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1
Knowledge
Representation and
Reasoning
Chapter 10.1-10.2, 10.6
CS 63
Adapted from slides by
Tim Finin and
Marie desJardins.
Some material adopted from notes
by Andreas Geyer-Schulz,
and Chuck Dyer.
2
Abduction
• Abduction is a reasoning process that tries to form plausible
explanations for abnormal observations
– Abduction is distinctly different from deduction and induction
– Abduction is inherently uncertain
• Uncertainty is an important issue in abductive reasoning
• Some major formalisms for representing and reasoning about
uncertainty
– Mycin’s certainty factors (an early representative)
– Probability theory (esp. Bayesian belief networks)
– Dempster-Shafer theory
– Fuzzy logic
– Truth maintenance systems
– Nonmonotonic reasoning
3
Abduction
• Definition (Encyclopedia Britannica): reasoning that derives
an explanatory hypothesis from a given set of facts
– The inference result is a hypothesis that, if true, could
explain the occurrence of the given facts
• Examples
– Dendral, an expert system to construct 3D structure of
chemical compounds
• Fact: mass spectrometer data of the compound and its
chemical formula
• KB: chemistry, esp. strength of different types of bounds
• Reasoning: form a hypothetical 3D structure that satisfies the
chemical formula, and that would most likely produce the
given mass spectrum
4
– Medical diagnosis
• Facts: symptoms, lab test results, and other observed findings
(called manifestations)
• KB: causal associations between diseases and manifestations
• Reasoning: one or more diseases whose presence would
causally explain the occurrence of the given manifestations
– Many other reasoning processes (e.g., word sense
disambiguation in natural language process, image
understanding, criminal investigation) can also been seen
as abductive reasoning
Abduction examples (cont.)
5
Comparing abduction, deduction,
and induction
Deduction: major premise: All balls in the box are black
minor premise: These balls are from the box
conclusion: These balls are black
Abduction: rule: All balls in the box are black
observation: These balls are black
explanation: These balls are from the box
Induction: case: These balls are from the box
observation: These balls are black
hypothesized rule: All ball in the box are black
A => B
A
---------
B
A => B
B
-------------
Possibly A
Whenever
A then B
-------------
Possibly
A => B
Deduction reasons from causes to effects
Abduction reasons from effects to causes
Induction reasons from specific cases to general rules
6
Characteristics of abductive
reasoning
• “Conclusions” are hypotheses, not theorems (may be
false even if rules and facts are true)
– E.g., misdiagnosis in medicine
• There may be multiple plausible hypotheses
– Given rules A => B and C => B, and fact B, both A and C
are plausible hypotheses
– Abduction is inherently uncertain
– Hypotheses can be ranked by their plausibility (if it can be
determined)
7
Characteristics of abductive
reasoning (cont.)
• Reasoning is often a hypothesize-and-test cycle
– Hypothesize: Postulate possible hypotheses, any of which would
explain the given facts (or at least most of the important facts)
– Test: Test the plausibility of all or some of these hypotheses
– One way to test a hypothesis H is to ask whether something that is
currently unknown–but can be predicted from H–is actually true
• If we also know A => D and C => E, then ask if D and E are
true
• If D is true and E is false, then hypothesis A becomes more
plausible (support for A is increased; support for C is
decreased)
8
Characteristics of abductive
reasoning (cont.)
