Download P(TP, C)

Advanced Artificial Intelligence
Lecture 4A: Probability Theory Review
Outline
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Axioms of Probability
Product and chain rules
Bayes Theorem
Random variables
PDFs and CDFs
Expected value and variance
Introduction
• Sample space
- set of all possible
outcomes of a random experiment
– Dice roll: {1, 2, 3, 4, 5, 6}
– Coin toss: {Tails, Heads}
• Event space
sample space
- subsets of elements in a
– Dice roll: {1, 2, 3} or {2, 4, 6}
– Coin toss: {Tails}
• 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑃:
• Axioms of Probability
→ ℝ
– 0 ≤ 𝑃 𝐴 ≤ 1, for all 𝐴 ∈
– 𝑃(Ω) = 1
• 𝑃 𝐴 =
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 ∈ 𝐴
lim𝑛→∞
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
examples
• Coin flip
– P(H)
– P(T)
– P(H,H,H)
– P(x1=x2=x3=x4)
– P({x1,x2,x3,x4} contains more than 3 heads)
Set operations
• Let 𝐴 = {1, 2, 3} and B = {2, 4, 6}
• 𝐴 ∩ 𝐵 = 2 and 𝐴 ∪ 𝐵 = 1, 2, 3, 4, 6
• 𝐴 − 𝐵 = 1, 3
• Properties:
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𝑃 𝐴 ∩ 𝐵 ≤ min 𝑃 𝐴 , 𝑃 𝐵
𝑃 𝐴∪𝐵 ≤𝑃 𝐴 +𝑃 𝐵
𝑃 Ω − 𝐴 = 1 − 𝑃(𝐴)
If 𝐴 ⊆ 𝐵 then 𝑃 𝐴 ≤ 𝑃(𝐵)
Conditional Probability
• 𝑃 𝐴|𝐵 =
𝑃 𝐴∩𝐵
𝑃 𝐵
=
𝑃 𝐴, 𝐵
𝑃 𝐵
• A and B are independent if
• 𝑃 𝐴|𝐵 = 𝑃(𝐴)
• A and B are conditionally independent given C if
• 𝑃 𝐴, 𝐵 𝐶) = 𝑃 𝐴 𝐶)𝑃 𝐵 𝐶)
Conditional Probability
• Joint probability:
• 𝑃 𝐴1 𝐴2 , … , 𝐴𝑛 ) =
𝑃 𝐴1 ,…,𝐴𝑛
𝑃 𝐴2 ,…,𝐴𝑛
• Product rule:
• 𝑃 𝐴1 , … , 𝐴𝑛 = 𝑃 𝐴1 𝐴2 , … , 𝐴𝑛 )𝑃 𝐴2 , … , 𝐴𝑛
• Chain rule of probability:
• 𝑃 𝐴1 , … , 𝐴𝑛 =
𝑛
𝑖=1 𝑃
𝐴𝑖 𝐴1 , 𝐴2 , … 𝐴𝑖−1 )
examples
• Coin flip
– P(x1=H)=1/2
– P(x2=H|x1=H)=0.9
– P(x2=T|x1=T)=0.8
– P(x2=H)=?
Conditional Probability
• 𝑃 𝐴|𝐵 − probability of A given B
• Example:
– 𝑃(𝐴) probability that school is closed
– 𝑃 𝐵 probability that it snows
– 𝑃 𝐴|𝐵 probability of school closing if it snows
– 𝑃 𝐵|𝐴 probability of snowing if school is closed
Conditional Probability
• 𝑃 𝐴|𝐵 − probability of A given B
• Example:
– 𝑃(𝐴) probability that school is closed
– 𝑃 𝐵 probability that it snows
P(A, B)
0.005
P(B)
0.02
P(A|B)
0.25
𝑃 𝐴|𝐵 =
𝑃 𝐴∩𝐵
𝑃 𝐵
Quiz
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P(D1=sunny)=0.9
P(D2=sunny|D1=sunny)=0.8
P(D2=rainy|D1=sunny)=?
