Week3 - Week 3 Bayesian Inference Modeling time series Last...

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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(j|a)P(m|a)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(B|j,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(B|j,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(j|a ) P(m|a ) ! Where do these come from? Exact inference by enumeration ! Now, push summations inwards ! " e " a P(B) P(e ) P(a |B,e ) P(j|a ) P(m|a ) = ! P(B) " e P(e ) " a P(a |B,e ) P(j|a ) P(m|a ) 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(j|a ) P(m|a ) ! e is true: (<.95,.29> * .9 * .7) + (<.05,.71> * .05 * .01) = <0.5985,0.1827> + <2.5e-05,3.55e-04> = <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(j|a ) P(m|a ) ! 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(j|a ) P(m|a ) = ! P(B) " e P(e ) < <0.599,0.1839> B|e , <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...
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Week3 - Week 3 Bayesian Inference Modeling time series Last...

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