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# Class Note for CMPSCI 683 at UMass(12)

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Date Created: 02/06/15

Victor Lesser CMPSCI 683 Fall 2004 Designed to Deal with the distinction between uncertainty and ignorance Ratherthan computing the probabilityol a proposition it computes the probability that evidence supports the proposition Applicability of 08 assume lack of sufficient data to accurately estimate the prior and conditional probabilities to use Bayes rule incomplete model gt ralherlhan estimating probabilities it uses belief intervals to estimate how close the evidence is to determining the truth ofa hypothesis vvmamim Alternative Models of Dealing with Uncertainty InformationEvidence DempsterShafler Theory of Evidence Fuzzy logic Logical ways of dealing with uncertainty vincamim Allows representation of ignorance about support provided by evidence allows reasoning system to be skeptical For example suppose we are informed that one of three terrorist groups A B or C has planted a bomb in a building We may have some evidence the group C is gulty P C 08 We would not want to say the probability of the othertwo groups being guilty is if in traditional theory forcedto regard belief and disbelief as functional opposites pa i pnot ah i and to distribute an equal amoun of the remaining pro ability to each group DS allows you to leave relative beliefs unspecified vincamim 04 RedGreeiiBlue 0 6 RedGreen RedBlue GreenBlue Red Green Blue Snwosethat memento summrreagremm manage a m minim suppm wll neasgm m mammal mile a hymn mm asunsum the reuniting suppnn IS HSQM DME negation mm mm or its permanent line Given a population F bluered green of mutually exclusive elements exactly one ofvvhich f is true a basic probability assignment m assigns a number in 01 to every subset of F such that the sum of the numbers is 1 7 Mass as a represenlalion ol evidence support There are 2l l propositions corresponding to the true value offis in subset Aquot 7 bluereogreenbluered bluegreenredgreenredbluegreenemoly sel A belief in a subset entails belief in subsets containingthat subset 7 Beliel in redenlails Beliel in red greenredblue red bluegreen uumicsumm a Random switch model Think oithe eidence as a sWitch that oscillates randomly between two or more positions Theiunction mrepresentstheiraction of the time spent by the sWitch in each position Voting model m represents the proportion of vctes castior each cl several results ofthe eldence possibly including that the evidence is inconclusive Envelope model Think cf the evidence as a sealed envelope and m as a probability distribution on What the contents maybe Belief or possibility is the probabilitythat B is provable supported bythe evidence BelA 2EMmB SupporlcommilledloA Mamas MMMWWM Plausib ty is the probabilitythat B is compatible with the available evidence cannot be disproved Upper belief limit on the proposition A 39 PllA 2 a n 4 mlB Supportthat can move intoA mm7mwwciuimmumimwirmmmgmiiwmwii PllA1BelaA uumicsumm x Belieflnterval BeAPIA confidence in A Interval width is good aid in deciding when you need more evidence 01 no belief in support of proposition total ignorance 00 belieflhe proposition is false 1 1 belieflhe proposition is true 31 partial belief in the proposition is true 08 partial disbelief in the proposition is true 27 belieffrom evidence both for and against propostion Given two basic probability assignment functions m1 and mZ howto combine them 7 Two diverantsoums of evidence 7 PxlerlPlxrlPlildmnecondi onalin ependznu Bel C Sum m1A m1Bl where A intersech C Normalized by amount of nonzero mass left after intersection Sum m1A m1B where A intersect Bl not empty mnmsllmn a D 9 DH DD 1 D s D 72 o D 08 aD o o Do D o DnD 2 D 18 DH DnD oz m2D 72180898 m2aD0 m12 FD02 Using intervals 81 and 91 981 vummxmm u Suppose that mD8 and m2aD9 Do ED 9 DaD1 D s Do 72 D be D o o aD0 aD 0 DD 2 Do D 18 DwD oz Need to normalize 180802 by 72 m2D29 m2aD64 m2DD07 Using intervals 81 and 01 2936 mnmsllmn 2 Let 8 he Tlmnwe can combine 11sz2 All allelgy