• Reasoning is non-monotonic
– That is, the plausibility of hypotheses can
increase/decrease as new facts are collected
– In contrast, deductive inference is monotonic: it never
change a sentence’s truth value, once known
– In abductive (and inductive) reasoning, some
hypotheses may be discarded, and new ones formed,
when new observations are made
9
Sources of uncertainty
• Uncertain inputs
– Missing data
– Noisy data
• Uncertain knowledge
– Multiple causes lead to multiple effects
– Incomplete enumeration of conditions or effects
– Incomplete knowledge of causality in the domain
– Probabilistic/stochastic effects
• Uncertain outputs
– Abduction and induction are inherently uncertain
– Default reasoning, even in deductive fashion, is uncertain
– Incomplete deductive inference may be uncertain
Probabilistic reasoning only gives probabilistic
results (summarizes uncertainty from various sources)
10
Decision making with uncertainty
• Rational behavior:
– For each possible action, identify the possible outcomes
– Compute the probability of each outcome
– Compute the utility of each outcome
– Compute the probability-weighted (expected) utility
over possible outcomes for each action
– Select the action with the highest expected utility
(principle of Maximum Expected Utility)
11
Bayesian reasoning
• Probability theory
• Bayesian inference
– Use probability theory and information about independence
– Reason diagnostically (from evidence (effects) to conclusions
(causes)) or causally (from causes to effects)
• Bayesian networks
– Compact representation of probability distribution over a set of
propositional random variables
– Take advantage of independence relationships
12
Other uncertainty representations
• Default reasoning
– Nonmonotonic logic: Allow the retraction of default beliefs if they
prove to be false
• Rule-based methods
– Certainty factors (Mycin): propagate simple models of belief
through causal or diagnostic rules
• Evidential reasoning
– Dempster-Shafer theory: Bel(P) is a measure of the evidence for P;
Bel(P) is a measure of the evidence against P; together they define
a belief interval (lower and upper bounds on confidence)
• Fuzzy reasoning
– Fuzzy sets: How well does an object satisfy a vague property?
– Fuzzy logic: “How true” is a logical statement?
13
Uncertainty tradeoffs
• Bayesian networks: Nice theoretical properties combined
with efficient reasoning make BNs very popular; limited
expressiveness, knowledge engineering challenges may
limit uses
• Nonmonotonic logic: Represent commonsense reasoning,
but can be computationally very expensive
• Certainty factors: Not semantically well founded
• Dempster-Shafer theory: Has nice formal properties, but
can be computationally expensive, and intervals tend to
grow towards [0,1] (not a very useful conclusion)
• Fuzzy reasoning: Semantics are unclear (fuzzy!), but has
proved very useful for commercial applications
14
Bayesian Reasoning
Chapter 13
CS 63
Adapted from slides by
Tim Finin and
Marie desJardins.
15
Outline
• Probability theory
• Bayesian inference
– From the joint distribution
– Using independence/factoring
– From sources of evidence
16
Sources of uncertainty
• Uncertain inputs
– Missing data
– Noisy data
• Uncertain knowledge
– Multiple causes lead to multiple effects
– Incomplete enumeration of conditions or effects
– Incomplete knowledge of causality in the domain
– Probabilistic/stochastic effects
• Uncertain outputs
– Abduction and induction are inherently uncertain
– Default reasoning, even in deductive fashion, is uncertain
– Incomplete deductive inference may be uncertain
Probabilistic reasoning only gives probabilistic
results (summarizes uncertainty from various sources)
17
Decision making with uncertainty
• Rational behavior:
– For each possible action, identify the possible outcomes
– Compute the probability of each outcome
– Compute the utility of each outcome
– Compute the probability-weighted (expected) utility
over possible outcomes for each action
– Select the action with the highest expected utility
(principle of Maximum Expected Utility)
18
Why probabilities anyway?
• Kolmogorov showed that three simple axioms lead to the
rules of probability theory
– De Finetti, Cox, and Carnap have also provided compelling
arguments for these axioms
1. All probabilities are between 0 and 1:
• 0 ≤ P(a) ≤ 1
2. Valid propositions (tautologies) have probability 1, and
unsatisfiable propositions have probability 0:
• P(true) = 1 ; P(false) = 0
3. The probability of a disjunction is given by:
• P(a  b) = P(a) + P(b) – P(a  b)
ab
a b
19
Probability theory
• Random variables
– Domain
• Atomic event: complete
specification of state
• Prior probability: degree
of belief without any other
evidence
• Joint probability: matrix
of combined probabilities
of a set of variables
• Alarm, Burglary, Earthquake
– Boolean (like these), discrete,
continuous
• (Alarm=True  Burglary=True 
Earthquake=False) or equivalently
(alarm  burglary  ¬earthquake)
• P(Burglary) = 0.1
• P(Alarm, Burglary) =
alarm ¬alarm
burglary 0.09 0.01
¬burglary 0.1 0.8
20
Probability theory (cont.)
