1. Probability and Information
A brief review
Copyright, 1996 © Dale Carnegie & Associates, Inc.

2. CSE 572, CBS572: Data Mining by H. Liu
Probability
Probability provides a way of summarizing uncertainty that comes from our laziness and ignorance!
Probability, belief of the truth of a sentence
1 - true, 0 - false, 0<P<1 - intermediate degrees of belief in the truth of the sentence
Degree of truth (fuzzy logic) vs. degree of belief
11/19/2018
CSE 572, CBS572: Data Mining by H. Liu

3. CSE 572, CBS572: Data Mining by H. Liu
All probability statements must indicate the evidence with respect to which the probability is being assessed.
Prior or unconditional probability
Posterior or conditional probability
11/19/2018
CSE 572, CBS572: Data Mining by H. Liu

4. Basic probability notation
Prior probability
Proposition - P(Sunny)
Random variable - P(Weather=Sunny)
Each Random Variable has a domain
Probability distribution P(weather) = <.7,.2,.08,.02>
Conditional probability
P(A|B) = P(A^B)/P(B)
Product rule - P(A^B) = P(A|B)P(B)
Probabilistic inference does not work like logical inference.
11/19/2018
CSE 572, CBS572: Data Mining by H. Liu

5. The axioms of probability
All probabilities are between 0 and 1
Necessarily true (valid) propositions have probability 1, false (unsatisfiable) 0
The probability of a disjunction
P(AvB)=P(A)+P(B)-P(A^B)
11/19/2018
CSE 572, CBS572: Data Mining by H. Liu

6. The joint probability distribution
Joint completely specifies an agent’s probability assignments to all propositions in the domain
A probabilistic model consists of a set of random variables (X1, …,Xn).
An atomic event is an assignment of particular values to all the variables.
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CSE 572, CBS572: Data Mining by H. Liu

7. CSE 572, CBS572: Data Mining by H. Liu
Joint
An example of two Boolean variables
Observations: mutually exclusive and
collectively exhaustive
What are P(Cavity), P(Cavity v Toothache),
P(Cavity|Toothache)?
Impractical to specify all the entries for the Joint over n Boolean variable.
If there is a Joint, we can read off any probability we need via marginalization.
Sidestep the Joint and work directly with conditional probability using Bayes rule
P(Cavity) =
P(Cavity or Toothache) = P(C) + P(T) - P(CT)
P(C|T) =P(CT)/P(T)
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CSE 572, CBS572: Data Mining by H. Liu

8. CSE 572, CBS572: Data Mining by H. Liu
Bayes’ rule
Deriving the rule via the product rule
P(B|A) = P(A|B)P(B)/P(A)
A more general case is
P(X|Y) = P(Y|X)P(X)/P(Y)
Bayes’ rule conditionalized on evidence E
P(X|Y,E) = P(Y|X,E)P(X|E)/P(Y|E)
Can you prove the above?
Applying the rule to medical diagnosis
page 426, meningitis P(M=1/50,000), stiff neck P(S)=1/20,
M causes S P(S|M) = 0.5, what is P(M|S)?
Relative likelihood
Comparing the relative likelihood of meningitis and whiplash, given a stiff neck
P(M|S)/P(W|S) = P(S|M)P(M)/P(S|W)P(W)
Avoiding direct assessment of the prior
P(M|S) + P(!M|S) = 1, P(S) = ? Can be solved by using product rules
Normalization - P(Y|X)=P(X|Y)P(Y)
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CSE 572, CBS572: Data Mining by H. Liu

9. CSE 572, CBS572: Data Mining by H. Liu
Independence
Independent events A, B
P(B|A)=P(B),
P(A|B)=P(A),
P(A,B)=P(A)P(B) – is it true in general?
Conditional independence
P(X|Y,Z)=P(X|Z)
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CSE 572, CBS572: Data Mining by H. Liu

10. CSE 572, CBS572: Data Mining by H. Liu
Information & Entropy
Entropy measures homogeneity of a collection of examples
In information theory, defined as the average number of bits needed to encode the class of an arbitrary example
With two classes in S, P and N, with p & n instances resp.; let t = p+n.
I(S) = - (p/t) log2 (p/t) - (n/t) log2 (n/t)
E.g., p=9, n=5, S=[9,5],
I(S) = - (9/14) log2 (9/14) - (5/14) log2 (5/14) = 0.940
I([14,0])=0; I([7,7])=1
In general, I([s1,s2,…,sk])= -  (si/s) log2 (si/s)
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CSE 572, CBS572: Data Mining by H. Liu

11. CSE 572, CBS572: Data Mining by H. Liu
Entropy curve
For p/(p+n) between 0 & 1, the 2-class entropy is
0 when p/(p+n) is 0
monotonically increasing between 0 and 0.5
1 when p/(p+n) is 0.5
monotonically decreasing between 0.5 and 1
0 when p/(p+n) is 1
When the data is pure, no need to send any bits
11/19/2018
CSE 572, CBS572: Data Mining by H. Liu