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Unit 1Probability ReviewWang, Yuan-Kai, 王元凱
[email protected]://www.ykwang.tw
Department of Electrical Engineering, Fu Jen Univ.輔仁大學電機工程系
2006~2011
Bayesian Networks
Fu Jen University Department of Electronic Engineering Yuan-Kai Wang Copyright
Reference this document as: Wang, Yuan-Kai, “Probability Review," Lecture Notes of Wang, Yuan-Kai,
Fu Jen University, Taiwan, 2008.
王元凱 Unit - Probability Review p.
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Goal of this Unit Review basic concepts of
probability in terms of Image processing terminologies
2
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Related Units Next units Statistics Review Uncertainty Inference (Discrete) Uncertainty Inference (Continuous)
3
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Self-Study Probability Theory textbook Artificial Intelligence: a modern
approach Russell & Norvig, 2nd, Prentice Hall,
2003. pp.462~474, Chapter 13, Sec. 13.1~13.3
4
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Contents1. Probability ......................................... 62. Random Variable .............................. 93. Probability Distribution .................... 194. Joint Distribution .............................. 265. Conditional Probability .................... 466. Bayes Theorem ................................. 597. Summary …………………………….. 648. References ………………………….. 67
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1. ProbabilityWe write P(A) as “the fraction of
possible worlds in which A is true”Event space of all possible worlds
Its area is 1Worlds in which A is False
Worlds in which A is true
P(A) = Area ofreddish oval
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The Axioms of Probability0 P(A) 1P(True) = 1P(False) = 0P(A B) = P(A) + P(B) - P(A B)
Where do these axioms come from?
Were they “discovered”? Answers coming up later.
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Theorems from the Axioms0 P(A) 1,
P(True) = 1, P(False) = 0
P(A B) = P(A) + P(B) - P(A B)From these we can prove:P(not A) = P(~A) = 1-P(A)P(A) = P(A B) + P(A ~B)
How?
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2. Random VariableA variable with randomnessprobability, degree of belief
An exampleRain : a random variableIts possible values: true, falseIts randomness
P(Rain=true)=0.8, P(Rain=false)=0.2
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Types of Random VariableBoolean random variableRain : true, false
Discrete random variable(Multivalued R.V.)Rain: cloudy, sunny, drizzle,
drenchContinuous random variableRain: rainfall in millimeter
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Discrete R.V. (1/4)Let A=Rain A can take on more than 2 valuesA is a random variable with arity k
if it can take on exactly one value out of {v1,v2, .. vk}Thus
jivAvAP ji if 0)(1)( 21 kvAvAvAP
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Discrete R.V. (2/4)
Using the axioms of probability…0 P(A) 1, P(True) = 1, P(False) = 0P(A B) = P(A) + P(B) - P(A B)
And assume that A obeysjivAvAP ji if 0)(
1)( 21 kvAvAvAP
)()(1
21
i
jji vAPvAvAvAP
Prove it?
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Discrete R.V. (3/4)
Using the axioms of probability…0 P(A) 1, P(True) = 1, P(False) = 0P(A or B) = P(A) + P(B) - P(A and B)
And assuming that A obeys…jivAvAP ji if 0)(
1)( 21 kvAvAvAP
)()(1
21
i
jji vAPvAvAvAP
1)(1
k
jjvAP
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Discrete R.V. (4/4)
Using the axioms of probability…0 P(A) 1, P(True) = 1, P(False) = 0P(A or B) = P(A) + P(B) - P(A and B)
And assuming that A obeys…
)(])[(1
21
i
jji vABPvAvAvABP
jivAvAP ji if 0)(1)( 21 kvAvAvAP
)()(1
k
jjvABPBP
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Random Vector A random vector is a vector of
random variables X = (X1, X2, ..., XN) X1 ... XN are random variables
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Example in Image Processing For a gray-level image,
random variable is “gray level”• Random variable X
(Gray level) has npossible values {x1, x2, ..., xn}, n=256
• N random data x1, x2, .., xN of X, N=Width*Height
Discrete v.s. continuous ?
