This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Week 3 Bayesian Inference Modeling time series Last week ! Talked about probability distributions ! Talked about Bayes’ Nets M J A B E This week ! How to do inference in Bayes nets ! Different ways of characterizing conditional probabilities ! How to look at things that change over time ! Dynamic Bayes’ Nets Reminder from last time ! Alarm example ! Compute: P(j,m,a,~b,~e)= P(ja)P(ma)P(a~b,~e) P(~b) P(~e) = .9 * .7 * .001 * .999 * .998 = 0.000628 B E A J M P(b) 0.001 P(b) 0.002 .001 F F .29 T F .94 F T .95 T T P(a) E B .01 F .70 T P(m) A .05 F .90 T P(j) A Missing information ! What’s the probability distribution over Burglary given that John calls and Mary calls? B E A J M P(b) 0.001 P(b) 0.002 .001 F F .29 T F .94 F T .95 T T P(a) E B .01 F .70 T P(m) A .05 F .90 T P(j) A Missing information ! What’s P(Bj,m)? ! The state of A, E are unknown. ! So we must “sum them out” (marginalize). ! These are called hidden variables . B E A J M P(b) 0.001 P(b) 0.002 .001 F F .29 T F .94 F T .95 T T P(a) E B .01 F .70 T P(m) A .05 F .90 T P(j) A Exact inference by enumeration ! First: redefine your quantity as the normalized sum over a joint distribution ! P(Bj,m) = P(B,j,m) / P(j,m) = ! P(B,j,m) = ! " e " a P(B,e ,a ,j,m) “e ” here means e or ~e Exact inference by enumeration ! Next, replace joint distribution by CPT entries ! " e " a P(B,e ,a ,j,m) = ! " e " a P(B) P(e ) P(a B,e ) P(ja ) P(ma ) ! Where do these come from? Exact inference by enumeration ! Now, push summations inwards ! " e " a P(B) P(e ) P(a B,e ) P(ja ) P(ma ) = ! P(B) " e P(e ) " a P(a B,e ) P(ja ) P(ma ) This reduces the number of products you need to do. Exact inference by enumeration ! Now, start computing summations. ! Need two cases for inner summation ! e is true, e is false ! Inner sum: " a P(a B,e ) P(ja ) P(ma ) ! e is true: • (<.95,.29> * .9 * .7) + (<.05,.71> * .05 * .01) = <0.5985,0.1827> + <2.5e05,3.55e04> = <0.598525,0.183055> (vector is over <b, ~b>) Exact inference by enumeration ! Now, start computing summations. ! Need two cases for inner summation ! e is true, e is false ! Inner sum: " a P(a B,e ) P(ja ) P(ma ) ! e is true: • <0.598525,0.183055> ! e is false: • (<.94,.001> * .9 * .7)+(<.06,.999> * .05 * .01>) = <0.592236,0.0011295> Exact inference by enumeration ! Bringing in the inner summation: ! P(B) " e P(e ) " a P(a B,e ) P(ja ) P(ma ) = ! P(B) " e P(e ) < <0.599,0.1839> Be , <0.593,0.001> B~e > ! Multiply each vector by P(e) or P(~e) ! P(B) ( 0.002*<0.599,0.1839> + 0.998*<0.593,0.001> ) = ! P(B) <.541,.0014> = ! <.001,.999>*<.541,.0014> = ! <.00054,.0014> = <.278,.722> (diff from book because of rounding) Hints for enumeration...
View
Full
Document
This note was uploaded on 04/13/2010 for the course CSE 730 taught by Professor Ericfoslerlussier during the Fall '08 term at Ohio State.
 Fall '08
 EricFoslerLussier
 Artificial Intelligence

Click to edit the document details