• Conditional probability:
probability of effect given causes
• Computing conditional probs:
– P(a | b) = P(a  b) / P(b)
– P(b): normalizing constant
• Product rule:
– P(a  b) = P(a | b) P(b)
• Marginalizing:
– P(B) = ΣaP(B, a)
– P(B) = ΣaP(B | a) P(a)
(conditioning)
• P(burglary | alarm) = 0.47
P(alarm | burglary) = 0.9
• P(burglary | alarm) =
P(burglary  alarm) / P(alarm)
= 0.09 / 0.19 = 0.47
• P(burglary  alarm) =
P(burglary | alarm) P(alarm) =
0.47 * 0.19 = 0.09
• P(alarm) =
P(alarm  burglary) +
P(alarm  ¬burglary) =
0.09 + 0.1 = 0.19
21
Example: Inference from the joint
alarm ¬alarm
earthquake ¬earthquake earthquake ¬earthquake
burglary 0.01 0.08 0.001 0.009
¬burglary 0.01 0.09 0.01 0.79
P(Burglary | alarm) = α P(Burglary, alarm)
= α [P(Burglary, alarm, earthquake) + P(Burglary, alarm, ¬earthquake)
= α [ (0.01, 0.01) + (0.08, 0.09) ]
= α [ (0.09, 0.1) ]
Since P(burglary | alarm) + P(¬burglary | alarm) = 1, α = 1/(0.09+0.1) = 5.26
(i.e., P(alarm) = 1/α = 0.109 Quizlet: how can you verify this?)
P(burglary | alarm) = 0.09 * 5.26 = 0.474
P(¬burglary | alarm) = 0.1 * 5.26 = 0.526
22
Exercise: Inference from the joint
• Queries:
– What is the prior probability of smart?
– What is the prior probability of study?
– What is the conditional probability of prepared, given
study and smart?
• Save these answers for next time! 
p(smart 
study  prep)
smart smart
study study study study
prepared 0.432 0.16 0.084 0.008
prepared 0.048 0.16 0.036 0.072
23
Independence
• When two sets of propositions do not affect each others’
probabilities, we call them independent, and can easily
compute their joint and conditional probability:
– Independent (A, B) ↔ P(A  B) = P(A) P(B), P(A | B) = P(A)
• For example, {moon-phase, light-level} might be
independent of {burglary, alarm, earthquake}
– Then again, it might not: Burglars might be more likely to
burglarize houses when there’s a new moon (and hence little light)
– But if we know the light level, the moon phase doesn’t affect
whether we are burglarized
– Once we’re burglarized, light level doesn’t affect whether the alarm
goes off
• We need a more complex notion of independence, and
methods for reasoning about these kinds of relationships
24
Exercise: Independence
• Queries:
– Is smart independent of study?
– Is prepared independent of study?
p(smart 
study  prep)
smart smart
study study study study
prepared 0.432 0.16 0.084 0.008
prepared 0.048 0.16 0.036 0.072
25
Conditional independence
• Absolute independence:
– A and B are independent if and only if P(A  B) = P(A) P(B);
equivalently, P(A) = P(A | B) and P(B) = P(B | A)
• A and B are conditionally independent given C if and only if
– P(A  B | C) = P(A | C) P(B | C)
• This lets us decompose the joint distribution:
– P(A  B  C) = P(A | C) P(B | C) P(C)
• Moon-Phase and Burglary are conditionally independent
given Light-Level
• Conditional independence is weaker than absolute
independence, but still useful in decomposing the full joint
probability distribution
26
Exercise: Conditional independence
• Queries:
– Is smart conditionally independent of prepared, given
study?