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Example in Image Processing For a color image,
random vector is (r,g,b)
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Example in Computer Vision Random vector is a vector of
“features” Face recognition
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3. Probability Distribution Probability distribution is a set of
probabilities Boolean R.V. P(A): P(A=true)=0.2, P(A=false)=0.8
Discrete R.V. P(A): P(A=v1)=0.2, P(A=v2)=0.4,
P(A=v3)=0.4, P(A=v4)=0.1, Continuous R.V.
Usually we plot the probability distribution as a function(Probability Distribution Function, pdf)
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Probability Distribution (1/3)Continuous R.V.
Rain (mm)0 100 200 300 400 500 600
0.80.60.40.2
0
• P(Rain) is a probability density function (pdf)
• P(Rain=200) is a probability
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Probability Distribution(2/3)Boolean R.V.P(Rain) is a probability distributionP(Rain=false) is a probability
Rain
P(Rain)
false true
0.8
0.2
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Probability Distribution(3/3)Discrete R.V.P(Rain) is a probability distributionP(Rain=drizzle) is a probability
Rain
P(Rain)
0.8
0.2
clou
dy
sunn
y
driz
zle
dren
ch
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Common Probability Distributions
Normal distribution (Gaussian)Parameters: , Discussed in Section 3
Uniform distributionExponential distributionPoisson distribution...
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The Uniform Distribution
-w/2 0 w/2
1/w
0][ XE12
]Var[2wX
2||if0
2||if1
)( wx
wxwxp
X
P(x)
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Example in Image Processing
• Random variable X (Gray level) has n possible values {x1, x2, ..., xn}, n=256
• Distribution P(xi) of the image•Is not a usual p.d.f.
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4. Joint Distribution When there are more than one
random variables: X, Y We want to know "the probabilities
of both random variables" Joint probability distribution P(X Y) P(X Y ...)
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An ExampleA set of random variables for the
typhoon worldRain, Wind, Speed, ...Every random variable Describes one facet of typhoonHas a probability distribution
A typhoon is a situation of joint probabilityP(Rain=200mm Wind=South
Speed=100km/hr) = 0.4
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P(X,Y) with X Y are Discrete P(X,Y) : (M x N) entries
P(X=x1,Y=y1) P(X=x2,Y=y1) P(X=x3,Y=y1)
P(X=x1,Y=y2) P(X=x2,Y=y2) P(X=x3,Y=y2)
X
Y
x1 x2 x3
y1
y2
0.2 0.1 0.10.1 0.2 0.3
X
Y
x1 x2 x3
y1
y2
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Atomic EventAtomic event is aCombination of values of all random
variablesSituation (state) of typhoon
(statistical world)For the typhoon example(Rain=200mm Wind=South
Speed=100km/hr) is an atomic event(Rain=200mm Wind=South) is not
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Joint Probability DistributionP(Rain=200mm Wind=South
Speed=100km/hr) is a joint probabilityP(Rain Wind Speed) is a probability
of all joint probabilitiesJoint probability function
A set of random variables can have mixed typesEx: Rain: Boolean, Wind: Discrete,
Speed: ContinuousWe will consider all-discrete case
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Prior and Joint Probability (Math)
Let {X1,...,Xn} be a set of random variablesP(Xi) is a probability functionP(Xi=xi) is a prior probabilityP(X1 X2 Xn) is a joint
probability functionP(X1=x1 X2=x2 Xn=xn) is
a joint probability
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For Boolean & Discrete R.V.Let X be a single R.V.P(X) is a vector
Boolean R.V.Rain: true, falseP(Rain) = <0.72, 0.28>
Discrete R.V.Rain: cloudy, sunny, drizzle, drenchP(Rain) = <0.72, 0.1, 0.08, 0.1>Normalized, i.e. sums to 1
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Full Joint Distribution (1/3)
For a set of random variables {X1, X2 , , Xn}
X1 X2 Xn are atomic eventsP(X1 X2 Xn) is A full joint probability distribution
(FJD)A table of all joint prob. of all atomic
events, if {X1, , Xn} are discrete
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Full Joint Distribution (2/3)Ex: X1: Rain, X2: WindX1: drizzle, drench, cloudy, X2: strong, weakX1 X2 are atomic eventsP(X1 X2) is a 3x2 matrix of values
RainWind Drizzle Drench Cloudy
Strong 0.15 0.12 0.06Weak 0.55 0.08 0.04
P(X1=drizzle X2=strong) = 0.15, ...