P(D2=sunny|D1=rainy)=0.6
P(D2=rainy|D1=rainy)=?
P(D2=sunny)=?
P(D3=sunny)=?
Joint Probability
• Multiple events: cancer, test result
P(has cancer, test positive)
Has cancer?
Test positive?
P(C,TP)
yes
yes
0.018
yes
no
0.002
no
yes
0.196
no
no
0.784
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Joint Probability
• The problem with joint distributions
It takes 2D-1 numbers to specify them!
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Conditional Probability
• Describes the cancer test:
P(test positive | has cancer) = 0.9
P(test positive | Øhas cancer) = 0.2
• Put this together with: Prior probability
P(has cancer) = 0.02
P(test negative | has cancer) = 0.1
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Conditional Probability
P(C) = 0.02
• We have:
P(ØC) = 0.98
P(TP | C) = 0.9 P(ØTP | C) = 0.1
P(TP |ØC) = 0.2 P(ØTP |ØC) = 0.8
• We can now calculate joint probabilities
Has cancer?
cancer?
Has
Test positive?
positive?
Test
P(TP, C)
P(TP,
C)
yes
yes
yes
yes
0.018
yes
yes
no
0.002
no
yes
yes
0.196
no
no
0.784
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Conditional Probability
• “Diagnostic” question: How likely do is cancer
given a positive test?
P(has cancer | test positive) = ?
Has cancer?
Test positive?
P(TP, C)
yes
yes
0.018
yes
no
0.002
no
yes
0.196
no
no
0.784
P(C | TP) = P(C, TP) / P(TP) = 0.018 / 0.214 = 0.084
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Bayes Theorem
• We can relate P(A|B) and P(B|A) through
Bayes’ rule:
𝑃 𝐵 𝐴)𝑃(𝐴)
𝑃 𝐴 𝐵 =
𝑃(𝐵)
Bayes Theorem
• We can relate P(A|B) and P(B|A) through
Bayes’ rule:
𝑃 𝐵 𝐴)𝑃(𝐴)
𝑃 𝐴 𝐵 =
𝑃(𝐵)
Posterior
Probability
Likelihood
Normalizing
Constant
Prior
Probability
Bayes Theorem
• We can relate P(A|B) and P(B|A) through
Bayes’ rule:
𝑃 𝐵 𝐴)𝑃(𝐴)
𝑃 𝐴 𝐵 =
𝑃(𝐵)
• 𝑃 𝐵 can be eliminated with law of total
probability:
𝑃 𝐵 =
𝑗𝑃
𝐵 𝐴𝑗 𝑃(𝐴𝑗 )
Random Variables
• Random variable X is a function s.t.
𝑋∶ Ω → ℝ
• Examples:
– Russian roulette: X = 1 if gun fires
and X = 0 otherwise
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𝑃 𝑋 = 1 = and 𝑃 𝑋 = 0 =
– X = # of heads in 10 coin tosses
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Cumulative Distribution Functions
