The document discusses probabilistic reasoning and probabilistic models. It introduces key concepts like representing knowledge with certainty factors rather than simple logic, defining sample spaces and probability distributions, calculating marginal and conditional probabilities, and using important probabilistic inference rules like the product rule and Bayes' rule. It provides examples of modeling problems with random variables and probabilities, like determining the probability of a disease given a positive test result.

1. Probabilistic Reasoning
Tameem Ahmad
Student
M.Tech.
Z.H.C.E.T.
A.M.U.,Aligarh
Copyright, 1996 © Dale Carnegie & Associates, Inc.

2. References:
NPTEL National Programme on Technology Enhanced Learning (NPTEL) is
a Government of India sponsored collaborative educational programme. By developing
curriculum-based video and web courses the programme aims to enhance the quality of
engineering education in India. It is being jointly carried out by 7 IITS and IISc Bangalore,
and is funded by the Ministry of Human Resources Development of the Governament of
India.
Computer Based Numerical & Statistical
Techniques by M.Goyal Laxmi Publications, Ltd., 01-Jan-2008
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3. Probabilistic Reasoning
What???
• Capturing uncertain knowledge
• Probabilistic inference
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4. Knowledge representation
bird(tweety).
fly(X) :- bird(X).
?- fly(tweety).
Yes
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5. Knowledge representation (Cont.)
But the real world is not so simple like this… There
are many other factors…
A way to handle knowledge representation in real
problems is to extend logic by using certainty factors.
IF condition with certainty x THEN fact with certainty
f(x)
Replace
smoking -> lung cancer
or
lotsofconditions, smoking -> lung cancer
With
P(lung cancer | smoking) = 0.6
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6. probabilistic model
A probabilistic model describes the
world in terms of a set S of possible
states - the sample space. We don’t
know the true state of the world, so we
(somehow) come up with a probability
distribution over S which gives the
probability of any state being the true
one. The world usually described by a
set of variables or attributes.
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7. Cont…
let the random variable Sum
(representing outcome of two die
throws) be defined thus:
Sum(die1, die2) = die1 +die2
P(Sum = 2) = 1/36,
P(Sum = 3) = 2/36, . . . ,
P(Sum = 12) = 1/36
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8. Cont…
Visit to Asia? A
Tuberculosis? T
Either tub. or lung cancer? E
Lung cancer? L
Smoking? S
Bronchitis? B
Dyspnoea? D
Positive X-ray? X
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9. Cont…
Sample Space
S = {(A = f, T = f,E = f,L = f, S = f,B = f,D = f,X = f),
(A = f, T = f,E = f,L = f, S = f,B = f,D = f,X = t), . . .
………
……….
(A = t, T = t,E = t,L = t, S = t,B = t,D = t,X = t)}
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10. Cont…
Marginal Probability Distribution
For example, P(A = t,D = f)
= P(A = t, T = f,E = f,L = f, S = f,B = f,D = f,X = f)
+ P(A = t, T = f,E = f,L = f, S = f,B = f,D = f,X = t)
+ P(A = t, T = f,E = f,L = f, S = f,B = t,D = f,X = f)
+ P(A = t, T = f,E = f,L = f, S = f,B = t,D = f,X = t)
...
P(A = t, T = t,E = t,L = t, S = t,B = t,D = f,X = t)
This has 64 summands!
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11. Cont…
Conditional Probablity
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12. Knowledge engineering for
uncertain reasoning
Decide what to talk about
Decide on a vocabulary of random variables
Encode general knowledge about the
dependence
Encode a description of the specific problem
instance
Pose queries to the inference procedure and get
answers
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13. Probabilistic Inference Rules
Two rules in probability theory are important for inferencing, namely, the
product rule and the Bayes' rule.
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14. Cont…
Suppose one has been tested positive for a disease; what is the
probability that you actually have the disease?
It depends on the accuracy and sensitivity of the test, and on
the background (prior) probability of the disease.
Let P(Test=+ve | Disease=true) = 0.95 (95%),
so the true negative rate, P(Test=-ve | Disease=true), is 0.05
(5%).
Let P(Test=+ve | Disease=false) = 0.05, so the false positive
rate is also 5%.
Suppose the disease is rare: P(Disease=true) = 0.01 (1%).
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15. Cont…
Applying Bayesian
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