Bayesian Networks: Approximate Inference


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1 pproches to inference yesin Networks: pproximte Inference xct inference Vrillimintion Join tree lgorithm pproximte inference Simplify the structure of the network to mkxct inferencfficient (vritionl methods, loopy elief propgtion) Proilistic methods Stochstic simultion / smpling methods Mrkov chin Monte rlo methods Network simplifiction Typicl simplifictions: Remove prts of the network Removdges Reduce the numer of vlues (vlue strction) Replce sunetwork with simpler one (model strction) These simplifictions re often wrt to the prticulr evidence nd query Inference y smpling Wnt to compute P(e) Suppose we cn smple instnces <x 1,,x n > ccording to P(X 1,,X n ) The proility tht rndom smple <x 1,,x n > stisfies e is pproximtely P(e) We cn view ech smple s tossing ised coin with proility P(e) of Heds Smpling yesin Network If P(X 1,,X n ) is represented y yesin network, how cn wfficiently smple from it? Ide: smple ccording to structure of the network Write distriution using the chin rule, nd then smplch vrile given its prents P(r) P() 003 e e e e 1
2 P() 003 P() 003 P(r) e e e e P(r) e e e e e e P() 003 P() 003 P(r) e e e e P(c) P(r) e e e e e c e c r Let X 1,,X n e order of vriles consistent with rc direction for i = 1,,n do smple x i from P(X i P(X i )) (Note: since P(X i ) {X 1,,X i1 }, we lredy ssigned vlues to them) return x 1,,x n Smpling complete instnce is liner in numer of vriles Regrdless of structure of the network However, if P(e) is smll, we need mny smples to get decent estimte 2
3 n we smple from P(X 1,,X n e)? If evidence is in roots of network, esily If evidence is in leves of network, we hve prolem Our smpling method proceeds ccording to order of nodes in grph Rejection smpling: keep those instntitions tht re consistent with the vlues of thvidence vriles stimte P(X e) y N(X,e) / N(e) where N() counts the numer of times n event ws smpled Mrkov chin Monte rlo smpling Genertes events y mking rndom chnges to the stte vrile The next stte is generted y smpling vlue for one of the nonevidence vriles conditioned on the current vlues Gis smpling exmple onsider 2 vrile network: T F F Initilize rndomly Smple vriles lterntely Lerning yesin Networks Lerning yesin networks Known Structure, omplete t t + Prior informtion R P(,) e 9 1 e 7 3 e 8 2 e P(,) e?? e?? e?? e??,, <Y,N,N> <Y,Y,Y> <N,N,Y> Network structure is specified Lerning lgorithm needs to estimte prmeters t does not contin missing vlues P(,) e 9 1 e 7 3 e 8 2 e
4 Unknown Structure, omplete t Known Structure, Incomplete t P(,) e?? e?? e?? e??,, <Y,N,N> <Y,Y,Y> <N,N,Y> Network structure is not specified lgorithm needs to select edges & estimte prmeters t does not contin missing vlues P(,) e 9 1 e 7 3 e 8 2 e P(,) e?? e?? e?? e??,, <Y,N,N> <Y,?,Y> <N,N,Y> <N,Y,?> <?,Y,Y> Network structure is specified t contins missing vlues P(,) e 9 1 e 7 3 e 8 2 e Known Structure / omplete t Given network structure G nd choice of prmetric fmily for P(X i P(X i )) Lerning Prmeters for yesin Network Trining dt hs the form: Lern prmeters for network Gol onstruct network tht is closest to proility tht generted the dt [1] = [ M] [1] [ M] [1] [ M] [1] [ M] enefits of Lerning Structure Why Worry out ccurte Structure? iscover structurl properties of the domin eg: Relevnce Identifying independencies fster inference Predict effect of ctions Involves lerning cusl reltionship mong vriles dding n edge Set Sound Set Sound Increses the numer of prmeters to e fitted Wrong ssumptions out cuslity nd domin structure Missing n edge Set Sound nnot e compensted y ccurte fitting of prmeters lso misses cuslity nd domin structure 4
5 pproches to Lerning Structure Serch for good structure Score sed efine score tht evlutes how well the (in)dependencies in structure mtch the oservtions Serch for structure tht mximizes the score Pros & ons + Sttisticlly motivted omputtionlly hrd efine serch spce: nodes re possile structures edges denote djcency of structures Trverse this spce looking for highscoring structures Serch techniques: Greedy hillcliming est first serch Simulted nneling Serch (cont) Greedy Hillliming Typicl opertions: S elete S dd Reverse S S Simplest heuristic locl serch Strt with network empty network rndom network t ech itertion vlute ll possile chnges pply chnge tht leds to est improvement in score Iterte Stop when no modifiction improves score ch step requires evluting pproximtely n new chnges Involves the stndrd pitflls of hillcliming pplictions of N Medicl dignosis Trouleshooting of hrdwre/softwre systems 5
6 6
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