Quantifying Uncertainty I Session 9

1. Course : Artificial Intelligence Effective Period : September 2018 Quantifying Uncertainty ISession 9 Prof. Dr. Widodo Budiharto2018

2. Learning Outcomes At the end of this session, students will be able to: • LO 4: Apply various techniques to an agent when acting under certainty

3. Outline Acting under Uncertainty Basic Probability Notation Exercise

4. Acting Under Uncertainty Agent may need to handle uncertainty, whether due to partial observability, nondeterminism, or combination of the two The agent’s knowledge can at best provide only a degree of belief in the relevant sentences Main tool for dealing with degree of belief is probability theory Probability provides a way of summarizing the uncertainty that come from laziness and ignorance, thereby solving the quantification problem

5. Acting Under Uncertainty • Uncertainty in logic sentence • Toothache ⇒ Cavity (True?) • Toothache ⇒ Cavity ∨ GumProblem ∨ Abscess . . . • Probability • From the statistical data, 80% of the toothache patients have had cavities

6. Acting Under Uncertainty • Let action At = leave for BINUS t minutes before class • Will At get me there on time? • Problems: • partial observability (road state, other drivers' plans, etc.) • uncertainty in action outcomes (at tire, etc.)

7. Acting Under Uncertainty • If we use purely logic to solve that problem, then • A25 will get the student arrive on time if “there is no accident or traffic jam and it does not rain” • No Rain AND No Accident  On Time

8. Acting Under Uncertainty

9. Acting Under Uncertainty • Preferences, as expressed by utilities, are combined with probabilities in the general theory of rational decisions called decision theory • Decision Theory = probability theory + utility theory • Fundamental of decision theory is that an agent is rational if only if it chooses the action that yields the highest expected utility, averaged over all the possible outcomes of the action called maximum expected utility(MEU).

10. Acting Under Uncertainty

11. Basic Probability Notation • In probability theory, the set of all possible worlds is called the sample space • For example, if we are about to roll two (distinguishable) dice, there are 36 possible worlds to consider: (1,1), (1,2), . . ., (6,6) • The probability for each possible world is as follows: • I.e. the probability of the rolled two dice are 1/36

12. Basic Probability Notation • The propositions is the set of two or more possible worlds • The probability is • I.e P(Total=11) = P((5,6)) + P((6,5)) = 1/36 + 1/36 = 1/18

13. Basic Probability Notation • There are two kinds of probabilities: • Unconditional or prior probabilities • Degrees of belief in propositions in the absence of any other information • Conditional or posterior probabilities • There is some evidence (information) to support the probability • If the first dice is 5, then what is the P(Total=11)?

14. Basic Probability Notation • Conditional probabilities • The “|” is pronounced “given” • Example • Product rule

15. Basic Probability Notation • Variables in probability theory are called random variables • Total, Die1 • Every random variable has a domain • Total = {2, …, 12} • Die1 = {1,…6}

16. Basic Probability Notation • Probabilities of all the possible values of a random variable: • P(Weather =sunny) = 0.6 • P(Weather =rain) = 0.1 • P(Weather =cloudy) = 0.29 • P(Weather =snow) = 0.01 • The probability distribution for the random variable Weather • P(Weather)=0.6, 0.1, 0.29, 0.01

17. Basic Probability Notation • For continuous variables, it is not possible to write out the entire distribution as a vector, because there are infinitely many values • The temperature at noon is distributed uniformly between 18 and 26 degrees Celcius • P(NoonTemp =x) = Uniform[18C,26C](x) • Called probability density function

18. Basic Probability Notation We need notation for distributions on multiple variables P(Weather , Cavity) denotes the probabilities of all combinations of the values of Weather and Cavity This is a 4×2 table of probabilities called the joint probability distributionof Weather and Cavity

19. References Stuart Russell, Peter Norvig. 2010. Artificial Intelligence : A Modern Approach. Pearson Education. New Jersey. ISBN:9780132071482

20. Exercise

21. Exercise Conditional Probability In a group of 100 sports car buyers, 30 bought alarm systems, 20 purchased bucket seats, and 10 purchased an alarm system and bucket seats. If a car buyer chosen at random, bought an alarm system, what is the probability he/she also bought bucket seats?