Reasoning under Uncertainty Reasoning under Uncertainty Probabilistic Reasoning Probabilistic Reasoning Fuzzy logic Fuzzy logic Dempster-Shafer Theory Dempster-Shafer Theory Summary Summary Where are we? Knowledge Engineering Semester 2, 2004-05 Last time . . . Michael Rovatsos ◮ Model-based reasoning mrovatso@inf.ed.ac.uk Today . . . ◮ Approaches to dealing with uncertainty ◮ Probabilistic Reasoning ◮ Fuzzy Logic ◮ Dempster-Shafer Theory Lecture 8 – Dealing with Uncertainty 8th February 2005 Informatics UoE Knowledge Engineering 1 Informatics UoE Knowledge Engineering 128 Reasoning under Uncertainty Reasoning under Uncertainty Probabilistic Reasoning Probabilistic Reasoning Fuzzy logic Fuzzy logic Dempster-Shafer Theory Dempster-Shafer Theory Summary Summary Reasoning under Uncertainty Probabilistic Reasoning ◮ Most general and widespread method of uncertainty ◮ So far, focus on certain knowledge reasoning How do we model what we know? ◮ Rests on mathematical foundations of probability theory ◮ But how do we model uncertainty? ◮ Two interpretations of probability: ◮ Different aspects: ◮ Subjective: belief about likelihood of a proposition ◮ Uncertainty regarding truthfulness of propositions ◮ Objective: frequency of observed events in which ◮ Vagueness in the way knowledge is captured proposition holds ◮ Questions of ignorance and confidence ◮ Major advances in 90s, today highly popular field in AI ◮ Different KR & R approaches for each of these ◮ Here: only very short overview (see PMR, LFD and similar courses) Informatics UoE Knowledge Engineering 129 Informatics UoE Knowledge Engineering 130
Reasoning under Uncertainty Reasoning under Uncertainty Probabilistic Reasoning Probabilistic Reasoning Fuzzy logic Fuzzy logic Dempster-Shafer Theory Dempster-Shafer Theory Summary Summary Probability Theory Example Why is Bayes’ rule useful? Assume m denotes “patient has ◮ Axioms of probability theory: P ( false ) = 0, P ( true ) = 1 meningitis”, s denotes “patient has a stiff neck” and we have P ( a ∨ b ) = P ( a ) + P ( b ) − P ( a ∧ b ) the following estimates: ◮ Other properties: P ( ¬ a ) = 1 − P ( a ), P ( a ) + P ( ¬ a ) = 1 P ( s | m ) = 0 . 5 ◮ For discrete random variable D with domain � d 1 , . . . , d n � : P ( m ) = 1 / 50000 � n i =1 P ( D = d i ) = 1 P ( s ) = 1 / 20 ◮ For atomic mutually exclusive events E such that a holds in E ( a ) ⊆ E : P ( a ) = � e ∈ E ( a ) P ( e ) We can infer: ◮ Bayes’ rule: P ( b | a ) = P ( a | b ) P ( b ) P ( a ) P ( m | s ) = P ( s | m ) P ( m ) = 0 . 5 × 1 / 50000 = 0 . 0002 ◮ Conditional independence: P ( a , b | c ) = P ( a | c ) P ( a | c ) P ( s ) 1 / 20 Informatics UoE Knowledge Engineering 131 Informatics UoE Knowledge Engineering 132 Reasoning under Uncertainty Reasoning under Uncertainty Probabilistic Reasoning Probabilistic Reasoning Fuzzy logic Fuzzy logic Dempster-Shafer Theory Dempster-Shafer Theory Summary Summary Belief Networks Belief Networks ◮ Nodes represent propositionsm, annotated with Using a graphical notation to represent probabilities of conditional probability tables propositions and conditional independence assumptions: ◮ Edges represent conditional dependencies P ( E ) P ( B ) ◮ Main idea: represent full joint probability distribution over Burglary Earthquake .002 .001 variables X 1 , . . . X n (to obtain probability of conjunction P ( X 1 = x 1 ∧ . . . ∧ X n = x n )) as product of independent B E P ( A ) probabilities using Bayes’ Rule T T .95 Alarm T F .94 F T .29 ◮ If parents ( X i ) are the parent nodes of X i , the joint F F .001 probability distribution is given by n A P ( J ) � A P ( M ) P ( x 1 , . . . x n ) = P ( x i | parents ( X i )) JohnCalls T .90 MaryCalls T .70 F .05 F .01 i =1 Informatics UoE Knowledge Engineering 133 Informatics UoE Knowledge Engineering 134