Flu u Cold com AM 08 e 02 Pneumneumonia mama 5 5C 048 Map 012 5 04 A50 032 6 008 Whan w begin with no mmmanon 11 IS Sn pm a m cmde 9 l m imam m Alumna n32 supposem colmpomis to am helm am ohselvmg level WW M m Cawmmn s W e um S mm m cnmamm hx tn m 1739de am the prohkm gues my ml Suppose m1 colmpomis to our helm aner observing a runny msfmmmzm39g quotm a my nose AH m Cold up 5 1 s um u mm H Applythhe numn39ztm39 nflhe cnmhinz nnnde yields A my 8 m1 Addressesquestions abontnecessity gm Egg gm and pOSSIbIIIty that BayeSIan approach ARE A ARE MP W U m MP W cannot answer 9 BER A BENZ 9 Hum Nnmzlizingwgtmannehelietnmsanmhuawimngimmj Prior probabilities not required but 7 FMCDM 101MJMEME be used Allemyflu old 100696 113216 p 39 FmCoIdPneu 1mm mzme Wm mmHm39Wquot Useful for reasoning in rulebased 8 10017tWE46 M31 is he helmY possibilily olAllelgy SyStems vunmsixmlm 5 mnmsllmm m Source of eyidence notindependenti can lead to misleading and counterintuitive results The normalization in Dempster s Rule loses some meta information and this treatment of conflicting eyidence is controversiaii r Dillicnlt to develop tlreory olntilitysince ael is not de ned precisely witn Mpect to decision making Bayesian approach can also do something akin to confidence interval by examining how much one39s belief would change it more eyidence acquired e lrnglicn uncertainty associated with various oossilrle clranges warming in Normalization process can produce strange conclusions when con icting evidence is involved a Aaci m1 Al 099 B 001 me Cl 099 B 001 W mrBl10 Certain of B even though neither piece of evidence supported it well No representation of inconsistency and resulting uncertaintyl ignorance mmlcsilmm ix Metlroolor reasoning with logical expressions oescrilring may set membership 7 Knowledge representation based on degreestimem rainerlnan a crisp membership ortnnary logic Ratheithan siornasnr pintesses 7 Degree arm in nropestren rarner inan degree or belief chie olgroonolproporrice as well as calcnlns lor incornglete inlornration e precision in reasoning is eosliy and should not be pursued more than necessary ApplicationFllzndlltveTtllory 7 Ixflu klmiMdgl coded as fuzzy rlllu 39 mile car39s quell Is slow lhmhbri ing hm isll39gM e campuluuontmlafawlderlnganfdwica washing Minn umsviueocm ac Warming iv Fuzziness is a way nfde ning concepts or categories that admit vagueness and degree nothing to do with degree of belief in something and need not be related to probabilities We believe 5 it Wlii rain today example of Was it a rainy dayquot fuzziness misty all day long but neer breaks into a shower 39 rain for a few minutes and then sunny heavy showers aii daylong mmlcsilmm 7n a l m 15121 Giventwo inputs 5 di erence between current temperature and target temperature normalixed hythetargettelnperatllre dE the time derivative 0 E idEl i Compute one output w the changein heat iorcoolingi source Fuzzy variahln NB ineg higi lls20es PB Example rule lie is 20 and us is Ms then Wis es we ISZO and us isPBthenWis NB Sign icant Reduction in number olnlles needed and handl noisy sensors vlnmsixmlm 1i Fuzzy uarialzlslalms on aiuuyset as avalue a Aluzzy set Diass Am x is characterized by a membership iurlUilnrl ya that assigns each pnlrlix m x a leai number between ol Height Examplequot in thetall class a ya 001 inlanypelsnn nvel ieeiiaii a ya x 0 im any pelsml underS ieetiaii 7 Wu x in between inl nalum gt5 and lt6 Piecsw39n LIIleerFunction I17 vlluls in between a veLinliaii ii406 16 i7 Hedge r systemaile medl eaimrl in a Dharamerlsileiurmilml in represent a ilrlgulsile speelailzatmn a very iaiiWNelyiaii xviaii m vlnmsixmlm 2 A different type of approximation related to vagueness rather than uncertainty Measures the degree of membership in certain sets or categories such as age height red several did many mnmsumm sf Genenitytrue 4i Mlyhetrue 4i Geneuitybixe u a a 7 HelngrlkA Example Several 23 35 411 91 Representatinn A ulaiu u E U 72 mall An ample dnlactzl39slic mmuluz umzzygtzapzsumug Willth mmmf 39 smegma whulmmylmueulummmluigm wnull lbemns39l kxel lhbemnnbusli