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Full Joint Distribution (3/3)
All questions about probability of joint events can be answered by the tableP(Wind=Strong),
P(Rain=Drizzle Wind=Strong), ...
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An Example with 3 Boolean R.V. (1/2)
1. Make a truth table listing all combinations of values of your variables• if there are M
Boolean variables then the table will have 2M rows
Boolean variables A, B, CA B C0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1
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An Example with 3 Boolean R.V. (2/2)
2. For each combination of values, say how probable it is
A B C Prob0 0 0 0.300 0 1 0.050 1 0 0.100 1 1 0.051 0 0 0.051 0 1 0.101 1 0 0.251 1 1 0.10
Sum 1.0
A
B
C0.050.25
0.10 0.050.05
0.10
0.100.30
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Using the FJD • Once you have the FJD • You can ask for the probability of
any logical expression: Inference
E
PEP matching rows
)row()(
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Examples of Inference (1/3)
P(Poor Male) = 0.4654 E
PEP matching rows
)row()(
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Examples of Inference (2/3)
P(Poor) = 0.7604 E
PEP matching rows
)row()(
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Examples of Inference (3/3)
2
2 1
matching rows
and matching rows
2
2121 )row(
)row(
)()()|(
E
EE
P
P
EPEEPEEP
P(Male | Poor) = 0.4654 / 0.7604 = 0.612
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Inference is a Big DealI’ve got this evidenceWhat’s the chance that this
conclusion is true?I’ve got a sore neck: how likely am
I to have meningitis?I see my lights are out and it’s
9pm. What’s the chance my spouse is already asleep?
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Where do Full Joint Distributions Come from?
(1/2)Idea One: Expert HumansIdea Two: Learn them from data!
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Where do FJD Come from? (2/2)
Build a JD table for your attributes in which the probabilities are unspecified
Then fill in each row with
records ofnumber totalrow matching records)row(ˆ P
A B C Prob0 0 0 ?0 0 1 ?0 1 0 ?0 1 1 ?1 0 0 ?1 0 1 ?1 1 0 ?1 1 1 ?
A B C Prob0 0 0 0.300 0 1 0.050 1 0 0.100 1 1 0.051 0 0 0.051 0 1 0.101 1 0 0.25
1 1 1 0.10Fraction of all records in whichA and B are True but C is False
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Example of Learning a FJDThis Joint was obtained by
learning from three attributes in the UCI “Adult” Census Database [Kohavi 1995]
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5. Conditional ProbabilityP(A|B) = Fraction of worlds in which B
is true that also have A true
F
H
H = “Have a headache”F = “Coming down with Flu”
P(H) = 1/10P(F) = 1/40P(H|F) = 1/2
“Headaches are rare and flu is rarer, but if you’re coming down with ‘flu there’s a 50-50 chance you’ll have a headache”
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Formula
F
H
H = “Have a headache”F = “Coming down with Flu”
P(H) = 1/10P(F) = 1/40P(H|F) = 1/2
)()()|(
FPFHPFHP
P(H|F) v.s. P(HF)
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Conditional, Joint, PriorRelationship among conditional,
joint, and prior probabilities
P(X1) P(X1 X2)P(X2)
)()()|(
2
2121 XP
XXPXXP
)()|()( 22121 XPXXPXXP
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Posterior v.s. Prior Probabilities
P(Cavity|Toothache)Conditional probabilityPosterior probability
(after the fact/evidence)P(Cavity)Prior probability (the fact)
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An Example (1/2)For a dental diagnosisLet {Cavity,Toothache} be a set
of Boolean random variablesDenotations for Boolean R.V.P(Cavity=true) = P(cavity)P(Cavity=false) = P(cavity)
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An Example (2/2)The full joint probability
distribution P(Toothache Cavity)
cavity 0.04 0.06cavity 0.01 0.89
toothache toothache
11.006.001.004.0)( toothachecavityP
80.001.004.0
04.0)(
