Note 14 for CMPSCI 683 at UMass
Note 14 for CMPSCI 683 at UMass
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Date Created: 02/06/15
Via Victor Lesser CMPSCI 683 Fall 2004 Easy Sast39n39rahleprnhlans wi39ere many wlminns More clauses torthe llard Sas ahleprnhlanswhereimsnlminns WENTle EasqumSatis ahle problems Val al lesmm constrains H yi 39 nutrenqu r y Mlnilibiirrlm nillblimlx39 r v Assun esonnourrent search in the satis ahle spaoeand the nnnrsatis ahle space i negation Mprnpnsitinnl Fhasetransitinn where ll39h satis ahleand Eli39h nnnsatis ahle mainline November 2 Tuesday in class Open book but no computers Covering only material through chapter 145 No material on utility theory or decision trees Style of questions Mix of Short essay and Technique Homework 3 is a good example Mammy Review ofHomework1 Making decisions under uncertainty using utility theory chapter 16 The value of information Decision Trees Mammy 3 25 Islamer Smh IslandrEased searnh is a teehntnne where tnsted nr ndtng a path dneetty tn the gna1nne rst ldmtl es an tstand that is a nnde mum n less hatrway hetwem the mmal nnde and the gual nnde The sereh than pmeeeds as rnunws rstanatlemptxs made tn nd anaeeeptahte path tn the gual that 1 25 4 CONSISWHCyt Monommcnyv Adm 551b1htY AmAZ 47 passesunnughmtststand Ths rst step aetuan than has twn subspans One rst nds a path 39umslannudem lslandand nail searchesfurapath fmmlslartd tn gual xrnnaeeeptahtepath Prove T4 391 a heunsuc 5 comment 139 639 monomrf c t quot159 thrnngnan tstand eanhe rnnnd we snnpty sulve the ungnal prnhtemtnsted xtts assumed that admlsslble Construct and demonstrate an admlsslble heuristh sarchcanunlybepa39fmmed rnrwardheeo e tuwards the gual unhkebxdlrechunal smr h that is not consistent a Assume that the ume needed tn sereha tree wtth bmnchmg entan and depth dts mm Ins snme arbitrary ennstant tn matntatn geneahty that the ttme required tn tdentary a snttahte tstand is s and the pmbabthty that the tstand ls nn an aeeeptah1enntneeeesen1y nptnnat path tn the gnat tsp Fmd the Eundmuns Buy and c sneh that the aveage Expected ume required by the 2 25 would bidirectional A search be a good idea islandrdnven appmach wtu he less than the ume needed by breadthr rst sereh If so under what conditions would ll be applicable When Hot n owe an Example nra serehprnntem wherexslandrdnven Sarah ts hkety tn save time Describe the algorithm39s workings and space time requirement e Dtsenss snme pnssthte extenstnns tn the tstand sereh patadtenand tharputmtlal nsks andbene ts 5 v tax tease end a s n g 4 25 Heuristic Sebcu39nn Selectinn made We WWW Combl m quot9 Bel Iefs and Des Ires whteh Vary W ym hnth the enstttmetn geneate a scureand the dehty ufthe hm secure The U nder U ncertai nty better heunsues are mum aeentate l e they nndeesunate by less hut they tnenr a mneh large enmpntattnnat enst e g the eatnattnn funcnm genemtes and searnhesthe almust entne wane a Desmhe In detad the densmn prunedure that yuur agEnt wuuld use tn seteet amung these heunsucs at eeh step m tts sereh tn mmm e nvean sereh ume Dr rather cumpu39ztmnal effm tn a npumam sntuunn rather than neustng nn the e emve branchmg eetnr urnumba39 ufnudes B as s of Theo ry expanded Yuur dennnn prunedure shnntd take mm aeennnt the fu uwmg pammela s Heunstn cumputatmn ttme nnde geneatann tame and nwnhe nrnndes geneatedh e the erreetwe henehtng t39aetnr that that heunsue makes fur A nn the wane and whatever else ynn thmk is necessary a e exphnt ahnnt any addttannat assumptmnsyuu nake ahnnt the wane b Dues swttehtng hetwem adrntssthte heunsues ted tn an admsslble heunsuc7 v Mme end 7 v Mme end Knowing the current state of the world requires perception learning knowledge representation and should choose an action that maximizes its inference The MEU principle says that a rational agent Computing P requires a complete causal model ofthe world Computing the utility ofa state often requires search or EUQIE maXA Z pResult AlDoAyE UResult A planning distinguish between