CS621 : Artificial Intelligence Pushpak Bhattacharyya CSE Dept., IIT Bombay Lecture 30 Uncertainty, Fuizziness

To Model Uncertainty Law of universe Heisenberg’s uncertainty principle: Position and momentum cannot be measured simultaneously with certainty. If one is measured with precision, the other is uncertain – Δp.Δx>=h/4π (h is plank’’s constant)

Uncertainty modeling (contd.) Also a law of life Real life inferencing modeled with Default Logic, Non Monotonic Reasoning, Modal Logic, Belief Worlds, Fuzzy logic, Abduction Other models: Bayesian Inferencing, Entropy and Information Theory

Bayesian Decision Theory Bayes Theorem : Given the random variables A and B, Posterior probability Prior probability Likelihood

Bayes Theorem Derivation Commutativity of “intersection”

To understand when and why to apply Bayes Theorem An example: it is known that in a population, 1 in has meningitis and 1 in 20 has stiff neck. It is also observed that 50% of the meningitis patients have stiff neck. A doctor observes that a patient has stiff neck. What is the probability that the patient has meningitis? (Mitchel, Machine Learning, 1997) Ans: We need to find P(m|s): probability of meningitis given the stiff neck

Apply Bayes Rule (why?) P(m|s) =[P(m). P(s|m)]/P(s) P(m)= prior probability of meningitis P(s|m)=likelihod of stiff neck given meningitis P(s)=Probability of stiff neck

Probabilities P(m)= P(s)= P(s|m)= 0.5 Prior posterior Hence meningitis is not likely Likelihood

Some Issues p(m|s) could have been found as Questions: – Which is more reliable to compute, p(s|m) or p(m|s)? – Which evidence is more sparse, p(s|m) or p(m|s)? – Test of significance : The counts are always on a sample of population. Which probability count has sufficient statistics?

Fuzziness

Fuzzy Logic Models Human Reasoning Works with imprecise statements such as: In a process control situation, “If the temperature is moderate and the pressure is high, then turn the knob slightly right” The rules have “Linguistic Variables”, typically adjectives qualified by adverbs (adverbs are hedges).

Underlying Theory: Theory of Fuzzy Sets Intimate connection between logic and set theory. Given any set ‘S’ and an element ‘e’, there is a very natural predicate, μ s (e) called as the belongingness predicate. The predicate is such that, μ s (e) = 1,iff e ∈ S = 0,otherwise For example, S = {1, 2, 3, 4}, μ s (1) = 1 and μ s (5) = 0 A predicate P(x) also defines a set naturally. S = {x | P(x) is true} For example, even(x) defines S = {x | x is even}

Fuzzy Set Theory (contd.) Fuzzy set theory starts by questioning the fundamental assumptions of set theory viz., the belongingness predicate, μ, value is 0 or 1. Instead in Fuzzy theory it is assumed that, μ s (e) = [0, 1] Fuzzy set theory is a generalization of classical set theory also called Crisp Set Theory. In real life belongingness is a fuzzy concept. Example: Let, T = set of “tall” people μ T (Ram) = 1.0 μ T (Shyam) = 0.2 Shyam belongs to T with degree 0.2.

Linguistic Variables Fuzzy sets are named by Linguistic Variables (typically adjectives). Underlying the LV is a numerical quantity E.g. For ‘tall’ (LV), ‘height’ is numerical quantity. Profile of a LV is the plot shown in the figure shown alongside. μ tall (h) height h

Example Profiles μ rich (w) wealth w μ poor (w) wealth w

Example Profiles μ A (x) x x Profile representing moderate (e.g. moderately rich) Profile representing extreme

Concept of Hedge Hedge is an intensifier Example: LV = tall, LV 1 = very tall, LV 2 = somewhat tall ‘very’ operation: μ very tall (x) = μ 2 tall (x) ‘somewhat’ operation: μ somewhat tall (x) = √(μ tall (x)) 1 0 h μ tall (h) somewhat tall tall very tall

Representation of Fuzzy sets Let U = {x 1,x 2,…..,x n } |U| = n The various sets composed of elements from U are presented as points on and inside the n-dimensional hypercube. The crisp sets are the corners of the hypercube. (1,0) (0,0) (0,1) (1,1) x1x1 x2x2 x1x1 x2x2 (x 1,x 2 ) A(0.3,0.4) μ A (x 1 )=0.3 μ A (x 2 )=0.4 Φ U={x 1,x 2 } A fuzzy set A is represented by a point in the n-dimensional space as the point {μ A (x 1 ), μ A (x 2 ),……μ A (x n )}

Degree of fuzziness The centre of the hypercube is the “ most fuzzy ” set. Fuzziness decreases as one nears the corners Measure of fuzziness Called the entropy of a fuzzy set Entropy Fuzzy setFarthest corner Nearest corner

(1,0) (0,0) (0,1) (1,1) x1x1 x2x2 d(A, nearest) d(A, farthest) (0.5,0.5) A

Definition Distance between two fuzzy sets L 1 - norm Let C = fuzzy set represented by the centre point d(c,nearest) = | | + |0.5 – 0.0| = 1 = d(C,farthest) => E(C) = 1

Definition Cardinality of a fuzzy set [generalization of cardinality of classical sets] Union, Intersection, complementation, subset hood

Note on definition by extension and intension S 1 = {x i |x i mod 2 = 0 } – Intension S 2 = {0,2,4,6,8,10,………..} – extension How to define subset hood?