Artificial Intelligence CMPSCI 683
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This 10 page Class Notes was uploaded by Roman McCullough on Friday October 30, 2015. The Class Notes belongs to CMPSCI 683 at University of Massachusetts taught by Staff in Fall. Since its upload, it has received 8 views. For similar materials see /class/232283/cmpsci-683-university-of-massachusetts in ComputerScienence at University of Massachusetts.
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Date Created: 10/30/15
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 Victor Lesser Mix of Short essay and Technique CMPSCI 683 Homework 3 is a good example Fall 2004 ylmyuamrm Z Easy Sast39n39rahleprnhlans wi39ere many wlminns More clauses tnrlhe llard Sas ahleprnhlansudrerelmsnlminns martinile Review of Homework 1 EasqumSatis ahle problems Vallablesmm canstrairts LT my 3 quot quotquotquotquot Making decisions under uncertainty using utility theory chapter 16 The value of information Decision Trees run H h 3 ryrmmm m v Assun esrnneurrem search in the satis ahle spareand the nnnrsatist rahle space i negation r71 prnpnsitinnl Phasetransitinn wirere ll39h satis ahleand Eli39h nnnsatis ahle y tummy r y lunamrm l 1 25 Consistency Monotonicity Admissibility AmAZ 47 Prove that if a heuristic is consistent i e monotonic it must be admissible Construct and demonstrate an admissible heuristic that is not consistent 2 25 Would bidirectional A search be a good idea If so under what conditions would it be applicable when not WUIIIH 7an 3 25 lslan leBasa l Smith u rn ll lP tl an island that is a nude mdie m less halfway between the initial nude and the gual nude The thruugh an island eah he rduhd we simply sulve the diighal pmhlem mstmd lt is assumed that l i islandrdnvm appmaeh Will he less than the time needed by hieadthhrst smmh b Give an Example dra samhrprublemywha39elslandrdnvm smreh is likelth save time i amuse mm 4 25 Heuristic seleednh Selaeu39nh meh step in its smrehtd mmmize dveall smreh umeurrathar edmputatidhal effm tn a r mm W liii p ed any additidhal assumptiemsydu rrake ahdut the maee b Dues smtehmg belwem admissible heimsties lmd td an admlsslble heunstlc7 Combining Beliefs and Desires Under Uncertainty 0 Basis of Utility Theory i amuse mm The MEU principle says that a rational agent should choose an action that maximizes its expected utility in the current state E EUotE maxA z PResultADoAE UResultA Why isn t the MEU principle all we need in order to build intelligent agents 7 Is it Difficult to Computer PE orU Why make decisions based on average or expected utility hy can one assume that utility functions exist Can an agent act rationally by expressing preferences between states without giving them numeric values Can every preference structure be captured by assigning a single number to every state Knowing the current state of the world requires perception learning knowledge representation and in erence Computing P requires a complete causal model ofthe world Computing the utility ofa state often requires search or planning distinguish between explicit and implicit utility calculation ofUtlllty of a partlcularstate may reoulre us to look a w at utllltles could be ac leved from that state All ofthe above can be computationally intractable hence one needs to distinguish between perfect rationalityquot and resourcebounded rationalityquot or boundedoptimalityquot Also Need to consider more than one action oneshot decisions versus sequential decisions v llama rm lo The MEU principle can be derived from a more basic set of assumptions Lotteries are used to describe scenarios of choice with probabilistic outcomes 7 Kev to he loea le rorrnallerlg preference siruclures arlo relallrlg them to MEU Different outcomes correspond to different prizes 7 L pA lapB Can have any number of outcomes an outcome ofa lottery can be another lottery Dnlcn s L DA lap Pllcll Darcy P an A lottery with only one outco e can be written as rELA or simply A Let A and B be two possible outcomes A gt B Outcome A is preferred to B A a B The agent is indifferent between A and A z B The agent prefers A to B or is indifferent between them v New m 13 Orderability the agent know what it wants AgtBVBgtA VAEB Transitivity AgtBABgtC AgtC Continuity Agt BgtC 2 EippA1pCs B Substitutability A E B VppA1pCEpB1pc H mm m 14 Monotonicity A gt B 2 p 2 q s pA 198 2 MA 1qBD Decomposability PA 1PqB 14110 5 PA 1PqB 1P1qC f Preference Structure Obeys Axioms Can be Mapped into a Lottery v New m 15 Theorem If an agent39s preferences obey the axioms of utility theory then there exists a realvalued function U that operates on states such that UA gt UB gt Agt B and UA UB gt As B H 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 vummm 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 alter n times you receive 2quotdollars EMVSt P 2 my 2 inf How much should you pay to participate in this mnmsixmim ix 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 vummm 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 mew 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 H 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 mew 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 U 39 1pU that makes the agent indifferent between S and the lottery H 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 Why multiattribute Example evaluating a new job offer salary commute time quality oflife etc Uabc ff1af2b where f is a simple function such as addition r In an of mutull nrlflrlnco indopendonu which occur Mun It i llwlyl pnfmlalo u henna cm In of In Itmhuu giqu nil nthquot mributn ll39l fiqu Dominance strict dominance vs stochastic dominance For every point Probablistic view t in mm m l visugru i mum l l JDklcrminnln opumA in mail dumuiiilul by E m gt domnmh39tl by B m not t r a W i um 2UA since UEXlX2ZUAX1x2 5 5 x x l z 5 my w m In mure m4 ilm bmm livmlrm rriixl luu limnhumm rm m mm 5i Um millAli ilmviumluk 32 in uni llw w 5 mi 92 PSl ELM P822Ul t in mm m 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 07500100O30 100 280 250 30 v New m 29 u we nml u Example 2 Suppose an oil company is hoping to What can the company do with the information buy one of n blocks of ocean drilling rights Exactly one block contains oil 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 WH 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 miiCh Shoiiidghe company be Wiiiing to Pay Now the overall expected profit is Cn for the information Q What is the value of information Case 1 block 3 contains oil p1n Company will buy it and make a profit of C Cn n1 Cn dollars v New m i u we nml 2 The general case We assume that exact evidence can be obtained about the value of some random variable E o The agent39s current knowledge is E o The value of the current best action or is defined by EUccE maxA z PResult ADoAE UResult A v mm m is With the information the value of the new best action will be EUaEJEE maxA z PResult A DoAEE UResult A But EJ is a random variable whose value is currently unknown so we must average over all possible values ek using our current belief VPIE E1 2k PEfeiu E EUWWI E E err EUOtl E v in mm m 4 In general VPIEEJEK 2 VPIEEJ VPIEEK But the order is not important VPIEEJEK VPIEEJ VPIEVEJEK VPIEEK VPIEJEKEJ What about the value of imperfect information v mm m 35 ill H r gm NEFIJ quot1m mu m m in gun167 nimgmm canalUr iii VKiluL ufln uithluu Iiixlh murmur m A mm ll 39l l 39 i am din infannmion is less aluublc Utility Distributions forActlonS Aland A over the range oflhe random variable E m Decision Trees and Networks Markov Decision Processes MDPs NonProbablistic Ways of Reasoning about Uncertainty v owe m 7