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CMSC 471 Fall 2002. Class #19 – Monday, November 4. Today’s 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

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cmsc 471 fall 2002

CMSC 471Fall 2002

Class #19 – Monday, November 4

today s class
Today’s 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
why probabilities anyway
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 <= P(a) <= 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

inference from the joint example
Inference from the joint: Example

P(Burglary | alarm) = α P(Burglary, alarm) = α [P(Burglary, alarm, earthquake) + P(Burglary, alarm, ¬earthquake) = α [ (.001, .01) + (.008, .09) ] = α [ (.009, .1) ]

Since P(burglary | alarm) + P(¬burglary | alarm) = 1, α = 1/(.009+.1) = 9.173 (i.e., P(alarm) = 1/α = .109 – quizlet: how can you verify this?)

P(burglary | alarm) = .009 * 9.173 = .08255

P(¬burglary | alarm) = .1 * 9.173 = .9173

independence
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
conditional independence
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
bayes rule
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)).
bayesian inference
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):
simple bayesian diagnostic reasoning
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)
bayesian diagnostic reasoning ii
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) = lj=1 P(Ej | Hi)
  • If we only care about relative probabilities for the Hi, then we have:
    • P(Hi | E1, …, El) = αP(Hi) lj=1 P(Ej | Hi)
limitations of simple bayesian inference
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) = αP(E1, …, El | H1  H2) P(H1  H2) = αP(E1, …, El | H1  H2) P(H1) P(H2) = αlj=1 P(Ej | H1  H2)P(H1) P(H2)
  • How do we compute P(Ej | H1  H2) ??
limitations of simple bayesian inference ii
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) << P(burglar | alarm)
  • Another limitation is that simple application of Bayes’ rule doesn’t allow us to handle causal chaining:
    • A: 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!
bayesian belief networks bns
Bayesian Belief Networks (BNs)
  • Definition: BN = (DAG, CPD)
    • DAG: directed acyclic graph (BN’s 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 (BN’s 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:
example bn

a

b c

d e

Example BN

P(A) = 0.001

P(C|A) = 0.2 P(C|~A) = 0.005

P(B|A) = 0.3 P(B|~A) = 0.001

P(D|B,C) = 0.1 P(D|B,~C) = 0.01

P(D|~B,C) = 0.01 P(D|~B,~C) = 0.00001

P(E|C) = 0.4 P(E|~C) = 0.002

Note that we only specify P(A) etc., not P(¬A), since they have to add to one

topological semantics
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 children’s 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
independence and chaining
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

chaining example

a

b c

d e

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)

direct inference with bns
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) = Σπi P(xi | πi) P(πi)
  • In this equation, P(xi | πi) is given in the CPT, but computing P(πi) is complicated
computing i example

a

b c

d e

Computing πi: 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
representational extensions
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)
inference tasks
Inference tasks
  • Simple queries: Computer posterior marginal P(Xi | E=e)
    • E.g., P(NoGas | Gauge=empty, Lights=on, Starts=false)
  • Conjunctive queries:
    • P(Xi, Xj | E=e) = P(Xi | e=e) P(Xj | Xi, E=e)
  • 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?
approaches to inference
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
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