chapter14b - Inference in Bayesian networks Chapter...

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Unformatted text preview: Inference in Bayesian networks Chapter 14.4–5 Chapter 14.4–5 1 Outline ♦ Exact inference by enumeration ♦ Exact inference by variable elimination ♦ Approximate inference by stochastic simulation ♦ Approximate inference by Markov chain Monte Carlo Chapter 14.4–5 2 Inference tasks Simple queries : compute posterior marginal P ( X i | E = e ) e.g., P ( NoGas | Gauge = empty, Lights = on,Starts = false ) Conjunctive queries : P ( X i ,X j | E = e ) = P ( X i | E = e ) P ( X j | X i , E = e ) Optimal decisions : decision networks include utility information; probabilistic inference required for P ( outcome | action,evidence ) Value of information : which evidence to seek next? Sensitivity analysis : which probability values are most critical? Explanation : why do I need a new starter motor? Chapter 14.4–5 3 Inference by enumeration Slightly intelligent way to sum out variables from the joint without actually constructing its explicit representation Simple query on the burglary network: B E J A M P ( B | j, m ) = P ( B,j, m ) /P ( j, m ) = α P ( B, j,m ) = α Σ e Σ a P ( B,e,a,j, m ) Rewrite full joint entries using product of CPT entries: P ( B | j, m ) = α Σ e Σ a P ( B ) P ( e ) P ( a | B,e ) P ( j | a ) P ( m | a ) = α P ( B ) Σ e P ( e ) Σ a P ( a | B,e ) P ( j | a ) P ( m | a ) Recursive depth-first enumeration: O ( n ) space, O ( d n ) time Chapter 14.4–5 4 Enumeration algorithm function Enumeration-Ask ( X , e , bn ) returns a distribution over X inputs : X , the query variable e , observed values for variables E bn , a Bayesian network with variables { X } ∪ E ∪ Y Q ( X ) ← a distribution over X , initially empty for each value x i of X do extend e with value x i for X Q ( x i ) ← Enumerate-All ( Vars [ bn ], e ) return Normalize ( Q ( X ) ) function Enumerate-All ( vars , e ) returns a real number if Empty? ( vars ) then return 1.0 Y ← First ( vars ) if Y has value y in e then return P ( y | Pa ( Y )) × Enumerate-All ( Rest ( vars ), e ) else return ∑ y P ( y | Pa ( Y )) × Enumerate-All ( Rest ( vars ), e y ) where e y is e extended with Y = y Chapter 14.4–5 5 Evaluation tree P(j|a) .90 P(m|a) .70 .01 P(m| a) .05 P(j| a) P(j|a) .90 P(m|a) .70 .01 P(m| a) .05 P(j| a) P(b) .001 P(e) .002 P( e) .998 P(a|b,e) .95 .06 P( a|b, e) .05 P( a|b,e) .94 P(a|b, e) Enumeration is inefficient: repeated computation e.g., computes P ( j | a ) P ( m | a ) for each value of e Chapter 14.4–5 6 Inference by variable elimination Variable elimination: carry out summations right-to-left, storing intermediate results ( factors ) to avoid recomputation P ( B | j, m ) = α P ( B ) B Σ e P ( e ) E Σ a P ( a | B,e ) A P ( j | a ) J P ( m | a ) M = α P ( B ) Σ e P ( e ) Σ a P ( a | B,e ) P ( j | a ) f M ( a ) = α P ( B ) Σ e P ( e ) Σ a P ( a | B,e ) f J ( a ) f M ( a ) = α P ( B ) Σ e P ( e ) Σ a f A ( a,b,e ) f J ( a ) f M ( a ) = α P ( B ) Σ e P ( e ) f ¯ AJM ( b,e ) (sum out A ) = α P ( B ) f ¯ E ¯ AJM ( b )...
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This note was uploaded on 02/07/2012 for the course CSCI 5512 taught by Professor Staff during the Spring '08 term at Minnesota.

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chapter14b - Inference in Bayesian networks Chapter...

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