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© C. Kemke Inexact Reasoning 1 1 Uncertainty and Rules We have already seen that expert systems can operate within the realm of uncertainty. There are several sources of uncertainty in rules: – Uncertainty related to individual rules – Uncertainty due to conflict resolution – Uncertainty due to incompatibility of rules
© C. Kemke Inexact Reasoning 2 2 Figure 5.1 Major Uncertainties in Rule-Based Expert Systems
© C. Kemke Inexact Reasoning 3 3 Figure 5.2 Uncertainty Associated with the Compatibilities of Rules
© C. Kemke Inexact Reasoning 4 4 Knowledge Engineer The knowledge engineer endeavors to minimize, or eliminate, uncertainty if possible. Minimizing uncertainty is part of the verification of rules. Verification is concerned with the correctness of the system’s building blocks – rules.
© C. Kemke Inexact Reasoning 5 5 Verification vs. Validation Even if all the rules are correct, it does not necessarily mean that the system will give the correct answer. Verification refers to minimizing the local uncertainties. Validation refers to minimizing the global uncertainties of the entire expert system. Uncertainties are associated with creation of rules and also with assignment of values.
© C. Kemke Inexact Reasoning 6 6 Sources of Uncertainty Potential contradiction of rules – the rules may fire with contradictory consequents, possibly as a result of antecedents not being specified properly. Subsumption of rules – one rules is subsumed by another if a portion of its antecedent is a subset of another rule.
© C. Kemke Inexact Reasoning 7 Information is partial Information is not fully reliable. Representation language is inherently imprecise. Information comes from multiple sources and it is conflicting. Information is approximate Non-absolute cause-effect relationships exist Cont’d…
© C. Kemke Inexact Reasoning 8 In many cases, our knowledge of the world is incomplete (not enough information) or uncertain (sensors are unreliable). Often, rules about the domain are incomplete or even incorrect We have to act in spite of this! Drawing conclusions under uncertainty
© C. Kemke Inexact Reasoning 9 9 Uncertainty When a fact is entered in the working memory, it receives a unique timetag – indicating when it was entered. The order that rules are entered may be a factor in conflict resolution – if the inference engine cannot prioritize rules, arbitrary choices must be made. Redundant rules are accidentally entered / occur when a rule is modified by pattern deletion.
© C. Kemke Inexact Reasoning 10 10 Uncertainty Deciding which redundant rule to delete is not a trivial matter. Uncertainty arising from missing rules occurs if the human expert forgets or is unaware of a rule. Data fusion is another cause of uncertainty – fusing of data from different types of information.
© C. Kemke Inexact Reasoning 11 11 State of Uncertainty There are two mountains – logic and uncertainty Expert systems are built on the mountain of logic and must reach valid conclusions given a set of premises – valid conclusions given that – – The rules were written correctly – The facts upon which the inference engine generates valid conclusions are true facts
© C. Kemke Inexact Reasoning 12 Knowledge & Inexact Reasoning inexact knowledge (truth of not clear) incomplete knowledge (lack of knowledge about ) defaults, beliefs (assumption about truth of ) contradictory knowledge ( true and false) vague knowledge (truth of not 0/1)
© C. Kemke Inexact Reasoning 13 Inexact Reasoning CF Theory - uncertainty uncertainty about facts and conclusions Fuzzy - vagueness truth not 0 or 1 but graded (membership fct.) Truth Maintenance - beliefs, defaults assumptions about facts, can be revised Probability Theory - likelihood of events statistical model of knowledge
© C. Kemke Inexact Reasoning 14 Inexact Reasoning not necessary... NOT necessary when assuming: complete knowledge about the "world" no contradictory facts or rules everything is either true or false This corresponds formally to a complete consistent theory in First-Order Logic, i.e. everything you have to model is contained in the theory, i.e. your theory or domain model is complete facts are true or false (assuming your rules are true) your sets of facts and rules contain no contradiction (are consistent)
© C. Kemke Inexact Reasoning 15 Exact Reasoning: Theories in First-Order Predicate Logic Theory (Knowledge Base) given as a set of well-formed formulae. Formulae include facts like mother (Mary, Peter) and rules like mother (x, y) child (y, x) Reasoning based on applying rules of inference of first-order predicate logic, like Modus Ponens: If p and p q given then q can be inferred (proven) p, p q q
© C. Kemke Inexact Reasoning 16 Forms of Inexact Knowledge uncertainty (truth not clear) –probabilistic models, multi-valued logic (true, false, don't know,...), certainty factor theory incomplete knowledge (lack of knowledge) –P true or false not known ( defaults) defaults, beliefs (assumptions about truth) –assume P is true, as long as there is no counter-evidence (i.e. that ¬P is true) –assume P is true with Certainty Factor contradictory knowledge (true and false) –inconsistent fact base; somehow P and ¬P true vague knowledge (truth value not 0/1; not crisp sets) –graded truth; fuzzy sets
© C. Kemke Inexact Reasoning 17 Inexact Knowledge - Example Person A walks on Campus towards the bus stop. A few hundred yards away A sees someone and is quite sure that it's his next-door neighbor B who usually goes by car to the University. A screams B's name. default - A wants to take a bus belief, (un)certainty - it's the neighbor B probability, default, uncertainty - the neighbor goes home by car default - A wants to get a lift default - A wants to go home Q: Which forms of inexact knowledge and reasoning are involved here?
