/ Informatica. Intelligent Systems. Agents dealing with uncertainty. Example: driving to the airport. Why logic fails


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1 Intelligent Systems Prof. dr. Paul De Bra Technische Universiteit Eindhoven Agents dealing with uncertainty Uncertainty Probability Syntax and Semantics Inference Independence and Bayes' Rule Intelligent Systems Intelligent Systems 3 Example: driving to the airport 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, engine failure, etc.) 4. immense complexity of modeling and predicting traffic Hence a purely logical approach either 1. risks falsehood: A 60 will get me there on time, or 2. leads to conclusions that are too weak for decision making: A 60 will get me there on time if there's no accident on the highway and it doesn't rain and my tires remain intact, and the parking lot isn t full, etc. (A 180 will get me there on time even if I have to abandon my car along the way and walk, but I risk having a very long wait before my flight.) Intelligent Systems 4 Why logic fails Example: dental diagnosis p Symptom(p, Toothache) Disease(p, Cavity) this rule is wrong: a cavity is not the only cause of a toothache p Symptom(p, Toothache) Disease(p, Cavity) Disease(p, GumDisease) Disease(p, Abscess) unfortunately the causes for toothache are almost unlimited p Disease(p, Cavity) Symptom(p, Toothache) this is also not true (after a root canal you do not feel a cavity) Three reasons why (firstorder) logic fails: Laziness: too much work to write all the rule that apply Theoretical ignorance: medical science is largely incomplete Practical ignorance: you cannot run all the test needed to be certain Intelligent Systems 5 Reasoning with uncertainty fuzzy logic We deal with sentences that are either true or false (in the real world) the patient either has a cavity or has no cavity we may use belief or probability: we are 80% certain that the patient has a cavity it is not 80% true that the patient has a cavity In fuzzy logic a sentence can be partially true: truth is a value between 0 and 1 T(A B) = min(t(a),t(b)), T(A B) = max(t(a),t(b), T( A) = 1 T(A) We deal with uncertainty, not with fuzzy logic Intelligent Systems 6 Methods for handling uncertainty Default or nonmonotonic logic: Assume my car does not have a flat tire Assume A 60 works unless contradicted by evidence Issues: Which assumptions are reasonable? How to handle contradiction? Rules with fudge factors : A get there on time Sprinkler 0.99 WetGrass WetGrass 0.7 Rain Issues: Problems with combination, e.g., Sprinkler causes Rain?? Probability Model agent's degree of belief Given the available evidence, A 45 will get me there on time with probability Intelligent Systems 7
2 Nonmonotonic logics Circumscription (closed world assumption): uses predicates that are false for every object unless known to be true (for abnormal objects) Bird(x) Abnormal(x) Flies(x) this is model preference logic: entailment when true in all preferred models of the KB Default logic (uses default rules): Bird(x) : Flies(x) Flies(x) if Bird(x) is true, and Flies(x) is consistent with KB then Flies(x) is concluded by default in general: P : J1,,Jn C, P is prerequisite, C is conclusion, Ji are justifications We are going to use probability instead of nonmonotonic logics Intelligent Systems 8 Probability Probabilistic assertions summarize effects of laziness : failure to enumerate exceptions, qualifications, etc. ignorance : lack of relevant facts, initial conditions, etc. Subjective probability: Probabilities relate propositions to agent's own state of knowledge e.g., P(A 45 no reported accidents) = 0.05 These are not assertions about the world Probabilities of propositions change with new evidence: e.g., P(A 45 no reported accidents, 5 a.m.) = Intelligent Systems 9 Making decisions under uncertainty Suppose I believe the following: P(A 45 gets me there on time ) = 0.05 P(A 60 gets me there on time ) = 0.70 P(A 90 gets me there on time ) = 0.95 P(A 270 gets me there on time ) = Which action to choose? 