Answering. Question Answering: Result. Application of the Theorem Prover: Question. Overview. Question Answering: Example.

Transcription

1 ! &0/ $% &) &% &0/ $% &) &% 5 % 4! pplication of the Theorem rover: Question nswering iven a database of facts (ground instances) and axioms we can pose questions in predicate calculus and answer them using resolution Resolution can answer Yes/o answers but it can be extended to answer more complex questions such as Who? or What? etc This is called nswer Extraction Question nswering: Result in the end but Resolution on the previous example generates what that answers is the question Is there a grandparent of Harmonia? Of course the answer is yes but the question is who? The negated question in the above examples was Clearly the binding which 1% %$ - ' ( # $ ultimately receives is the desired answer! Observation: one substitution along the way starting from the negated conclusion is 1% %$ - ' ( # $ must be an answer % )$ thus )$ 3 in the example in the previous Exercise: use resolution to derive slide 4 Overview pplication of theorem proving: question answering Uncertainty ecision theory example robability basics Conditional probability xioms of probability Joint probability distribution Bayes rule Bayes rule: Example 1 Question nswering: Example Example: Question: Who is a grandparent of Harmonia? 1 egated: 3

2 98 $) $% &) &%! 98 $) 98 $) irst-order Logic: Summary Standard forms: prenex normal form skolemization C Resolution: negated conclusion substitution unification factors and resolvents Theorem provers: two-pointer method various deletion strategies various speed up strategies pplication of theorem provers: question answering 6 Example: Trying to Catch a light minutes before the flight departure time : plan to leave home : The traveler needs to make a decision in an uncertain environment: car can break down traffic can be extremely congested natural disaster etc Such worst-case scenarios are hard to explicitly enumerate: the list goes on ran out of gas spouse/children in an emergency flight crews goes on a strike etc etc Thus the traveler only has an incomplete understanding of the situation but this can <<= ; The traveler can play safe by going with plan cause the traveler to wait for a long time at the airport before 8 departure nswer Extraction We can introduce special predicates to extract the answers nswer predicate: & 6 1% &0/ %$ - ' ( # $ The answer predicate has these properties: It does not resolve with anything but it keeps track of variable bindings The theorem prover recognize a clause consisting only of the predicate as & or example resolution on the previous example results in: % )$ & as the final clause 5 Uncertainty roblem with first-order logic: agents almost never have full access to the whole truth about their environment Therefore the agent must act under uncertainty Uncertainty can also arise because of incompleteness and incorrectness in the agent s understanding of the properties in the environment Incomplete because there are too many conditions to explicitly enumerate There are trade-offs (playing safe can result in other annoyances) thus the right thing to do depends on both the relative importance of various goals and the likelihood (and degree to which) they will be achieved

3 ifficulties in pplying -O-L in Uncertain omains or example application of first-order logic in medical diagnosis domain can fail because of these reasons: Laziness: cannot list the complete set of antecedents and consequents needed to ensure an exceptionless rule and too hard to use the enormous rules that result Theoretical ignorance: medical science has no complete theory ractical ignorance: even though we have all the rules it is practically impossible to run all the tests Similar situation arises in law business dating etc The agent s knowledge can at best provide only a degree of belief robability theory is well suited for such a domain 9 Example When playing black jack as new cards are drawn and shown your degree of belief in the fact that you need more cards can change What about poker? or slot machine? 11 cquisition of ew Information and robability The degree of belief changes as an agent perceives or acquires new information from the world: we call this the evidence This is analogous to saying whether or not a given logical sentence is entailed by (ie is a logical consequence of) the knowledge base because the truth value can change when new facts are added to the KB Before the evidence is received we talk about prior or unconditional probability fter the evidence is obtained we talk about posterior or conditional probability 10 Rational ecisions Under Uncertainty: ecision Theory There are trade-offs and an agent must first have preferences between different results when a certain plan was executed Utility theory deals with such preferences: how useful is such and such result to the agent? ecision theory is a general theory of rational decision under uncertainty combining probability theory and utility theory 1

4 ecision Theory n agent is rational iff it chooses the action that yields the highest expected utility averaged over all possible outcomes of the action: rinciple of Maximum Expected Utility Example: backgammon (discussed earlier) min-max trees with probabilistic levels 13 ecision Theoretic gent function T-gent (percept) returns action static: a set probabilistic belief about the state of the world calculate updated probabilities for current state based on percept and past actions calculate outcome probabilities for actions given action descriptions and prob of current states select action with highest expected utility given prob of outcomes and utility information return action 14 ecision Theory: Example ecision theory = robability theory Utility theory Utility of Resulting State robability ction ction ecision Theory: Example ction Which action would an optimal ecision Theoretic gent take? 15 ecision theory = robability theory Utility theory robability Expected Utility Utility of Resulting State ction 1 10 ction 1000 ction 3 5 ction 3 has the maximum expected utility thus action 3 will be carried out 16

