SP11 cs188 lecture 16 -- bayes nets IV 6PP

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 08/26/2011 for the course CS 188 taught by Professor Staff during the Spring '08 term at Berkeley.

Ask a homework question - tutors are online