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 Intelligent Systems 10 Syntax Basic element: random variable Similar to propositional logic: possible worlds defined by assignment of values to random variables. Three kinds: Boolean, discrete and continuous random variables e.g., cavity (tf), weather (sunny,rainy,cloudy,snow>, time, speed, size Domain values must be exhaustive and mutually exclusive Elementary proposition constructed by assignment of a value to a random variable: e.g., Weather = sunny, Cavity = false Complex propositions formed from elementary propositions and standard logical connectives e.g., Weather = sunny Cavity = false Intelligent Systems 11 Probability of continuous variables With continuous variables the probability of any specific value is always 0 We use a probability density function: assume is distributed uniformly between 5 and 15 P( = 10.0) = U[5,15](10.0) = 0.1C. This does not mean the chance that will be 10.0 is 0.1 This means: lim P(10.0 dx 10.0+dx) 2dx = 0.1C. dx 0 Syntax Atomic event: A complete specification of the state of the world about which the agent is uncertain E.g., if the world consists of only two Boolean variables Cavity and Toothache, then there are 4 distinct atomic events: Cavity = false Toothache = false Cavity = false Toothache = true Cavity = true Toothache = false Cavity = true Toothache = true Atomic events are mutually exclusive and exhaustive Intelligent Systems Intelligent Systems 13
3 Axioms of probability For any propositions A, B 1. 0 P(A) 1 2. P(true) = 1 and P(false) = 0 3. P(A B) = P(A) + P(B) P(A B) Exercise: show that P( A) = 1 P(A) using the above axioms This exercise can be easily extended to the discrete case Exercise: why is it impossible that: P(A) = 0.4, P(B) = 0.3, P(A B) = 0.0 and P(A B) = Intelligent Systems 14 Prior probability Prioror unconditional probabilities of propositions e.g., P(Cavity = true) = 0.25 and P(Weather = sunny) = 0.72 correspond to belief prior to arrival of any (new) evidence Probability distribution gives values for all possible assignments: P(Weather) = <0.72,0.1,0.08,0.1> (normalized, i.e., sums to 1) Joint probability distribution for a set of random variables gives the probability of every atomic event on those random variables P(Weather, Cavity) = a 4 2 matrix of values: Weather = sunny rainy cloudy snow Cavity = true Cavity = false Every question about a domain can be answered by the joint distribution Intelligent Systems 15 Conditional probability Conditional or posterior probabilities e.g., P(cavity toothache ) = 0.8 i.e., given that toothache is all I know (Notation for conditional distributions: P(Cavity Toothache) = 2element vector of 2element vectors) If we know more, e.g., cavity is also given, then we have P(cavity toothache, cavity ) = 1 New evidence may be irrelevant, allowing simplification, e.g., P(cavity toothache, sunny) = P(cavity toothache) = 0.8 This kind of inference, sanctioned by domain knowledge, is crucial Intelligent Systems 16 Conditional probability Definition of conditional probability: P(a b) = P(a b) P(b) if P(b ) > 0 Product rule gives an alternative formulation: P(a b) = P(a b) P(b) = P(b a) P(a ) A general version holds for whole distributions, e.g., P(Weather, Cavity) = P(Weather Cavity) P(Cavity) (View as a set of 4 2 equations, not matrix mult.) Chain rule is derived by successive application of product rule: P( 1,, n ) = P( 1,..., n1 ) P( n 1,..., n1 ) = P( 1,..., n2 ) P( n1 1,..., n2 ) P( n 1,..., n1 ) = = π i= 1^n P( i 1,, i1 ) Intelligent Systems 17 Start with the full joint probability distribution: Start with the joint probability distribution: For any proposition φ, sum the atomic events where it is true: P(φ) = Σ ω:ω φ P(ω) For any proposition φ, sum the atomic events where it is true: P(φ) = Σ ω:ω φ P(ω ) P(toothache cavity) = = Intelligent Systems Intelligent Systems 19
4 Start with the joint probability distribution: Start with the joint probability distribution: For any proposition φ, sum the atomic events where it is true: P(φ) = Σ ω:ω φ P(ω ) P(toothache ) = = 0.2 also called marginal probability of toothache Intelligent Systems 20 Can also compute conditional probabilities: P( cavity toothache) = P( cavity toothache) P(toothache) = = Intelligent Systems 21 Normalization Denominator can be viewed as a normalization constant α P(Cavity toothache) = α P(Cavity,toothache) = α [P(Cavity,toothache,catch) + P(Cavity,toothache, catch)] = α [<0.108,0.016> + <0.0,0.064>] = α <0.,0.08> = <0.6,0.4> General idea: compute distribution on query variable by fixing evidence variables and summing over hidden variables Intelligent Systems 22, contd. Typically, we are interested in the posterior joint distribution of the query variable(s) given specific values e for the