Reasoning under Uncertainty Reasoning under Uncertainty Probabilistic Reasoning Probabilistic Reasoning Fuzzy logic Fuzzy logic Dempster-Shafer Theory Dempster-Shafer Theory Summary Summary Example Critique P ( j ∧ m ∧ a ∧ ¬ b ∧ ¬ e ) = P ( j | a ) P ( m | a ) P ( a |¬ b ¬ e ) P ( ¬ b ) P ( ¬ e ) = 0 . 9 × 0 . 7 × 0 . 001 × 0 . 999 × 0 . 998 = 0 . 00062 ◮ Lots of methods for exact and approximate inference P ( E ) P ( B ) Burglary Earthquake .002 .001 ◮ Area of Bayesian Learning ◮ Where do these probabilities come from? B E P ( A ) ◮ Where do the independence assumptions come from? T T .95 Alarm T F .94 F T .29 ◮ Worst case: all variables depend on each other F F .001 no gain A P ( J ) A P ( M ) JohnCalls T .90 MaryCalls T .70 F .05 F .01 Informatics UoE Knowledge Engineering 135 Informatics UoE Knowledge Engineering 136 Reasoning under Uncertainty Reasoning under Uncertainty Probabilistic Reasoning Probabilistic Reasoning Fuzzy logic Fuzzy logic Dempster-Shafer Theory Dempster-Shafer Theory Summary Summary Fuzzy Logic Fuzzy Sets Fuzzy sets (unlike crisp ones) based on notion of “degree” ◮ Method for expressing vagueness ◮ Uncertainty about degree of appropriateness of a statement, not about its truthfulness ◮ Foundation: notion of fuzzy sets ◮ Allows for expressing degree with which an object belongs to a set and applying mathematical methods to manipulate these statements ◮ Fuzzy control: extremely successful in industrial applications Informatics UoE Knowledge Engineering 137 Informatics UoE Knowledge Engineering 138
Reasoning under Uncertainty Reasoning under Uncertainty Probabilistic Reasoning Probabilistic Reasoning Fuzzy logic Fuzzy logic Dempster-Shafer Theory Dempster-Shafer Theory Summary Summary Fuzzy sets Operations on fuzzy sets ◮ When describing concepts using subset relationships crisp ◮ Logical operations define membership often inflexible ◮ Let T ( A ), T ( B ) fuzzy truth values of A and B ◮ Characteristic function : members have value 1, ◮ T ( A ∧ B ) = min( T ( A ) , T ( B )) non-members have value 0 ◮ T ( A ∨ B ) = max( T ( A ) , T ( B )) ◮ Take example of “young person” in terms of age ◮ T ( ¬ A ) = 1 − T ( A ) ◮ Naive definition: Use, for example [0,20] as a crisp ◮ Truth-functional approach problems with interval correlations and anti-correlations between propositions ◮ Is someone one day after his 20th birthday not young ? ◮ Example: Fuzzy truth value of “tall and heavy” will be ◮ Note that this problem appears regardless of the bound unreasonably high for someone who is extremely tall ◮ Solution: Allow more intermediate values for although “heavy” should be less strict for very tall people characteristic function (gradual membership) Informatics UoE Knowledge Engineering 139 Informatics UoE Knowledge Engineering 140 Reasoning under Uncertainty Reasoning under Uncertainty Probabilistic Reasoning Probabilistic Reasoning Fuzzy logic Fuzzy logic Dempster-Shafer Theory Dempster-Shafer Theory Summary Summary Further Issues Critique ◮ Devise rules of the form “if person is young and not overweight then blood pressure is normal” to make decisions ◮ Success in practical applications often attributed to: ◮ Defuzzification : how to make crisp choices after ◮ Use in limited, controllable domains evaluation of fuzzy rules (e.g. take center of gravity of a ◮ Fine-tuning of parameters for a particular use fuzzy set) ◮ No chaining of inferences ◮ Attempts to map fuzzy logic to probabilistic concepts ◮ Hard to combine with other kinds of KBS ◮ Discrete observation interpretation: ◮ However, so far the only AI technology that has found its P ( Observer says person is tall and heavy | Height , Weight ) way to (almost) every washing machine! solves truth-functionality problems ◮ Random set interpretation: view Tall as a random variable (denoting a set), P ( Tall = S ) is probability that set S of persons would be identified as tall Informatics UoE Knowledge Engineering 141 Informatics UoE Knowledge Engineering 142
Recommend
More recommend
