Uncertainty Copyright, 1996 © Dale Carnegie & Associates, Inc. Chapter 13

CSE 471/598, CBS 598 by H. Liu2 Uncertainty Evolution of an intelligent agent: problem solving, planning, uncertainty Dealing with uncertainty is an unavoidable problem in reality. An agent must act under uncertainty. To make decision with uncertainty, we need Probability theory Utility theory Decision theory

CSE 471/598, CBS 598 by H. Liu3 Sources of uncertainty No access to the whole truth No categorical answer Incompleteness The qualification problem - impossible to explicitly enumerate all conditions Incorrectness of information about conditions The rational decision depends on both the relative importance of various goals and the likelihood of its being achieved.

CSE 471/598, CBS 598 by H. Liu4 Handling uncertain knowledge Difficulties in using FOL to cope with UK A dental diagnosis system using FOL  Symptom (p, Toothache) =>Disease (p, Cavity)  Disease (p, Cavity) => Symptom (p, Toothache)  Are they correct? Reasons Laziness - too much work! Theoretical ignorance - we don’t know everything Practical ignorance - we don’t want to include all Represent UK with a degree of belief The tool for handling UK is probability theory

CSE 471/598, CBS 598 by H. Liu5 Probability provides a way of summarizing the uncertainty that comes from our laziness and ignorance - how wonderful it is! Probability, belief of the truth of a sentence 1 - true, 0 - false, 0<P<1 - intermediate degrees of belief in the truth of the sentence Degree of truth (fuzzy logic) vs. degree of belief Alternatives to probability theory? Yes, to be discussed in later chapters.

CSE 471/598, CBS 598 by H. Liu6 All probability statements must indicate the evidence w.r.t. which the probability is being assessed. Prior or unconditional probability before evidence is obtained Posterior or conditional probability after new evidence is obtained

CSE 471/598, CBS 598 by H. Liu7 Uncertainty & rational decisions Without uncertainty, decision making is simple - achieving the goal or not With uncertainty, it becomes uncertain - three plans A90, A120 and A1440 We need first have preferences between the different possible outcomes of the plans Utility theory is used to represent and reason with preferences.

CSE 471/598, CBS 598 by H. Liu8 Rationality Decision Theory = Probability T + Utility T Maximum Expected Utility Principle defines rationality An agent is rational iff it chooses the action that yields the highest utility, averaged over all possible outcomes of the action A decision-theoretic agent(Fig 13.1, p 466) Is it any different from other agents we learned?

CSE 471/598, CBS 598 by H. Liu9 Basic probability notation Prior probability Proposition - P(Sunny) Random variable - P(Weather=Sunny)  Boolean, discrete, and continuous random variables Each RV has a domain (sunny,rain,cloudy,snow) Probability distribution P(weather) = Joint probability P(A^B) probabilities of all combinations of the values of a set of RVs more later

CSE 471/598, CBS 598 by H. Liu10 Conditional probability P(A|B) = P(A^B)/P(B) Product rule - P(A^B) = P(A|B)P(B) Probabilistic inference does not work like logical inference “P(A|B)=0.6” != “when B is true, P(A) is 0.6”  P(A)  P(A|B), P(A|B,C),...

CSE 471/598, CBS 598 by H. Liu11 The axioms of probability All probabilities are between 0 and 1 Necessarily true (valid) propositions have probability 1, false (unsatisfiable) 0 The probability of a disjunction P(AvB)=P(A)+P(B)-P(A^B) A Venn diagram illustration Ex: Deriving the rule of Negation from P(a v !a)

CSE 471/598, CBS 598 by H. Liu12 The joint probability distribution Joint completely specifies an agent’s probability assignments to all propositions in the domain A probabilistic model consists of a set of random variables (X1, …,Xn). An atomic event is an assignment of particular values to all the variables Given Boolean random variables A and B, what are atomic events?

CSE 471/598, CBS 598 by H. Liu13 Joint probabilities An example of two Boolean variables Observations: mutually exclusive and collectively exhaustive What are P(Cavity), P(Cavity v Toothache), P(Cavity|Toothache)?

CSE 471/598, CBS 598 by H. Liu14 Joint (2) If there is a Joint, we can read off any probability we need. Is it true? How? Impractical to specify all the entries for the Joint over n Boolean variables. Sidestep the Joint and work directly with conditional probability

CSE 471/598, CBS 598 by H. Liu15 Inference using full joint distributions Marginal probability (Fig 13.3) P(cavity) = Maginalization – summing out all the variables other than cavity  P(Y) = Sum-over z P(Y,z) Conditioning – a variant of maginalization using the product rule  P(Y) = Sum-over z P(Y|z)P(z)

CSE 471/598, CBS 598 by H. Liu16 Normalization Method 1 using the def of conditional prob  P(cavity|toothache) = P(c^t)/P(t)  P(!cavity|toothache) = P(!c^t)/P(t) Method 2 using normalization  P(cavity|toothache) = αP(cavity,toothache) = α[P(cavity, T, catch) + P(cavity, T, !catch)] = α[ + ] = α  What is α?