– Is study conditionally independent of prepared, given
smart?
p(smart 
study  prep)
smart smart
study study study study
prepared 0.432 0.16 0.084 0.008
prepared 0.048 0.16 0.036 0.072
27
Bayes’s rule
• Bayes’s rule is derived from the product rule:
– P(Y | X) = P(X | Y) P(Y) / P(X)
• Often useful for diagnosis:
– If X are (observed) effects and Y are (hidden) causes,
– We may have a model for how causes lead to effects (P(X | Y))
– We may also have prior beliefs (based on experience) about the
frequency of occurrence of effects (P(Y))
– Which allows us to reason abductively from effects to causes (P(Y |
X)).
28
Bayesian inference
• In the setting of diagnostic/evidential reasoning
– Know prior probability of hypothesis
conditional probability
– Want to compute the posterior probability
• Bayes’ theorem (formula 1):
ons
anifestati
evidence/m
hypotheses
1 m
j
i
E
E
E
H
)
(
/
)
|
(
)
(
)
|
( j
i
j
i
j
i E
P
H
E
P
H
P
E
H
P 
)
( i
H
P
)
|
( i
j H
E
P
)
|
( i
j H
E
P
)
|
( j
i E
H
P
)
( i
H
P
… …
29
Simple Bayesian diagnostic reasoning
• Knowledge base:
– Evidence / manifestations: E1, …, Em
– Hypotheses / disorders: H1, …, Hn
• Ej and Hi are binary; hypotheses are mutually exclusive (non-
overlapping) and exhaustive (cover all possible cases)
– Conditional probabilities: P(Ej | Hi), i = 1, …, n; j = 1, …, m
• Cases (evidence for a particular instance): E1, …, Em
• Goal: Find the hypothesis Hi with the highest posterior
– Maxi P(Hi | E1, …, Em)
30
Bayesian diagnostic reasoning II
• Bayes’ rule says that
– P(Hi | E1, …, Em) = P(E1, …, Em | Hi) P(Hi) / P(E1, …, Em)
• Assume each piece of evidence Ei is conditionally
independent of the others, given a hypothesis Hi, then:
– P(E1, …, Em | Hi) = m
j=1 P(Ej | Hi)
• If we only care about relative probabilities for the Hi, then
we have:
– P(Hi | E1, …, Em) = α P(Hi) m
j=1 P(Ej | Hi)
31
Limitations of simple
Bayesian inference
• Cannot easily handle multi-fault situation, nor cases where
intermediate (hidden) causes exist:
– Disease D causes syndrome S, which causes correlated
manifestations M1 and M2
• Consider a composite hypothesis H1  H2, where H1 and H2
are independent. What is the relative posterior?
– P(H1  H2 | E1, …, Em) = α P(E1, …, Em | H1  H2) P(H1  H2)
= α P(E1, …, Em | H1  H2) P(H1) P(H2)
= α m
j=1 P(Ej | H1  H2) P(H1) P(H2)
• How do we compute P(Ej | H1  H2) ??
32
Limitations of simple Bayesian
inference II
• Assume H1 and H2 are independent, given E1, …, Em?
– P(H1  H2 | E1, …, Em) = P(H1 | E1, …, Em) P(H2 | E1, …, Em)
• This is a very unreasonable assumption
– Earthquake and Burglar are independent, but not given Alarm:
• P(burglar | alarm, earthquake) << P(burglar | alarm)
• Another limitation is that simple application of Bayes’s rule doesn’t
allow us to handle causal chaining:
– A: this year’s weather; B: cotton production; C: next year’s cotton price
– A influences C indirectly: A→ B → C
– P(C | B, A) = P(C | B)
• Need a richer representation to model interacting hypotheses,
conditional independence, and causal chaining
• Next time: conditional independence and Bayesian networks!