• Defined as 𝐹𝑋 ∶ ℝ → 0, 1 such that 𝐹𝑋 𝑥 =
𝑃(𝑋 ≤ 𝑥)
• Properties:
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0 ≤ 𝐹𝑋 𝑥 ≤ 1
lim𝑥→−∞ 𝐹𝑋 𝑥 = 0
lim𝑥→∞ 𝐹𝑋 𝑥 = 1
𝑥 ≤ 𝑦 ⇒ 𝐹𝑋 𝑥 ≤ 𝐹𝑋 𝑦
Probability Density Functions
• For discrete random variables, defined as:
• 𝑓 𝑥𝑗 = 𝑃 𝑥𝑗−1 < 𝑋 ≤ 𝑥𝑗 = 𝑃(𝑥𝑗 )
• Relates to CDFs:
• 𝐹𝑋 𝑥 =
𝑥𝑗 ≤𝑥 𝑓(𝑥𝑗 )
• 𝑓(𝑥𝑗 ) = 𝐹𝑋 𝑥𝑗 − 𝐹𝑋 𝑥𝑗−1
Probability Density Functions
• For continuous random variables, defined as:
• 𝑓 𝑥𝑗 △ 𝑥 ≈ 𝑃 𝑥 < 𝑋 ≤ 𝑥 +△ 𝑥
• Relates to CDFs:
• 𝐹𝑋 𝑥 =
• 𝑓𝑋 𝑥 =
𝑥
𝑓
−∞ 𝑋
𝑑𝐹𝑋 𝑥
𝑑𝑥
𝑥 𝑑𝑥
Probability Density Functions
• Example:
• Let X be the angular position of freely spinning pointer on
a circle
• 𝑓(𝑥)
Probability Density Functions
• Example:
• Let X be the angular position of freely spinning pointer on
a circle
f(X)
• 𝑓 𝑥 =
1
2π
1/2π
X
2π
Probability Density Functions
• Example:
• Let X be the angular position of freely spinning pointer on
a circle
f(X)
• 𝑓 𝑥 =
1
2π
1/2π
• 𝐹 𝑥
X
2π
Probability Density Functions
• Example:
• Let X be the angular position of freely spinning pointer on
a circle
f(x)
1
2π
1
=
𝑥
2π
• 𝑓 𝑥 =
• 𝐹 𝑥
1/2π
for 0 ≤ 𝑥 ≤ 2π
x
• 𝑃 0 ≤ 𝑥 ≤ π/3
2π
F(x)
1
x
2π
Probability Density Functions
• Example:
• Let X be the angular position of freely spinning pointer on
a circle
f(x)
1
2π
1
=
𝑥
2π
• 𝑓 𝑥 =
• 𝐹 𝑥
1/2π
for 0 ≤ 𝑥 ≤ 2π
• 𝑃 0 ≤ 𝑥 ≤ π/3 =
1
6
x
2π
F(x)
1
x
2π
Expectation
• Given a discrete r.v. X and a function 𝑔 ∶ ℝ → ℝ,
𝐸𝑔 𝑋
=
𝑔 𝑥 𝑓𝑋 (𝑥)
𝑥∈𝑉𝑎𝑙(𝑋)
• For a continuous r.v. X and a function 𝑔 ∶ ℝ → ℝ,
∞
𝐸𝑔 𝑋
=
𝑔(𝑥)𝑓𝑋 𝑥 𝑑𝑥
−∞
Expectation
• Expectation of a r.v. X is known as the first
moment, or the mean value, of that variable >X = 𝑔 𝑋
• Properties:
• 𝐸 𝑎 = 𝑎 for any constant 𝑎 ∈ ℝ
• 𝐸 𝑎𝑓(𝑋) = 𝑎𝐸[𝑓 𝑋 ] for any constant 𝑎 ∈ ℝ
• 𝐸 𝑓 𝑋 + 𝑔(𝑋) = 𝐸 𝑓(𝑋) + 𝐸 𝑔(𝑋)
Variance
• For a random variable X, define:
𝑉𝑎𝑟 𝑋 = 𝐸
𝑋 −𝐸 𝑋
• Another common form:
𝑉𝑎𝑟 𝑋 = 𝐸 𝑋 2 − 𝐸 𝑋
• Properties:
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• 𝑉𝑎𝑟 𝑎 = 0 for any constant 𝑎 ∈ ℝ
• 𝑉𝑎𝑟 𝑎𝑓(𝑋) = 𝑎2 𝑉𝑎𝑟[𝑓 𝑋 ] for any constant 𝑎 ∈ ℝ
Gaussian Distributions
• Let 𝑋 ~ 𝑁𝑜𝑟𝑚𝑎𝑙 μ, σ2 , then
1
1
− 2 𝑥−μ 2
𝑓 𝑥 =
𝑒𝑥𝑝 2σ
2𝜋σ
where μrepresents the mean and σ2 the variance
• Occurs naturally in many phenomena
– Noise/error
– Central limit theorem