
Introduction
•
•
Probability is the study of randomness and uncertainty.
In the early days, probability was associated with games
of chance (gambling).
Pattern Classification, Chapter 1
3
Simple Games Involving Probability
Game: A fair die is rolled. If the result is 2, 3, or 4, you win
$1; if it is 5, you win $2; but if it is 1 or 6, you lose $3.
Should you play this game?
Pattern Classification, Chapter 1
4
Random Experiment
•
a random experiment is a process whose outcome is uncertain.
•
•
•
•
Examples:
Tossing a coin once or several times
Picking a card or cards from a deck
Measuring temperature of patients
...
Pattern Classification, Chapter 1
5
Events & Sample Spaces
Sample Space
The sample space is the set of all possible outcomes.
Simple Events
The individual outcomes are called simple events.
Event
An event is any collection
of one or more simple events
Pattern Classification, Chapter 1
6
Example
Experiment: Toss a coin 3 times.
•
Sample space 
•
 = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.
•
Examples of events include
•
•
A = {HHH, HHT,HTH, THH}
= {at least two heads}
B = {HTT, THT,TTH}
= {exactly two tails.}
Pattern Classification, Chapter 1
7
Basic Concepts (from Set Theory)
•
The union of two events A and B, A  B, is the event consisting of
all outcomes that are either in A or in B or in both events.
•
The complement of an event A, Ac, is the set of all outcomes in 
that are not in A.
•
The intersection of two events A and B, A  B, is the event
consisting of all outcomes that are in both events.
•
When two events A and B have no outcomes in common, they are
said to be mutually exclusive, or disjoint, events.
Pattern Classification, Chapter 1
8
Example
Experiment: toss a coin 10 times and the number of heads is observed.
•
Let A = { 0, 2, 4, 6, 8, 10}.
•
B = { 1, 3, 5, 7, 9}, C = {0, 1, 2, 3, 4, 5}.
•
A  B= {0, 1, …, 10} = .
•
A  B contains no outcomes. So A and B are mutually exclusive.
•
Cc = {6, 7, 8, 9, 10}, A  C = {0, 2, 4}.
Pattern Classification, Chapter 1
9
Rules
•
•
•
Commutative Laws:
•
A  B = B  A, A  B = B  A
Associative Laws:
•
•
(A  B)  C = A  (B  C )
(A  B)  C = A  (B  C) .
Distributive Laws:
•
•
(A  B)  C = (A  C)  (B  C)
(A  B)  C = (A  C)  (B  C)
• DeMorgan’s Laws:
c


c
  Ai    Ai ,
i 1
 i 1 
n
n
c
n


c
  Ai    Ai .
i 1
 i 1 
n
Pattern Classification, Chapter 1
10
Venn Diagram

A
A∩B
B
Pattern Classification, Chapter 1
11
Probability
•
•
A Probability is a number assigned to each subset (events) of a sample
space .
Probability distributions satisfy the following rules:
Pattern Classification, Chapter 1
12
Axioms of Probability
•
For any event A, 0  P(A)  1.
•
P() =1.
•
If A1, A2, … An is a partition of A, then
P(A) = P(A1) + P(A2) + ...+ P(An)
(A1, A2, … An is called a partition of A if A1  A2  … An = A and A1,
A2, … An are mutually exclusive.)
Pattern Classification, Chapter 1
13
Properties of Probability
•
For any event A, P(Ac) = 1 - P(A).
•
If A  B, then P(A)  P(B).
•
For any two events A and B,
P(A  B) = P(A) + P(B) - P(A  B).
For three events, A, B, and C,
P(ABC) = P(A) + P(B) + P(C) P(AB) - P(AC) - P(BC) + P(AB C).
Pattern Classification, Chapter 1
14
Example
•
In a certain population, 10% of the people are rich, 5% are famous,
and 3% are both rich and famous. A person is randomly selected
from this population. What is the chance that the person is
•
•
•
not rich?
rich but not famous?
either rich or famous?
Pattern Classification, Chapter 1
15
Intuitive Development
(agrees with axioms)
• Intuitively, the probability of an event a could be
defined as:
Where N(a) is the number that event a happens in n trials
Pattern Classification, Chapter 1
Here We Go Again: Not So Basic
Probability
17
More Formal:
•
•
•
 is the Sample Space:
•
Contains all possible outcomes of an experiment
w in  is a single outcome
A in  is a set of outcomes of interest
Pattern Classification, Chapter 1
18
Independence
• The probability of independent events A, B and C is
given by:
P(A,B,C) = P(A)P(B)P(C)
A and B are independent, if knowing that A has happened
does not say anything about B happening
Pattern Classification, Chapter 1
19
Bayes Theorem
• Provides a way to convert a-priori probabilities to aposteriori probabilities:
Pattern Classification, Chapter 1
20
Conditional Probability
• One of the most useful concepts!

