Overview Decision Theory: Example - TAMU Computer Science Faculty

Overview Decision Theory: Example - TAMU Computer Science Faculty

5 Pages · 2002 · 70 KB · English

Overview Decision theory example Probability basics Conditional probability Axioms of probability Joint probability distribution Bayes rule Bayes rule: Example

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Overview  Decision theory example  Probability basics  Conditional probability  Axioms of probability  Joint probability distribution  Bayes rule  Bayes rule: Example Tip: for efcient Lisp code, see http://www.cs.utexas.edu/users/novak/lispeff.html 1 Decision Theory: Example Decision theory = Probability theory + Utility theory Utility of Resulting State Probability Action 1 10 0.2 Action 2 10000 0.001 Action 3 5 0.799 Which action would an optimal Decision Theoretic Agent take? 2 Decision Theory: Example Decision theory = Probability theory + Utility theory Utility of Resulting State  Probability Expected Utility Action 1 10  0.2 2 Action 2 1000  0.001 1 Action 3 5 0.799 3.995 Action 3 has the maximum expected utility, thus action 3 will be carried out. 3 Probability: Notations a  Random variable: variable that can take on different values – A;B;::: : boolean values ( T or F ). – X;Y;::: : numerical values or other multivalued enumerations (1, 2, 0.5, Cloudy, Rainy, Sunny, ...) P(X=v) :probabilityof the variable X having value v. –This can be viewed as an event . –For boolean variables, P(A) means P(A=T) , and P(:A) means P(A=F) . P(X) :probability distribution, a full list of probabilities for all possible values that X can take (note that P is inbold. aAll conventions follow Russel & Norvig 4 Examples  Boolean: P(I nf ected)=0:01 ,P(:I nf ected)=0:99 .  Multivalued: P(D ice=1)= 1 6 ;P(D ice=2)= 1 6 ;:::  Multivalued: P(W eather=S unny)=0:7; P(W eather=R ainy)=0:2;::: 5 Logical Connectives and Conditional Probability  Logical connectives can be used: P(A_B);P(A^:B);P(C av ity^:Insured); etc.  Conditional Probability P(AjB) (read probability of A given B ): P(CavityjT oothache)=0:8  As new evidence comes in, the conditional probability gets updated: P(C av ityjT oothache^B adB r eath | {z } ) 6 Conditional Probability A B A/\B = P(A|B) = P(A/\B) P(B) U  Think about theareaoccupied by each event.  The bounding rectangle U has an area of 1, thus P(A)= Area of A Area of U = Area of A 1 = Area of A P(AjB) means B now takes on the role of U. Within this limited event space, what is the probability of A. 7 The Axioms of Probability All axioms 1. All probabilities are between 0 and 1 0P(A)1 2. For a valid proposition A ( T under all interpretations): P(A)=1; and for a inconsistent proposition A ( F under all interpretations): P(A)=0 . 3. P(A_B)=P(A)+P(B)P(A^B) Other properties follow from these three axioms. 8 Other Properties  From the axioms, P(A_:A)=P(A)+P(:A)P(A^:A) P(T)=P(A)+P(:A)P(F) 1=P(A)+P(:A) P(:A)=1P(A)  More generally, thesumof probabilities P(X=v) is 1, for all values vthe random variable X can take: 2 4 X v2V P(X=v) 3 5=1; where V is the set of all possible values X can take. 9 Joint Probability Distribution For random variables X1 ;X2 ;:::;Xn ,  Anatomic eventis an assignment of particular values to each random variable.  Thejoint probability distribution P(X1 ;X2 ;:::;Xn ) completely species the probabilities of allatomic events.  Thus, 2 4 X (v1 ;v2 ;:::;vn )2V P(X1 =v1 ;X2 =v2 ;:::;Xn =vn ) 3 5=1; where V is a set of all possible n vectors that the vector (X1 ;X2 ;::;Xn ) can assume.. 10 Joint Probability Distribution: Example Toothache :Toothache Sum Cavity 0:040:06 P(C)=0:1 :Cavity 0:010:89 P(:C)=0:9 Sum P(T)=0:05P(:T)=0:95 P=1:0 Abbreviations: C=C av ity ,T=T oothache P(C_T)=P(C)+P(T)P(C^T) =0:1+0:050:04=0:11 P(CjT)= P(C^T) P(T) = 0:04 0:05 =0:8 P(TjC)= P(C^T) P(C) = 0:04 0:1 =0:5 In practice,

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