CMSC 471 Fall 2002

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CMSC 471Fall 2002

• Class 19 Monday, November 4

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Todays class

• (Probability theory)
• Bayesian inference
• From the joint distribution
• Using independence/factoring
• From sources of evidence
• Bayesian networks
• Network structure
• Conditional probability tables
• Conditional independence
• Inference in Bayesian networks

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Bayesian Reasoning /Bayesian Networks

• Chapters 14, 15.1-15.2

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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
• All probabilities are between 0 and 1
• 0 lt P(a) lt 1
• Valid propositions (tautologies) have probability
  1, and unsatisfiable propositions have
  probability 0
• P(true) 1 P(false) 0
• The probability of a disjunction is given by
• P(a ? b) P(a) P(b) P(a ? b)

a
a?b
b
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Inference from the joint Example
alarm alarm alarm alarm
earthquake earthquake earthquake earthquake
burglary .001 .008 .0001 .0009
burglary .01 .09 .001 .79
P(Burglary alarm) a P(Burglary, alarm)
a P(Burglary, alarm, earthquake) P(Burglary,
alarm, earthquake) a (.001, .01)
(.008, .09) a (.009, .1) Since
P(burglary alarm) P(burglary alarm) 1, a
1/(.009.1) 9.173 (i.e., P(alarm)
1/a .109 quizlet how can you verify
this?) P(burglary alarm) .009 9.173
.08255 P(burglary alarm) .1 9.173 .9173
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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 theres a new
  moon (and hence little light)
• But if we know the light level, the moon phase
  doesnt affect whether we are burglarized
• Once were burglarized, light level doesnt
  affect whether the alarm goes off
• We need a more complex notion of independence,
  and methods for reasoning about these kinds of
  relationships

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Conditional independence

• Absolute independence
• A and B are independent 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
• 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

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Bayes rule

• Bayes 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)).

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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)

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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,
  , El
• Goal Find the hypothesis Hi with the highest
  posterior
• Maxi P(Hi E1, , El)

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Bayesian diagnostic reasoning II

• Bayes rule says that
• P(Hi E1, , El) P(E1, , El Hi) P(Hi) /
  P(E1, , El)
• Assume each piece of evidence Ei is conditionally
  independent of the others, given a hypothesis Hi,
  then
• P(E1, , El Hi) ?lj1 P(Ej Hi)
• If we only care about relative probabilities for
  the Hi, then we have
• P(Hi E1, , El) a P(Hi) ?lj1 P(Ej Hi)

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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, , El) a P(E1, , El H1 ? H2)
  P(H1 ? H2) a P(E1, , El H1 ? H2) P(H1)
  P(H2) a ?lj1 P(Ej H1 ? H2) P(H1) P(H2)
• How do we compute P(Ej H1 ? H2) ??

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Limitations of simple Bayesian inference II

• Assume H1 and H2 are independent, given E1, ,
  El?
• P(H1 ? H2 E1, , El) P(H1 E1, , El) P(H2
  E1, , El)
• This is a very unreasonable assumption
• Earthquake and Burglar are independent, but not
  given Alarm
• P(burglar alarm, earthquake) ltlt P(burglar
  alarm)
• Another limitation is that simple application of
  Bayes rule doesnt allow us to handle causal
  chaining
• A years weather B cotton production C next
  years 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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Bayesian Belief Networks (BNs)

• Definition BN (DAG, CPD)
• DAG directed acyclic graph (BNs structure)
• Nodes random variables (typically binary or
  discrete, but methods also exist to handle
  continuous variables)
• Arcs indicate probabilistic dependencies between
  nodes (lack of link signifies conditional
  independence)
• CPD conditional probability distribution (BNs
  parameters)
• Conditional probabilities at each node, usually
  stored as a table (conditional probability table,
  or CPT)
• Root nodes are a special case no parents, so
  just use priors in CPD

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Example BN
P(A) 0.001
P(CA) 0.2 P(CA) 0.005
P(BA) 0.3 P(BA) 0.001
P(DB,C) 0.1 P(DB,C) 0.01 P(DB,C)
0.01 P(DB,C) 0.00001
P(EC) 0.4 P(EC) 0.002
Note that we only specify P(A) etc., not P(A),
since they have to add to one
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Topological semantics

• A node is conditionally independent of its
  non-descendants given its parents
• A node is conditionally independent of all other
  nodes in the network given its parents, children,
  and childrens parents (also known as its Markov
  blanket)
• The method called d-separation can be applied to
  decide whether a set of nodes X is independent of
  another set Y, given a third set Z

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Independence and chaining

• Independence assumption
• where q is any set of variables
• (nodes) other than and its successors
• blocks influence of other nodes on
• and its successors (q influences only
• through variables in )
• With this assumption, the complete joint
  probability distribution of all variables in the
  network can be represented by (recovered from)
  local CPD by chaining these CPD

q
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Chaining Example

• Computing the joint probability for all variables
  is easy
• P(a, b, c, d, e)
• P(e a, b, c, d) P(a, b, c, d) by Bayes
  theorem
• P(e c) P(a, b, c, d) by indep. assumption
• P(e c) P(d a, b, c) P(a, b, c)
• P(e c) P(d b, c) P(c a, b) P(a, b)
• P(e c) P(d b, c) P(c a) P(b a) P(a)

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Direct inference with BNs

• Now suppose we just want the probability for one
  variable
• Belief update method
• Original belief (no variables are instantiated)
  Use prior probability p(xi)
• If xi is a root, then P(xi) is given directly in
  the BN (CPT at Xi)
• Otherwise,
• P(xi) S pi P(xi pi) P(pi)
• In this equation, P(xi pi) is given in the CPT,
  but computing P(pi) is complicated

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Computing pi Example

• P (d) P(d b, c) P(b, c)
• P(b, c) P(a, b, c) P(a, b, c)
  (marginalizing) P(b a, c) p (a, c) p(b
  a, c) p(a, c) (product rule) P(b a)
  P(c a) P(a) P(b a) P(c a) P(a)
• If some variables are instantiated, can plug
  that in and reduce amount of marginalization
• Still have to marginalize over all values of
  uninstantiated parents not computationally
  feasible with large networks

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Representational extensions

• Compactly representing CPTs
• Noisy-OR
• Noisy-MAX
• Adding continuous variables
• Discretization
• Use density functions (usually mixtures of
  Gaussians) to build hybrid Bayesian networks
  (with discrete and continuous variables)

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Inference tasks

• Simple queries Computer posterior marginal P(Xi
  Ee)
• E.g., P(NoGas Gaugeempty, Lightson,
  Startsfalse)
• Conjunctive queries
• P(Xi, Xj Ee) P(Xi ee) P(Xj Xi, Ee)
• Optimal decisions Decision networks include
  utility information probabilistic inference is
  required to find P(outcome action, evidence)
• Value of information Which evidence should we
  seek next?
• Sensitivity analysis Which probability values
  are most critical?
• Explanation Why do I need a new starter motor?

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Approaches to inference

• Exact inference
• Enumeration
• Variable elimination
• Clustering / join tree algorithms
• Approximate inference
• Stochastic simulation / sampling methods
• Markov chain Monte Carlo methods
• Genetic algorithms
• Neural networks
• Simulated annealing
• Mean field theory