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Artificial Intelligence Bayesian Reasoning

  • 1. 1 Knowledge Representation and Reasoning Chapter 10.1-10.2, 10.6 CS 63 Adapted from slides by Tim Finin and Marie desJardins. Some material adopted from notes by Andreas Geyer-Schulz, and Chuck Dyer.
  • 2. 2 Abduction • Abduction is a reasoning process that tries to form plausible explanations for abnormal observations – Abduction is distinctly different from deduction and induction – Abduction is inherently uncertain • Uncertainty is an important issue in abductive reasoning • Some major formalisms for representing and reasoning about uncertainty – Mycin’s certainty factors (an early representative) – Probability theory (esp. Bayesian belief networks) – Dempster-Shafer theory – Fuzzy logic – Truth maintenance systems – Nonmonotonic reasoning
  • 3. 3 Abduction • Definition (Encyclopedia Britannica): reasoning that derives an explanatory hypothesis from a given set of facts – The inference result is a hypothesis that, if true, could explain the occurrence of the given facts • Examples – Dendral, an expert system to construct 3D structure of chemical compounds • Fact: mass spectrometer data of the compound and its chemical formula • KB: chemistry, esp. strength of different types of bounds • Reasoning: form a hypothetical 3D structure that satisfies the chemical formula, and that would most likely produce the given mass spectrum
  • 4. 4 – Medical diagnosis • Facts: symptoms, lab test results, and other observed findings (called manifestations) • KB: causal associations between diseases and manifestations • Reasoning: one or more diseases whose presence would causally explain the occurrence of the given manifestations – Many other reasoning processes (e.g., word sense disambiguation in natural language process, image understanding, criminal investigation) can also been seen as abductive reasoning Abduction examples (cont.)
  • 5. 5 Comparing abduction, deduction, and induction Deduction: major premise: All balls in the box are black minor premise: These balls are from the box conclusion: These balls are black Abduction: rule: All balls in the box are black observation: These balls are black explanation: These balls are from the box Induction: case: These balls are from the box observation: These balls are black hypothesized rule: All ball in the box are black A => B A --------- B A => B B ------------- Possibly A Whenever A then B ------------- Possibly A => B Deduction reasons from causes to effects Abduction reasons from effects to causes Induction reasons from specific cases to general rules
  • 6. 6 Characteristics of abductive reasoning • “Conclusions” are hypotheses, not theorems (may be false even if rules and facts are true) – E.g., misdiagnosis in medicine • There may be multiple plausible hypotheses – Given rules A => B and C => B, and fact B, both A and C are plausible hypotheses – Abduction is inherently uncertain – Hypotheses can be ranked by their plausibility (if it can be determined)
  • 7. 7 Characteristics of abductive reasoning (cont.) • Reasoning is often a hypothesize-and-test cycle – Hypothesize: Postulate possible hypotheses, any of which would explain the given facts (or at least most of the important facts) – Test: Test the plausibility of all or some of these hypotheses – One way to test a hypothesis H is to ask whether something that is currently unknown–but can be predicted from H–is actually true • If we also know A => D and C => E, then ask if D and E are true • If D is true and E is false, then hypothesis A becomes more plausible (support for A is increased; support for C is decreased)
  • 8. 8 Characteristics of abductive reasoning (cont.) • Reasoning is non-monotonic – That is, the plausibility of hypotheses can increase/decrease as new facts are collected – In contrast, deductive inference is monotonic: it never change a sentence’s truth value, once known – In abductive (and inductive) reasoning, some hypotheses may be discarded, and new ones formed, when new observations are made