explicit and implicit utility Calculation ofUtlllty of a partlcularstate may require us to look at what utilities could be achieved from that state i All ofthe above can be computationally intractable Why lsn t the MEU prmmple a we need In hence one needs to distinguish between perfect order to bund intelligent agents rationalityquot and resourcebounded rationalityquot or 7 Is it Difficult to Computer PE orU 7 boundedoptimalityquot Also Need to consider more than one action oneshot decisions versus sequential decisions v Wrasse nmt g v omega nmt in expected utility in the current state E Why make decisions based on average or expected Utility basic set of assumptions Lotteries are used to describe scenarios of choice Why can one assume that utility functions with probabilistic outcomes exist 7 Kev to the idea or formallzlr lg preterence structures and relating therrl to MEU can an agent aCt ratlonally by expressmg Different outcomes correspond to different prizes preferences between states without giving 7 L pA lea them numeric values Can have any number of outcomes an outcome ofa lottery can be another lottery Can every preference structure be a L ipiiciipzczi one captured by assigning a single number to Mi 3 WM 9292 prior A lottery with only one outcome can be written as every State 1A or simply A v Wrasse nmt ii v omega nmt i2 Let A and B be two possible outcomes Orderability the agent know what it wants AgtBVBgtA VAEB A gt B Outcome A is preferred to B Transitivity AgtBABgtC AgtC A a B The agent is indifferent continuity between A and B A gt B gt C 2 ElppA 1CgtCE B A z B The agent prefers A to B or I SUbsmmability is indifferent between them A E B gt Vp p Ai 1 p c 5 0B 1 p c v mm m is v mm m 14 M quot t quoti ity Theorem If an agent39s preferences obey A gt B 2 P 2 q PW 139PYB the axioms of utility theory then there exists 2 WA 139quD a realvalued function U that operates on Decomposability states such that PyAi 1Piqui 141101 5 PA 1Pqu PW CDC UA gt UB A gt B and UA UB gt A5 B If Preference Structure Obeys Axioms Can be Mapped into a Lottery v mm m 15 v mm m a Theorem The utility of a lottery is the sum of probabilities of each outcome times the utility of that outcome UPi13i P2132 Pmsnl 2 Pi USi Q Does the existence ofa utility function that captures the agent39s preference structure imply that a rational agent must act by maximizing expected utility msiwzm n Example You can take a 1000000 prize or gamble on it by ipping a coin lfyou gamble you will either triple the prize or loose it EMV expected monetary value ofthe lottery is 1500000 but does it have higher utility Bernoulli39s 1738 St Petersburg Paradox Toss a coin until it comes up heads If it happens after n times you receive 2quotdollars EMVSt P 2 my 2 inf How much should you pay to participate in this game mnmsixmim ix Um m U large m ltlt m mmcney Decreasing marginal utility for money Will buy affordable insurance Will only take gambles with substantial positive expected monetary payoff xcsiwm w m m U large m gtgt m Increasing marginal utility for money Will not buy insurance Will sometimes participate in unfavorable gamble having negative expected monetary payoff mnmsixmim 2n Riskneutral agents linear curve Regardless of the attitude towards risk the utility function can always be approximated by a straight line over a small range of monetary outcome The certainty equivalent of a lottery Example Most people will accept about 400 in lieu of a gamble that gives 1000 half the times and 0 the other half v mm m 21 ls decision theory compatible with human judgment under uncertainty Does itoutperform humanjudgment in micromacro worlds Are people experts in reasoning under uncertainty How well do they perform What kind of heuristics do they use The impact of automated techniques for reasoning under uncertainty on our capability in future forecasting policy formation etc v mm m 22 Choose between lotteries A and B and then between C and D A 80 chance of4000 C 20 chance of 4000 B 100 chance