)()|(
toothacheP
toothachecavityPtoothachecavityP
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Conditional Probability (Math)P(X1=x1i| X2 =x2j) is a conditional
probabilityP(X1 | X2) is a conditional
distribution functionAll P(X1=x1i| X2 =x2j) for all possible i, j
For Boolean & discrete R.V.s, conditional distribution function is a conditional probability tableConditional distribution of continuous
R.V. will not used in our discussions
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Conditional Probability Table (CPT)
For the dental diagnosis problem,Toothache & Cavity are Boolean R.V.sP(Toothache|Cavity) is a CPT
Cavity P(toothache|Cavity)T 0.90F 0.05
Cavity P(toothache|Cavity) P(toothache|Cavity)T 0.90 0.1F 0.05 0.95
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CPT v.s. FJD
Cavity P(toothache|Cavity) P(toothache|Cavity)T 0.90 0.1F 0.05 0.95
cavity 0.04 0.06cavity 0.01 0.89
toothache toothache
Sum of all atomic events = 1
Sum of a row = 1
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More Than 2 Random Variables in
Conditional ProbabilityJoint conditional probability
)()(
)|(
43
4321
4321
XXPXXXXP
XXXXP
)|()|( 4324321 XXXPXXXXP
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Chain Rule
)()()|(
2
2121 XP
XXPXXP
)()|()|()|()()|(
))(()(
4434324321
4324321
4321
4321
XPXXPXXXPXXXXPXXXPXXXXP
XXXXPXXXXP
)|()( 11
21 nii
n
in XXXPXXXP
• Joint probability can be calculated from a chain of conditional probability
)()|()( 22121 XPXXPXXP
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Useful Easy-to-prove Facts
1)|()|( BAPBAP
1)|(1
An
kk BvAP
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Where Are We?We have recalled the fundamentals
of probabilityWe know what JDs are and how to
use themNext two sectionsBayes ruleGaussian distribution
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6. Bayes Theorem
)()|()()|( APABPBPBAP
)()|()( BPBAPBAP
)()|()( APABPABP
Bayes, Thomas (1763) An essay towards solving a problem in the doctrine of chances. Philosophical Transactions of the Royal Society of London, 53:370-418
)()(
)|()|(
BPAP
ABPBAP
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Useful Forms of Bayes Rule
)()()|()|(
BPAPABPBAP
)()(
BPBAP
)()()|()|(
CBPCAPCABPCBAP
)|()|(),|(
),(),(),|(),|(
CBPCAPCABP
CBPCAPCABPCBAP
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More General Forms of Bayes Rule
AA n
kkk
iin
kk
ii
vAPvABP
vAPvABP
BvAP
BvAPBvAP
11)()|(
)()|(
)(
)()|(
)()()(
)()()|(
BAPBAPBAP
BPBAPBAP
If A is a Boolean R.V.
If A is a discrete R.V.
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Example –Bayesian Detection (1/2)
2-class recognition Ex.: Skin color detection
Let A be skin color pixel A be non-skin color pixel c be the color (R,G,B) of a pixel
We can get : P(c|A), P(c|A) For a pixel in a new image, is it a skin
color pixel?
)()()|()|(
cPAPAcPcAP
)()()|()|(
cPAPAcPcAP
>
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Example –Bayesian Detection (2/2)
2-class recognition Ex.: Background detection
Let A be background pixel A be non-background pixel c be the color (R,G,B) of a pixel
We can get : P(c|A), P(c|A) For a pixel in a new image, is it a
background pixel?
)()()|()|(
cPAPAcPcAP
)()()|()|(
cPAPAcPcAP
>
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7. SummaryContinuous variable
Scalar* Scalar
Function of two variables MxN matrix
Function of two variables MxN matrix
Function of one variable M vector
Function of one variable N vector
Scalar* Scalar
Discrete variable
Function of one variable M vector
P(X=x)
P(X,Y)
P(X|Y)
P(X|Y=y)
P(X=x|Y)
P(X=x|Y=y)
P(X)(X has M values, Y has N values)
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Ex (1/2)
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Ex (2/2)
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8. Reference Artificial Intelligence: a modern
approach Russell & Norvig, 2nd, Prentice Hall, 2003. Sections 13.1~13.3, pp.462~474.
統計學的世界 墨爾著,鄭惟厚譯 天下文化,2002
深入淺出統計學 D. Grifiths, 楊仁和譯,2009 O’ Reilly