  • 9. 9 Sources of uncertainty • Uncertain inputs – Missing data – Noisy data • Uncertain knowledge – Multiple causes lead to multiple effects – Incomplete enumeration of conditions or effects – Incomplete knowledge of causality in the domain – Probabilistic/stochastic effects • Uncertain outputs – Abduction and induction are inherently uncertain – Default reasoning, even in deductive fashion, is uncertain – Incomplete deductive inference may be uncertain Probabilistic reasoning only gives probabilistic results (summarizes uncertainty from various sources)
  • 10. 10 Decision making with uncertainty • Rational behavior: – For each possible action, identify the possible outcomes – Compute the probability of each outcome – Compute the utility of each outcome – Compute the probability-weighted (expected) utility over possible outcomes for each action – Select the action with the highest expected utility (principle of Maximum Expected Utility)
  • 11. 11 Bayesian reasoning • Probability theory • Bayesian inference – Use probability theory and information about independence – Reason diagnostically (from evidence (effects) to conclusions (causes)) or causally (from causes to effects) • Bayesian networks – Compact representation of probability distribution over a set of propositional random variables – Take advantage of independence relationships
  • 12. 12 Other uncertainty representations • Default reasoning – Nonmonotonic logic: Allow the retraction of default beliefs if they prove to be false • Rule-based methods – Certainty factors (Mycin): propagate simple models of belief through causal or diagnostic rules • Evidential reasoning – Dempster-Shafer theory: Bel(P) is a measure of the evidence for P; Bel(P) is a measure of the evidence against P; together they define a belief interval (lower and upper bounds on confidence) • Fuzzy reasoning – Fuzzy sets: How well does an object satisfy a vague property? – Fuzzy logic: “How true” is a logical statement?
  • 13. 13 Uncertainty tradeoffs • Bayesian networks: Nice theoretical properties combined with efficient reasoning make BNs very popular; limited expressiveness, knowledge engineering challenges may limit uses • Nonmonotonic logic: Represent commonsense reasoning, but can be computationally very expensive • Certainty factors: Not semantically well founded • Dempster-Shafer theory: Has nice formal properties, but can be computationally expensive, and intervals tend to grow towards [0,1] (not a very useful conclusion) • Fuzzy reasoning: Semantics are unclear (fuzzy!), but has proved very useful for commercial applications
  • 14. 14 Bayesian Reasoning Chapter 13 CS 63 Adapted from slides by Tim Finin and Marie desJardins.
  • 15. 15 Outline • Probability theory • Bayesian inference – From the joint distribution – Using independence/factoring – From sources of evidence
  • 16. 16 Sources of uncertainty • Uncertain inputs – Missing data – Noisy data • Uncertain knowledge – Multiple causes lead to multiple effects – Incomplete enumeration of conditions or effects – Incomplete knowledge of causality in the domain – Probabilistic/stochastic effects • Uncertain outputs – Abduction and induction are inherently uncertain – Default reasoning, even in deductive fashion, is uncertain – Incomplete deductive inference may be uncertain Probabilistic reasoning only gives probabilistic results (summarizes uncertainty from various sources)
  • 17. 17 Decision making with uncertainty • Rational behavior: – For each possible action, identify the possible outcomes – Compute the probability of each outcome – Compute the utility of each outcome – Compute the probability-weighted (expected) utility over possible outcomes for each action – Select the action with the highest expected utility (principle of Maximum Expected Utility)
  • 18. 18 Why probabilities anyway? • Kolmogorov showed that three simple axioms lead to the rules of probability theory – De Finetti, Cox, and Carnap have also provided compelling arguments for these axioms 1. All probabilities are between 0 and 1: • 0 ≤ P(a) ≤ 1 2. Valid propositions (tautologies) have probability 1, and unsatisfiable propositions have probability 0: • P(true) = 1 ; P(false) = 0 3. The probability of a disjunction is given by: • P(a  b) = P(a) + P(b) – P(a  b) ab a b