CSE 471/598, CBS 598 by H. Liu17 Independence P(toothache, catch, cavity, weather) A total of 32 entries, given W has 4 values How is one’s tooth problem related to weather? P(T,Ch,Cy,W=cloudy) = P(W=Clo|T...)P(T…)?  Whose tooth problem can influence our weather? P(W=Clo|T…) = P(W=Clo) Hence, P(T,Ch,Cy,W=clo) = P(W=Clo)P(T…) How many joint distribution tables? Two - (4, 8) Independence between X and Y means P(X|Y) = P(X) or P(Y|X) = P(Y) or P(XY) = P(X)P(Y)

CSE 471/598, CBS 598 by H. Liu18 Bayes’ rule Deriving the rule via the product rule P(B|A) = P(A|B)P(B)/P(A) A more general case is P(X|Y) = P(Y|X)P(X)/P(Y) Bayes’ rule conditionalized on evidence E P(X|Y,E) = P(Y|X,E)P(X|E)/P(Y|E) Applying the rule to medical diagnosis meningitis (P(M)=1/50,000)), stiff neck (P(S)=1/20), P(S|M)=0.5, what is P(M|S)? Why is this kind of inference useful?

CSE 471/598, CBS 598 by H. Liu19 Applying Bayes’ rule Relative likelihood Comparing the relative likelihood of meningitis and whiplash, given a stiff neck, which is more likely? P(M|S)/P(W|S) = P(S|M)P(M)/P(S|W)P(W) Avoiding direct assessment of the prior P(M|S) =? P(!M|S) =? And P(M|S) + P(!M|S) = 1, P(S) = ? P(S|!M) = ?

CSE 471/598, CBS 598 by H. Liu20 Using Bayes’ rule Combining evidence from P(Cavity|Toothache) and P(Cavity|Catch) to P(Cavity|Toothache,Catch) Bayesian updating from P(Cavity|T)=P(Cavity)P(T|Cavity)/P(T)  P(A|B) = P(B|A)P(A)/P(B) to P(Cavity|T,Catch)=P(Catch|T,Cavity)/P(Catch|T)  P(A|B,C) = P(B|A,C)P(A|C)/P(B|C)

CSE 471/598, CBS 598 by H. Liu21 Recall that independent events A, B P(B|A)=P(B), P(A|B)=P(A), P(A,B)=P(A)P(B) Conditional independence (X and Y are ind given Z) P(X|Y,Z)=P(X|Z) and P(Y|X,Z)=P(Y|Z) P(XY|Z)=P(X|Z)P(Y|Z) derived from absolute indepence Given Cavity, Toothache and Catch are indpt P(T,Ch,Cy) = P(T,Ch|Cy)P(Cy) = P(T|Cy)P(Ch|Cy)P(Cy) One large table is decomposed into 3 smaller tables: vs. 5 (= 2*(2 1 -1)+2*(2 1 -1) ) T|CyT|!Cy Cy

CSE 471/598, CBS 598 by H. Liu22 Independence, decomposition, Naïve Bayes If all n symptoms are conditionally indpt given Cavity, the size of the representation grows as O(n) instead of O(2 n ) The decomposition of large probabilistic domains into weakly connected subsets via conditional independence is one important development in modern AI Naïve Bayes model (Cause and Effects) P(Cause,E1,…,En) = P(Cause)  P(Ei|Cause) An amazingly successful classifier as well

CSE 471/598, CBS 598 by H. Liu23 Where do probabilities come from? There are three positions: The frequentist - numbers can come only from experiments The objectivist - probabilities are real aspects of the universe The subjectivist - characterizing an agent’s belief The reference class problem – intrusion of subjectivity A frequentist doctor wants to consider similar patients  How similar two patients are? Laplace’s principle of indifference Propositions that are syntactically “symmetric” w.r.t. the evidence should be accorded equal probability

CSE 471/598, CBS 598 by H. Liu24 Summary Uncertainty exists in the real world. It is good (it allows for laziness) and bad (we need new tools) Priors, posteriors, and joint Bayes’ rule - the base of Bayesian Inference Conditional independence allows Bayesian updating to work effectively with many pieces of evidence. But...