5 ! C ' ' E BBB - R H H! I I ' $V8 ) Z 00 c c c T %) ' Z 00 b _ a` Examples Boolean: ) K )L J & ) K )L J & Multivalued: BBB ; Q- 1L ) ; Q- 1L ) Multivalued: X M B- U V &&W T )$ %) S M R B- B BB Y 1&W% T )$ %) S 18 Conditional robability U B ( B) = = (/B) (B) /B Think about the area occupied by each event has an area of 1 thus The bounding rectangle rea of rea of rea of rea of Within this now takes on the role of means limited event space what is the probability of 0 robability: otations a Random variable: variable that can take on different values ) or : boolean values ( BBB - - : numerical values or other multivalued - enumerations (1 05 Cloudy Rainy Sunny ) having value : probability of the variable This can be viewed as an event C and means or boolean variables means : probability distribution a full list of probabilities for all is in bold can take (note that possible values that a ll conventions follow Russel & orvig 1 Logical Connectives and Conditional robability Logical connectives can be used: - etc J & 1WH % [ - Z - 6 ): given (read probability of Conditional robability ^MB T ) T %L ] 1WH % [ s new evidence comes in the conditional probability gets updated: $ % T ) T %L ] 1WH % [ 19

6 ! C d d C! H M H H m l p { ƒ { Œ ƒ œ ƒ i œ -BBB t s y z & Other roperties rom the axioms e e e is 1 for all More generally the sum of probabilities can take: the random variable values - nho ikj fhg can take is the set of all possible values where Joint robability istribution: Example Toothache Sum Toothache ƒ Cavity ˆ { ˆ Cavity } Šˆ { Š } Sum Ž ƒœ ƒ bbreviations: ƒ ƒ Š } ž Ÿ ž œ Š } ž Ÿ ž ƒ In practice writing a full joint probability table like this is impossible (or entries rr boolean random variables you need too much effort): for 4 The xioms of robability ll axioms 1 ll probabilities are between 0 and 1 under all interpretations): or a valid proposition ( under all ( proposition inconsistent a for and - interpretations): e 6 3 Other properties follow from these three axioms 1 Joint robability istribution r q - -BBB ; - or random variables n atomic event is an assignment of particular values to each random variable r q - -BBB ; - I The joint probability distribution completely specifies the probabilities of all atomic events Thus - nho H r r Hq - q H; - ; fhg xlj jvuwww u jtu vectors that the vector is a set of all possible where can assume r q - -BB ; - 3

7 x x x s «««± ] ] ] Extended Bayes Rule This rule follows from : «««ote: Exercise: text book exercise 145b and 146 (p 434) 6 Solution: ood ews and Bad ews These are given: ] MMMM B MMM M- We want to calculate the probability that you have the disease given a postive test result: ] We can use Bayes rule to derive this probability 8 Bayes Rule s s we and x s rom get and in turn from which we get the Bayes Rule: 5 Example: pplication of Bayes Rule Exercise 143 (p 433): fter your yearly checkup the doctor has bad news and good news The bad news is that you tested positive for a serious disease and that the test is 99% accurate (ie the probability of testing positive given that you have the disease is 099 as is the probability of testing negative given you don t have the disease) The good news is that this is a rare disease striking only 1 in people Why is it good news that the disease is rare? What are the chances that you actually have the disease? : : have disease : tested negative : tested positive clean

8 ] ] Solution: ood ews and Bad ews (cont d) ] e ] ] Observation a : ^OMM O OOOMB e MMMMB ] Thus and with this ^ - OMMMB MMMM B ^OMM ] which is slightly less than 1% is greater ] Exercise: how accurate should the test be so that than 095 (ie 95%)? See page 11 of this lecture ]Z e ]Z ] a 30 Key oints pplication of theorem proving: question answering Uncertainty ecision theory example: how prob theory and decision theory are combined robability basics: terminology notations Solution: ood ews and Bad ews (cont d) ] ] ] MMMM B are and ] given OO rom these we can get and ] ] ] Since Joint probability distribution: concept Conditional probability: definition various ways of representing conditional prob xioms of probability: basic axioms and using them to prove simple equalities 3 Bayes rule: definition and application are give we only need to calculate and 9 What s The Big eal? may be easier to obtain: you can run the test on a ] small pool of known patients (say 100) at a hospital is much harder to obtain directly Since the test makes ] 1 mistake out of 100 tests if you run the test on people you ll get 100 false-positives and one genuine patient who tests ) So just to get ^OMM ] positive (consider that about 100 people testing positive you have to run the tests on people serves as a prior in this case In many cases the prior represents subjective belief of the person calculating the is not directly measurable probability in case 31

9 ext Time More on Bayes rule robabilistic reasoning: chapter 15 33