  • 19. 19 Probability theory • Random variables – Domain • Atomic event: complete specification of state • Prior probability: degree of belief without any other evidence • Joint probability: matrix of combined probabilities of a set of variables • Alarm, Burglary, Earthquake – Boolean (like these), discrete, continuous • (Alarm=True  Burglary=True  Earthquake=False) or equivalently (alarm  burglary  ¬earthquake) • P(Burglary) = 0.1 • P(Alarm, Burglary) = alarm ¬alarm burglary 0.09 0.01 ¬burglary 0.1 0.8
  • 20. 20 Probability theory (cont.) • Conditional probability: probability of effect given causes • Computing conditional probs: – P(a | b) = P(a  b) / P(b) – P(b): normalizing constant • Product rule: – P(a  b) = P(a | b) P(b) • Marginalizing: – P(B) = ΣaP(B, a) – P(B) = ΣaP(B | a) P(a) (conditioning) • P(burglary | alarm) = 0.47 P(alarm | burglary) = 0.9 • P(burglary | alarm) = P(burglary  alarm) / P(alarm) = 0.09 / 0.19 = 0.47 • P(burglary  alarm) = P(burglary | alarm) P(alarm) = 0.47 * 0.19 = 0.09 • P(alarm) = P(alarm  burglary) + P(alarm  ¬burglary) = 0.09 + 0.1 = 0.19
  • 21. 21 Example: Inference from the joint alarm ¬alarm earthquake ¬earthquake earthquake ¬earthquake burglary 0.01 0.08 0.001 0.009 ¬burglary 0.01 0.09 0.01 0.79 P(Burglary | alarm) = α P(Burglary, alarm) = α [P(Burglary, alarm, earthquake) + P(Burglary, alarm, ¬earthquake) = α [ (0.01, 0.01) + (0.08, 0.09) ] = α [ (0.09, 0.1) ] Since P(burglary | alarm) + P(¬burglary | alarm) = 1, α = 1/(0.09+0.1) = 5.26 (i.e., P(alarm) = 1/α = 0.109 Quizlet: how can you verify this?) P(burglary | alarm) = 0.09 * 5.26 = 0.474 P(¬burglary | alarm) = 0.1 * 5.26 = 0.526
  • 22. 22 Exercise: Inference from the joint • Queries: – What is the prior probability of smart? – What is the prior probability of study? – What is the conditional probability of prepared, given study and smart? • Save these answers for next time!  p(smart  study  prep) smart smart study study study study prepared 0.432 0.16 0.084 0.008 prepared 0.048 0.16 0.036 0.072
  • 23. 23 Independence • When two sets of propositions do not affect each others’ probabilities, we call them independent, and can easily compute their joint and conditional probability: – Independent (A, B) ↔ P(A  B) = P(A) P(B), P(A | B) = P(A) • For example, {moon-phase, light-level} might be independent of {burglary, alarm, earthquake} – Then again, it might not: Burglars might be more likely to burglarize houses when there’s a new moon (and hence little light) – But if we know the light level, the moon phase doesn’t affect whether we are burglarized – Once we’re burglarized, light level doesn’t affect whether the alarm goes off • We need a more complex notion of independence, and methods for reasoning about these kinds of relationships
  • 24. 24 Exercise: Independence • Queries: – Is smart independent of study? – Is prepared independent of study? p(smart  study  prep) smart smart study study study study prepared 0.432 0.16 0.084 0.008 prepared 0.048 0.16 0.036 0.072
  • 25. 25 Conditional independence • Absolute independence: – A and B are independent if and only if P(A  B) = P(A) P(B); equivalently, P(A) = P(A | B) and P(B) = P(B | A) • A and B are conditionally independent given C if and only if – P(A  B | C) = P(A | C) P(B | C) • This lets us decompose the joint distribution: – P(A  B  C) = P(A | C) P(B | C) P(C) • Moon-Phase and Burglary are conditionally independent given Light-Level • Conditional independence is weaker than absolute independence, but still useful in decomposing the full joint probability distribution
  • 26. 26 Exercise: Conditional independence • Queries: – Is smart conditionally independent of prepared, given study? – Is study conditionally independent of prepared, given smart? p(smart  study  prep) smart smart study study study study prepared 0.432 0.16 0.084 0.008 prepared 0.048 0.16 0.036 0.072
  • 27. 27 Bayes’s rule • Bayes’s rule is derived from the product rule: – P(Y | X) = P(X | Y) P(Y) / P(X) • Often useful for diagnosis: – If X are (observed) effects and Y are (hidden) causes, – We may have a model for how causes lead to effects (P(X | Y)) – We may also have prior beliefs (based on experience) about the frequency of occurrence of effects (P(Y)) – Which allows us to reason abductively from effects to causes (P(Y | X)).