evidence variables E Let the remaining unobserved variables be Y Then the required summation of joint entries is done by summing out the hidden variables: P( E = e) = αp(, E = e) = ασ y P(, E= e, Y = y ) The terms in the summation are joint entries because, E and Y together exhaust the set of random variables Obvious problems: 1. Worstcase time complexity O(d n ) where d is the largest arity 2. Space complexity O(d n ) to store the joint distribution 3. How to find the numbers for O(d n ) entries? Intelligent Systems 23 Independence A and B are independent iff P(A B) = P(A) or P(B A) = P(B) P(Toothache, Catch, Cavity, Weather) = P(Toothache, Catch, Cavity) P(Weather) or P(A B) = P(A) P(B ) 32 entries reduced to ; for n independent biased coins, O(2 n ) O(n) Absolute independence is powerful but rare Dentistry is a large field with hundreds of variables, none of which are independent. What to do? Intelligent Systems 24 Conditional independence P(Toothache, Cavity, Catch) has = 7 independent entries If I have a cavity, the probability that the probe catches in it doesn't depend on whether I have a toothache: (1) P(catch toothache, cavity) = P(catch cavity) The same independence holds if I haven't got a cavity: (2) P(catch toothache, cavity) = P(catch cavity) Catch is conditionally independent of toothache given cavity : P(catch toothache,cavity) = P(catch cavity ) Equivalent statements: P(toothache catch, cavity) = P(toothache cavity ) P(toothache, catch cavity) = P(toothache cavity) P(catch cavity ) Intelligent Systems 25
5 Conditional independence contd. Write out full joint distribution using chain rule: P(toothache, catch, cavity) = P(toothache catch, cavity) P(catch, cavity ) = P(toothache catch, cavity) P(catch cavity) P(cavity ) = P(toothache cavity) P(catch cavity) P(cavity ) i.e., = 5 independent numbers In most cases, the use of conditional independence reduces the size of the representation of the joint distribution from exponential in n to linear in n. Conditional independence is our most basic and robust form of knowledge about uncertain environments Intelligent Systems 26 Bayes' Rule Product rule P(a b) = P(a b) P(b) = P(b a) P(a ) Bayes' rule: P(a b) = P(b a) P(a) P(b ) or in distribution form P(Y ) = P( Y) P(Y) P() = α P( Y) P(Y) Useful for assessing diagnostic probability from causal probability: P(Cause Effect) = P(Effect Cause) P(Cause) P(Effect ) E.g., let M be meningitis, S be stiff neck: P(m s) = P(s m) P(m) P(s) = = Note: posterior probability of meningitis still very small! Intelligent Systems 27 Bayes' Rule and conditional independence P(cavity toothache catch) = α P(toothache catch Cavity) P(Cavity) = α P(toothache Cavity) P(catch Cavity) P(Cavity) This is an example of a naïve Bayes model: P(Cause,Effect 1,,Effect n ) = P(Cause) π i P(Effect i Cause ) Total number of parameters is linear in n The Wumpus World Revisited Propositional logic cannot solve all cases A breeze in [1,2] and [2,1]. which move to make? (no move is safe) Calculate best move, knowing that each square except [1,1] contains a pit with probability 0.2. Can you guess P(p1,3), P(p2,2) and P(p3,1)?? B? B? Intelligent Systems Intelligent Systems 29 The Wumpus World Revisited (cont.) We do not need the full joint distribution because all the squares marked with do not matter. Let us query [1,3]. Let fringe be the other squares adjacent to visited squares. known is p1,1 p1,2 p2,1 evidence b is b1,1 b1,2 b2,1 probabilities for pits in a pair of squares is = 0.04 probabilities for a pit in one square of a pair and not in the other square of the pair is 0.2 x 0.8 = 0.16 P(p1,3 known, b ) = α 0.2( ), 0.8( ) 0.31, 0.69 likewise P(p2,2 known, b ) = 0.86, 0.14 so moving to [2,2] is a very bad decision Intelligent Systems 30 Example: Burglary Alarm Two types of event set off the alarm. Two people might call when the alarm goes off Intelligent Systems 31
6 Order is important in a Bayesian Network Network becomes very strange if we start from the other end Intelligent Systems 32 Summary Probability is a rigorous formalism for uncertain knowledge Joint probability distribution specifies probability of every atomic event Queries can be answered by summing over atomic events For nontrivial domains, we must find a way to reduce the joint size Independence and conditional independence provide the tools Intelligent Systems 33
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