This preview shows pages 1–3. Sign up to view the full content.
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 ! Whats 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 ! Whats 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
 Fall '08
 EricFoslerLussier
 Artificial Intelligence

Click to edit the document details