  • 28. 28 Bayesian inference • In the setting of diagnostic/evidential reasoning – Know prior probability of hypothesis conditional probability – Want to compute the posterior probability • Bayes’ theorem (formula 1): ons anifestati evidence/m hypotheses 1 m j i E E E H ) ( / ) | ( ) ( ) | ( j i j i j i E P H E P H P E H P  ) ( i H P ) | ( i j H E P ) | ( i j H E P ) | ( j i E H P ) ( i H P … …
  • 29. 29 Simple Bayesian diagnostic reasoning • Knowledge base: – Evidence / manifestations: E1, …, Em – Hypotheses / disorders: H1, …, Hn • Ej and Hi are binary; hypotheses are mutually exclusive (non- overlapping) and exhaustive (cover all possible cases) – Conditional probabilities: P(Ej | Hi), i = 1, …, n; j = 1, …, m • Cases (evidence for a particular instance): E1, …, Em • Goal: Find the hypothesis Hi with the highest posterior – Maxi P(Hi | E1, …, Em)
  • 30. 30 Bayesian diagnostic reasoning II • Bayes’ rule says that – P(Hi | E1, …, Em) = P(E1, …, Em | Hi) P(Hi) / P(E1, …, Em) • Assume each piece of evidence Ei is conditionally independent of the others, given a hypothesis Hi, then: – P(E1, …, Em | Hi) = m j=1 P(Ej | Hi) • If we only care about relative probabilities for the Hi, then we have: – P(Hi | E1, …, Em) = α P(Hi) m j=1 P(Ej | Hi)
  • 31. 31 Limitations of simple Bayesian inference • Cannot easily handle multi-fault situation, nor cases where intermediate (hidden) causes exist: – Disease D causes syndrome S, which causes correlated manifestations M1 and M2 • Consider a composite hypothesis H1  H2, where H1 and H2 are independent. What is the relative posterior? – P(H1  H2 | E1, …, Em) = α P(E1, …, Em | H1  H2) P(H1  H2) = α P(E1, …, Em | H1  H2) P(H1) P(H2) = α m j=1 P(Ej | H1  H2) P(H1) P(H2) • How do we compute P(Ej | H1  H2) ??
  • 32. 32 Limitations of simple Bayesian inference II • Assume H1 and H2 are independent, given E1, …, Em? – P(H1  H2 | E1, …, Em) = P(H1 | E1, …, Em) P(H2 | E1, …, Em) • This is a very unreasonable assumption – Earthquake and Burglar are independent, but not given Alarm: • P(burglar | alarm, earthquake) << P(burglar | alarm) • Another limitation is that simple application of Bayes’s rule doesn’t allow us to handle causal chaining: – A: this year’s weather; B: cotton production; C: next year’s cotton price – A influences C indirectly: A→ B → C – P(C | B, A) = P(C | B) • Need a richer representation to model interacting hypotheses, conditional independence, and causal chaining • Next time: conditional independence and Bayesian networks!