emzzymlmupusuus Height in Set m Fig 639 Tluaa puma lluau chancu39st39t mums Express prelumceson oossiirisvarrss o1 amiable mime em vans is not known 7 mrruepi mmemum 7 Usmmrmsmirgmmzzy miesels Cmismncyulambsetwithmpedtoamnoept tallpeoplequot 7 Hartman BEnh mower 7 Pninymnsrrsm mAmaxmvIaiiwerAgi moasrromweoogooiurmoirmsoi lali W W39imssirriyinA 99nh5 irrplies Al 7 ammoMum lint NlAl1 lcnmnEmemmM mmemtudeg enc momhvsel Mpawwisnawssmyim nompiorronrorkiso swoms 11 Moms mammals now 55 oomssmr Represartation A ualul i rr 5 u so oouaors ova sarrmriairle A U E llmaxim hul us U A n E umir l hu in EH a A ui 7auiueU For examlet Young aid Ridl v n Rt mites39al product ovutwo variants A x a luvlnirrlalul ier in e UvEV other set operators ae mmairresuseruor exanple A U B uau r nor 7am hul u eU A n E Mann not i u e U etc lumuslmzm 7o L a c39 n o39 osmnysss nrar turgumllud quotmos porras39ns xisc klscmllys visd s9 msrmirly rsmiow it visibility is immhen cmdmm is gov Cnmmm is mnl Howais matted e39u mnaw dismmkdby d39 lhe uzylmoelshzsedmtwocmepts i luzzyimpiimnon 2 me mrrrposrrorrai rule mimermce Fuzzylrmllntlm norms as a 7 h aandhaeluzzysels oomssmr A fuzzy relation is represented by a matrix A fuzzy relation 3 associated with the implication a a o is a fuzzy set of the oartesian product U K V R u vlmuv uE U1 VEV One ofthe most commonly used fuzzy implications is based on the min op leurv mmaur bv lumuslmzm 7o l1 Ris aleldion om A 0 B S is a Idatioll lmmB to C Hspeed s nnrma men brakmg mm s medmm Thecompos ion 01R and S is alelaion mm A to C denmm R S F quot3 IACDI39IIIIII39MWWJL AMEND 5quot a u Nurmabm WHO S40150 women Simila y an heused as ml Mimrmca mmquot mm mm Risamzzyrelanmmmmnv mamms MemumOD 51 W m 24 05 vusavuzzysnsauv lian WXMRWWX 39R moduwoms When uharamnsm Amnth are mecewxsa hymen nne waym carry vmmwm M m quotWm M E U um maxrmm mverenua sm name a mamxmnne Dnmpnsmnn The n 7 mame anunmpnsmnna Meranue 5 mm snmat no oethat mm aw ma mimaiy 1 ma THE malnxfnr ms EXamp E is ash mws Mummy 2 mnmsllmm an Exam my mmap 10 mmmammarymaxmmmmmmzyvmmmgmmnpxmme ThemalrlximmmexammesasmHnWs WMM W W quot39 quotquotquot M Bah bmmn nm M alike SP n I I 3 5 Tnegmevatsmwsgwenhv 5w 1 H mm mm mm mm mm mm m mm mm mm mm mm mm mm m quotHrle WHILE nunle mman mmll mum Thus anmegvenwaedvedm mnave 0 39l m mm mm mm mm mm mm mum10mm mummnmmnmnw quot 1 1m mm mm mm mm Mm Mmm r Immmwvhmmmm5mquotlt v Jmquotlt 15 39 1 mummwpumgpx m m my 1 n Emphfymgterms wahavethe fu nwmg WWWHWH hmn vx HIM 1 mm mm mmmmm 0mm mum 2 mum 1 nix mm 2 1 quotI quot1 2 J maxlxmnw mmm lmnu u mnnmnmm manna u s s s 2 n 515 555 a 2m 5 1 Equw em y m we vmm nmannn we mveme munwng mzzv repvssmannn Vanna hyang n 1 1 I I n 01mm he appueu 5 mm in m 311 mm Induced fuzzyser n n n n n H mm m mnmslmm 2 Nnrvmi Med urn speed brake force 391 7 Fuzzy mum Fuzzy mmm 1 i M 2 u M 7 U a 2149 so so Giveispeed mmn How to decide wnanddo W what breaking force to appm mummmrmm g v hlavumrmt 34 G on bid Tame GIVEquot tWD InPUtS Re reaemmg NE NS 1 P5 pa E difference between current temperature and target AH e RMBS NB Pa 6 temperature normalized by the target temperature E 23 PE P5 3 NS NE dE the time derivative of E dEdt N5 NBS Compute one output W the change in heat or cooling source NB NS 20 PS PB Fuzzy variables NB neg big NS 20 PS PB 1312 4 Example rule if E is 20 and dE is quotS then Wis PS assmated if E IS 20 and dE is PB thenWis NB Wm W95 0 71 0 1 mummmrmm 35 v hlavumrmt m 919m fj l iljl C is Suppose that E75 and dE0 Hence E is PB with degree 5 E is P5 with degree 5 E is 20 with degree 1 Two rules are applicable a mm A mme 5 A 1 5 g mPSE A mmdE 5 A 1 5 merVl 61 A mulel muwr a A mauwi Warm 71 w 0 139 v neruzzincation trans iating a ruuy category to a precise output neruzzincation using tire center at granny wnyisiuuyiogicsusuccessrui uunnrsumn ax Easily and succinctly represent expert system rules that involve continuous variables Model environment variables in terms of piece wise linear characteristic functions Warm More on Logical Reasoning about Uncertainty Learning uunnrsumn m

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