SP11 cs188 lecture 16 -- bayes nets IV 6PP

# 08 002 009 081 t t 017 083 32 5 multiple elimination r

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Unformatted text preview: g variable   Build a new factor over the union of the variables involved +r +r  ­r  ­r +t  ­t +t  ­t   E.g. if we know   First basic operation: joining factors   Combining factors: 0.1 0.9 +r +r  ­r  ­r   Any known values are selected Operation 1: Join Factors +r  ­r 0.1 0.9 0.024 0.056 0.002 0.018 0.027 0.063 0.081 0.729 31   Shrinks a factor to a smaller one   A projection operation   Example: +r +r  ­r  ­r +t  ­t +t  ­t 0.08 0.02 0.09 0.81 +t  ­t 0.17 0.83 32 5 Multiple Elimination R, T, L +r +r +r +r  ­r  ­r  ­r  ­r +t +t  ­t  ­t +t +t  ­t  ­t +l  ­l +l  ­l +l  ­l +l  ­l P(L) : Marginalizing Early! T, L 0.024 0.056 0.002 0.018 0.027 0.063 0.081 0.729 +r  ­r L Sum out T Sum out R +t +t  ­t  ­t +l  ­l +l  ­l R 0.051 0.119 0.083 0.747 +l 0.134  ­l 0.886 T L 33 0.1 0.9 +r +r  ­r  ­r +t  ­t +t  ­t +l  ­l +l  ­l +r +r  ­r  ­r 0.8 0.2 0.1 0.9 +t +t  ­t  ­t 0.3 0.7 0.1 0.9 0.08 0.02 0.09 0.81 +t  ­t 0.17 0.83 T +t +t  ­t  ­t +l  ­l +l  ­l 0.3 0.7 0.1 0.9 +t +t  ­t  ­t L +l  ­l +l  ­l 0.3 0.7 0.1 0.9 L 34 Evidence   If evidence, start with factors that select that evidence T T, L Join T L +t +t  ­t  ­t +t  ­t +t  ­t R, T Marginalizing Early (aka VE*) +t  ­t Sum out R Join R Sum out T   No evidence uses these initial factors: L +r  ­r 0.17 0.83 +l  ­l +l  ­l +t +t  ­t  ­t 0.3 0.7 0.1 0.9 +l  ­l +l  ­l 0.1 0.9 +r +r  ­r  ­r +t  ­t +t  ­t   Computing 0.051 0.119 0.083 0.747 +l 0.134  ­l 0.886 +r 0.1 0.8 0.2 0.1 0.9 +t +t  ­t  ­t +l  ­l +l  ­l 0.3 0.7 0.1 0.9 , the initial factors become: +r +r +t  ­t 0.8 0.2 +t +t  ­t  ­t +l  ­l +l  ­l 0.3 0.7 0.1 0.9   We eliminate all vars other than query + evidence 36 * VE is variable elimination Evidence II General Variable Elimination   Result will be a selected joint of query and evidence   Query:   E.g. for P(L | +r), we d end up with:   Start with initial factors: Normalize +r +l 0.026 +r  ­l 0.074   Local CPTs (but instantiated by evidence) +l 0.26  ­l 0.74   While there are still hidden variables (not Q or evidence):   Pick a hidden variable H   Join all factors mentioning H   Eliminate (sum out) H   To get our answer, just normalize this!   Join all remaining factors and normalize   That s it! 37 38 6 Example Example Choose E Choose A Finish with B Normalize 39 Example 2: P(B|a) Start / Select Join on B B B 0.9 Normalize a   We will see special cases of VE later A B A P +a B P A B P +a +b +b 0.08 +a +b 8/17 +a ¬b 0.09 +a ¬b 9/17 0.8 b ¬a +a 0.1 ¬b ¬a 0.9   On tree-structured graphs, variable elimination runs in polynomial time, like tree-structured CSPs   You ll have to implement a tree-structured special case to track invisible ghosts (Project 4) 0.2 ¬b   What you need to know:   Should be able to run it on small examples, understand the factor creation / reduction flow   Better than enumeration: saves time by marginalizing variables as soon as possible rather than at the end a, B 0.1 ¬b Variable Elimination P +b 40 41 7...
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## This note was uploaded on 08/26/2011 for the course CS 188 taught by Professor Staff during the Spring '08 term at Berkeley.

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