of 3000 D 25 chance of 3000 The majority of the subjects choose B over A and C over D But if Um m we get 08 U4000 lt u3000 and 02 U4000 gt 025 u3000 contradicts the axioms a400020lt1300000 253000750gt24000a0 v mm m 23 Utility functions are not unique for a given preference structure U39S a b US Normalized utility U 0 Utilityworst possible catastrophe U 1 Utilitybest possible prize Can find the utility of a state S by adjusting the probability p of a standard lottery pU 1pU that makes the agent indifferent between S and the lottery v mm m 24 Several standard currencies are used Micromort a one in a million chance of immediate death 1 micromort 20 in 1980 dollars QALY Quality Adjusted Life Year a year in good health with no infirmities These measures are useful for decision making With small incremental risks and rewards 25 Why multiattribute Example evaluating a new job offer salary commute time quality oflife etc Uabc ff1af2b where f is a simple function such as add39 39 n r ln can of mutull putmic independence which occur Whirl It i Ilwln Prif39r39hl to lncrelse on VIII of In Ittrihull 3quotquot III Mr lttributil Irl xll Dominance strict dominance vs stochastic dominance For every point Probablistic view v mm m za l in m i im M A l i l a c l u l i i 7 7 7 7 7 AL V A i 3 i m m gun la Sinuumiimm udelcrminmm UpliunA Mimi Himalaya but ml by Ler mi Unumim A it lntll timmiiiti by a mum t c strict dominance occurs irari uptiun lS ufluweryalue on all attributes than some Either umiun um 2UA since UEgtltlgtlt2 2u Agtltix2 27 rl5 J 35 Mgv w m lb39i miliii iimiimigx 32 mi um lle ii min umlwlrihulimhHrrliiuiiqidukumHiSiHrillf z PSi ELM P822Ui v mm m 22 39 Example 11 YOU COHSider buying a Program to Expected utility given information manage yourfinances that costs 100 There is o75oo100030 a prior probability of 07 that the program is suitable in which case it will have a positive effect 0 yourwork Worth 500 There iS a probability Expected utility not given information of 03 that the program is not suitable in which 07500100030100 case it will have no effect What is the value of knowing whether the Value Of Information program is suitable before buying it 07500100030 07500100030 100 280 250 30 v we m 29 v we m u What can the company do with the information Example 2 Suppose an oil company is hoping to buy one of n blocks of ocean drilling rights Exactly one block contains 0 worth C dollars The price of each block is Cn dollars If the company is riskneutral it will be indifferent between buying a block or not WHY Case 2 block 3 contains no on p n1n A seismologist offers the company a survey Company will buy different block and make indicating whether block 3 contains oil Cn 1 Cn Cn n 1 dollars How much should the company be willing to pay for the information Case 1 block 3 contains 0 p1n Company will buy it and make a profit of C Cn n1 Cn dollars Now the overall expected profit is Cn Q What is the value of information v we m 31 v we m 2 The general case We assume that With the information the value of the new exact evidence can be obtained about beSt amen Wi be3 EUaElEiE the value of some random variable Ej maXA Z F ResultiA D0AEE UResult A But EJ is a random variable whose value is The value ofthe current best action a is currently unknown so we must average over all possible values ek using our current d r d b 8 me y belief VPEEJ EUC E maxiE i ADoAE Zk pKEFelkl E EUUerkI EY EJ elk EUa E o The agent39s current knowledge is E v mm m 4 v mm m 33 In general ii MW VPIEEJEK VPIEEJ VPIEEK gt But the order is not important a A VPIEEJEK L J i i J Er Urine um VPIEEJ VPIEVEJEK VPIEEK m VPIE EKEj quot 5 572 lii 3 llflig quotquot5233 2 mm What about the value of imperfect um immune 7 Informatlon39 Utility Distributions forActioi iS Aland A over the range oflhe random variable E v mm m a v mm m 35 Decision Trees and Networks Markov Decision Processes MDPs NonProbablistic Ways of Reasoning about Uncertainty v mew m 7
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