A
B
Pattern Classification, Chapter 1
21
Bayes Theorem
• Provides a way to convert a-priori probabilities to aposteriori probabilities:
Pattern Classification, Chapter 1
22
Using Partitions:
• If events Ai are mutually exclusive and partition 
Pattern Classification, Chapter 1
23
Random Variables
• A (scalar) random variable X is a function that maps
the outcome of a random event into real scalar
values

X(w)
w
Pattern Classification, Chapter 1
24
Random Variables Distributions
• Cumulative Probability Distribution (CDF):
• Probability Density Function (PDF):
Pattern Classification, Chapter 1
25
Random Distributions:
• From the two previous equations:
Pattern Classification, Chapter 1
26
Uniform Distribution
• A R.V. X that is uniformly distributed between x1 and
x2 has density function:
X1
X2
Pattern Classification, Chapter 1
27
Gaussian (Normal) Distribution
• A R.V. X that is normally distributed has density
function:
m
Pattern Classification, Chapter 1
28
Statistical Characterizations
• Expectation (Mean Value, First Moment):
•Second Moment:
Pattern Classification, Chapter 1
29
Statistical Characterizations
• Variance of X:
• Standard Deviation of X:
Pattern Classification, Chapter 1
30
Mean Estimation from Samples
• Given a set of N samples from a distribution, we can
estimate the mean of the distribution by:
Pattern Classification, Chapter 1
31
Variance Estimation from Samples
• Given a set of N samples from a distribution, we can
estimate the variance of the distribution by:
Pattern Classification, Chapter 1
Pattern
Classification
Chapter 1: Introduction to Pattern
Recognition (Sections 1.1-1.6)
• Machine Perception
• An Example
• Pattern Recognition Systems
• The Design Cycle
• Learning and Adaptation
• Conclusion
34
Machine Perception
• Build a machine that can recognize patterns:
• Speech recognition
• Fingerprint identification
• OCR (Optical Character Recognition)
• DNA sequence identification
Pattern Classification, Chapter 1
35
An Example
• “Sorting incoming Fish on a conveyor according to
species using optical sensing”
Sea bass
Species
Salmon
Pattern Classification, Chapter 1
36
• Problem Analysis
• Set up a camera and take some sample images to extract
features
• Length
• Lightness
• Width
• Number and shape of fins
• Position of the mouth, etc…
• This is the set of all suggested features to explore for use in our
classifier!
Pattern Classification, Chapter 1
37
•
Preprocessing
• Use a segmentation operation to isolate fishes from one
another and from the background
• Information from a single fish is sent to a feature
extractor whose purpose is to reduce the data by
measuring certain features
• The features are passed to a classifier
Pattern Classification, Chapter 1
38
Pattern Classification, Chapter 1
39
• Classification
• Select the length of the fish as a possible feature for
discrimination
Pattern Classification, Chapter 1
40
Pattern Classification, Chapter 1
41
The length is a poor feature alone!
Select the lightness as a possible feature.
Pattern Classification, Chapter 1
42
Pattern Classification, Chapter 1
43
• Threshold decision boundary and cost relationship
• Move our decision boundary toward smaller values of
lightness in order to minimize the cost (reduce the number
of sea bass that are classified salmon!)
Task of decision theory
Pattern Classification, Chapter 1
44
• Adopt the lightness and add the width of the fish
Fish
xT = [x1, x2]
Lightness
Width
Pattern Classification, Chapter 1
45
Pattern Classification, Chapter 1
46
• We might add other features that are not correlated
with the ones we already have. A precaution should be
taken not to reduce the performance by adding “noisy
features”
• Ideally, the best decision boundary should be the one
which provides an optimal performance such as in the
following figure:
Pattern Classification, Chapter 1
47
Pattern Classification, Chapter 1
48
• However, our satisfaction is premature because
the central aim of designing a classifier is to
correctly classify novel input
Issue of generalization!
Pattern Classification, Chapter 1
49
Pattern Classification, Chapter 1