© C. Kemke Inexact Reasoning 18 Examples of Inexact Knowledge Person A walks on Campus towards the bus stop. A few hundred yards away A sees someone and is quite sure that it's his next- door neighbor B who usually goes by car to the University. A screams B's name. Fuzzy - a few hundred yards define a mapping from "#hundreds" to 'few', 'many',... not uncertain or incomplete but graded, vague Probabilistic - the neighbor usually goes by car probability based on measure of how often he takes car; calculates always p(F) = 1 - p(¬F) Belief - it's his next-door neighbor B "reasoned assumption", assumed to be true Default - A wants to take a bus assumption based on commonsense knowledge
© C. Kemke Inexact Reasoning 19 Dealing with Inexact Knowledge Methods for representing and handling: 1.incomplete knowledge: defaults, beliefs Truth Maintenance Systems (TMS); non-monotonic reasoning 2.contradictory knowledge: contradictory facts or different conclusions, based on defaults or beliefs TMS, Certainty Factors,..., multi-valued logics 3.uncertain knowledge: hypotheses, statistics Certainty Factors, Probability Theory 4.vague knowledge: "graded" truth Fuzzy, rough sets 5.inexact knowledge and reasoning involves 1-4; clear 0/1 truth value cannot be assigned
© C. Kemke Inexact Reasoning 20 In many cases, our knowledge of the world is incomplete (not enough information) or uncertain (sensors are unreliable). Often, rules about the domain are incomplete or even incorrect We have to act in spite of this! Drawing conclusions under uncertainty
© C. Kemke Inexact Reasoning 21 Example Goal: The agent wants to drive someone to air port to catch a flight Let action A t = leave for airport t minutes before flight Will A t get me there on time? Problems: 1. partial observability (road state, other drivers' plans, etc.) 2. noisy sensors (traffic reports) 3. uncertainty in action outcomes (flat tire, etc.) 4. immense complexity of modeling and predicting traffic Hence a purely logical approach either 1. risks falsehood: “A 25 will get me there on time”, or 2. leads to conclusions that are too weak for decision making: “A 25 will get me there on time if there's no accident on the bridge and it doesn't rain and my tires remain intact etc etc.” (A 1440 might reasonably be said to get me there on time but I'd have to stay overnight in the airport …)
© C. Kemke Inexact Reasoning 22 Making decisions under uncertainty Suppose I believe the following: P(A 25 gets me there on time | …) = 0.04 P(A 90 gets me there on time | …) = 0.70 P(A 120 gets me there on time | …) = 0.95 P(A 1440 gets me there on time | …) = 0.9999 Which action to choose? Which one is rational? Depends on my preferences for missing flight vs. time spent waiting, etc. Utility theory is used to represent and infer preferences Decision theory = probability theory + utility theory The fundamental idea of decision theory is that an agent is rational if and only if it chooses the action that yields that highest expected utility, averaged over all the possible outcomes of the action.
© C. Kemke Inexact Reasoning 23 Uncertainty in logical rules
© C. Kemke Inexact Reasoning 24 Imagine an urn containing 1500 red, pink, yellow, blue and white marbles. Take one ball from the urn. What is: P(black) = P(~black) = ~ = NOT 0 1 Probabilities are all greater than or equal to zero and less than or equal to one. Probability
© C. Kemke Inexact Reasoning 25 Same urn: Suppose the number of balls is as follows: Red400 Pink100 Yellow400 Blue500 White100 Total 1500 What is: P(Red) = P(Pink) = P(Yellow) = P(Blue) = P(White) = Total = 400/1500 =.267 100/1500 =.067 400/1500 =.267 500/1500 =.333 100/1500 =.067 1
© C. Kemke Inexact Reasoning 26 Joint probabilities and independence Define A as the event “draw a red or a pink marble.” We know 500 marbles are either red or pink. What are:P(A) = P(~A) = (1 - P(A)) =.67 =.33
© C. Kemke Inexact Reasoning 27 Joint probabilities and independence (we’re getting there) Define B as the event, “draw a pink or white marble.” We know 200 marbles are pink or white. What are: P(B) = P(~B) =.133.867
© C. Kemke Inexact Reasoning 28 Joint probabilities and independence Define A as the event “draw a red or a pink marble.” Define B as the event “draw a pink or white marble.” What is: P(A, B) = P(A B) This is the joint probability of A and B. What color is the marble? Pink P(A, B) = P(pink) ==.0667
© C. Kemke Inexact Reasoning 29 Conditional probabilities What is P(A | B) = P(~A | ~B) = P(A | ~B) =P(~B | A) = The probability that a particular event will occur, given we already know that another event has occurred. We have information to bring to bear on the base rate probability of the event 1500
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