Download Probability and Information Theory

Probability and Information
A brief review
Copyright, 1996 © Dale Carnegie & Associates, Inc.
Probability
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Probability provides a way of summarizing
uncertainty that comes from our laziness and
ignorance - how wonderful it is!
Probability, belief of the truth of a sentence
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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
Data Mining -- Probability
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All probability statements must indicate the
evidence wrt which the probability is being
assessed.
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Prior or unconditional probability
Posterior or conditional probability
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Basic probability notation
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Prior probability
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Proposition: P(Sunny)
Random variable: P(Weather=Sunny)
Each Random Variable has a domain
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Sunny, Cloudy, Rain, Snow
Probability distribution P(Weather) = <.7,.2,.08,.02>
A random variable is not a number; a number
may be obtained by observing a RV.
A random variable can be continuous or discrete
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Conditional Probability
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Definition
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Product rule
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P(A|B) = P(A^B)/P(B)
P(A^B) = P(A|B)P(B)
Probabilistic inference does not work like
logical inference.
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The axioms of probability
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All probabilities are between 0 and 1
Necessarily true (valid) propositions have
probability 1; false (unsatisfiable) have 0
The probability of a disjunction
P(AvB)=P(A)+P(B)-P(A^B)
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The joint probability distribution
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Joint completely specifies 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.
Marginalization rule for RV Y and Z:
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P(Y) = ΣP(Y,z) over z
Let’s see an example next.
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Joint Probability
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An example of two Boolean variables
Toothache
Cavity
!Cavity
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0.04
0.01
!Toothache
0.06
0.89
Observations: mutually exclusive and collectively exhaustive
What are
P(Cavity) =
P(Cavity V Toothache) =
P(Cavity ^ Toothache) =
P(Cavity|Toothache) =
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Bayes’ rule
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Deriving the rule via the product rule
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P(B|A) = P(A|B)P(B)/P(A)
P(A) can be viewed as a normalization factor that
makes P(B|A) + (!B|A) = 1
P(A) = P(A|B)P(B)+P(A|!B)P(!B)
A more general case is
P(X|Y) = P(Y|X)P(X)/P(Y)
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Bayes’ rule conditionalized on evidence E
P(X|Y,E) = P(Y|X,E)P(X|E)/P(Y|E)
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Independence
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Independent events A, B
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P(B|A)=P(B),
P(A|B)=P(A),
P(A,B)=P(A)P(B)
Conditional independence
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P(X|Y,Z)=P(X|Z) – given Z, X and Y are independent
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Entropy
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Entropy measures homogeneity/purity of sets of
examples
Or as information content: the less you need to
know (to determine class of new case), the more
information you have
With two classes (P,N) in S, p & n instances; let
t=p+n. View [p, n] as class distribution of S.
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Entropy(S) = - (p/t) log2 (p/t) - (n/t) log2 (n/t)
E.g., p=9, n=5; Entropy(S) = Entropy([9,5]) = - (9/14)
log2 (9/14) - (5/14) log2 (5/14) = 0.940
E.g., Entropy([14,0])=0; Entropy([7,7])=1
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Entropy curve
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For p/(p+n) between 0 & 1,
the 2-class entropy is
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0 when p/(p+n) is 0
1 when p/(p+n) is 0.5
0 when p/(p+n) is 1
monotonically increasing
between 0 and 0.5
monotonically decreasing
between 0.5 and 1
When the data is pure, only
need to send 1 bit
1
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0.5
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