Ramtin Raji Kermani, Mehrdad Rashidi, Hadi Yaghoobian

1. Uncertaintyin Artificial Intelligence Ramtin Raji Kermani, Mehrdad Rashidi, Hadi Yaghoobian

2. Overview • Uncertainty • Probability • Syntax and Semantics • Inference • Independence and Bayes' Rule

3. sensors ? environment ? ? agent ? actuators model Uncertain Agent

4. Uncertainty !!! • Let action At = leave for airport t minutes before flight • Will At get me there on time? • Problems: • partial observability (road state, other drivers' plans, etc.) • noisy sensors (traffic reports) • uncertainty in action outcomes (flat tire, etc.) • immense complexity of modeling and predicting traffic • Hence a purely logical approach either • risks falsehood: “A25 will get me there on time”, or • leads to conclusions that are too weak for decision making: • “A25 will get me there on time if there's no accident on the bridge and it doesn't rain and my tires remain intact etc etc.” • (A1440 might reasonably be said to get me there on time but I'd have to stay overnight in the airport …)

5. Truth & Belief • Ontological Commitment: What exists in the world — TRUTH • Epistemological Commitment: What an agent believes about facts — BELIEF

6. Types of Uncertainty • Uncertainty in prior knowledgeE.g., some causes of a disease are unknown and are not represented in the background knowledge of a medical-assistant agent • Uncertainty in actionsE.g., actions are represented with relatively short lists of preconditions, while these lists are in fact arbitrary long • Uncertainty in perceptionE.g., sensors do not return exact or complete information about the world; a robot never knows exactly its position

7. Questions • How to represent uncertainty in knowledge? • How to perform inferences with uncertain knowledge? • Which action to choose under uncertainty?

8. How do we represent Uncertainty? We need to answer several questions: • What do we represent & how we represent it? • What language do we use to represent our uncertainty? What are the semantics of our representation? • What can we do with the representations? • What queries can be answered? How do we answer them? • How do we construct a representation? • Can we ask an expert? Can we learn from data?

9. Uncertainty? What we call uncertainty is a summary of all that is not explicitly taken into account in the agent’s KB

10. Uncertainty? Sources of uncertainty: • Ignorance • Laziness (efficiency?)

11. Methods for handling uncertainty • Default or nonmonotonic logic: • Assume my car does not have a flat tire • Assume A25 works unless contradicted by evidence • Issues: What assumptions are reasonable? How to handle contradiction? • Rules with fudge factors: • A25 |→0.3 get there on time • Sprinkler |→0.99WetGrass • WetGrass |→0.7Rain • Issues: Problems with combination, e.g., Sprinkler causes Rain?? • Probability • Model agent's degree of belief • Given the available evidence, • A25 will get me there on time with probability 0.04 Creed: The world is fairly normal. Abnormalities are rare So, an agent assumes normality, until there is evidence of the contrary E.g., if an agent sees a bird x, it assumes that x can fly, unless it has evidence that x is a penguin, an ostrich, a dead bird, a bird with broken wings, … Creed: Just the opposite! The world is ruled by Murphy’s Law Uncertainty is defined by sets, e.g., the set possible outcomes of an action, the set of possible positions of a robot The agent assumes the worst case, and chooses the actions that maximizes a utility function in this case Example: Adversarial search Creed: The world is not divided between “normal” and “abnormal”, nor is it adversarial. Possible situations have various likelihoods (probabilities) The agent has probabilistic beliefs – pieces of knowledge with associated probabilities (strengths) – and chooses its actions to maximize the expected value of some utility function

12. More on deal with uncertainty? • Implicit: • Ignore what you are uncertain of when you can • Build procedures that are robust to uncertainty • Explicit: • Build a model of the world that describe uncertainty about its state, dynamics, and observations • Reason about the effect of actions given the model

13. Making decisions under uncertainty Suppose I believe the following: P(A25 gets me there on time | …) = 0.04 P(A90 gets me there on time | …) = 0.70 P(A120 gets me there on time | …) = 0.95 P(A1440 gets me there on time | …) = 0.9999 • Which action to choose? Depends on my preferences for missing flight vs. time spent waiting, etc.

14. Probability • A well-known and well-understood framework for uncertainty • Clear semantics • Provides principled answers for: • Combining evidence • Predictive & Diagnostic reasoning • Incorporation of new evidence • Intuitive (at some level) to human experts • Can be learned

15. Probability Probabilistic assertions summarize effects of • laziness: failure to enumerate exceptions, qualifications, etc. • ignorance: lack of relevant facts, initial conditions, etc. Subjective probability: • Probabilities relate propositions to agent's own state of knowledge e.g., P(A25 | no reported accidents) = 0.06 These are not assertions about the world Probabilities of propositions change with new evidence: e.g., P(A25 | no reported accidents, 5 a.m.) = 0.15

16. Decision Theory • Decision Theory develops methods for making optimal decisions in the presence of uncertainty. • Decision Theory = utility theory + probability theory • Utility theory is used to represent and infer preferences: Every state has a degree of usefulness • An agent is rational if and only if it chooses an action that yields the highest expected utility, averaged over all possible outcomes of the action.

17. Random variables A discrete random variableis a function that • takes discrete values from a countable domain and • maps them to a number between 0 and 1 • Example: Weather is a discrete (propositional) random variable that has domain <sunny,rain,cloudy,snow>. • sunny is an abbreviation for Weather = sunny • P(Weather=sunny)=0.72, P(Weather=rain)=0.1, etc. • Can be written: P(sunny)=0.72, P(rain)=0.1, etc. • Domain values must be exhaustive and mutually exclusive • Other types of random variables: • Boolean random variablehas the domain <true,false>, • e.g., Cavity (special case of discrete random variable) • Continuous random variableas the domain of real numbers, e.g., Temp

18. Propositions • Elementary proposition constructed by assignment of a value to a random variable: • e.g., Weather =sunny, Cavity = false (abbreviated as cavity) • Complex propositions formed from elementary propositions & standard logical connectives • e.g., Weather = sunny  Cavity = false

19. Atomic Events • Atomic event: • A complete specification of the state of the world about which the agent is uncertain • E.g., if the world consists of only two Boolean variables Cavity and Toothache, then there are 4 distinct atomic events: Cavity = false Toothache = false Cavity = false  Toothache = true Cavity = true  Toothache = false Cavity = true  Toothache = true • Atomic events are mutually exclusive and exhaustive

20. A  B A B U Atomic Events, Events & the Universe • The universeconsists of atomic events • An event is a set of atomic events • P: event  [0,1] • Axioms of Probability • P(true) = 1 = P(U) • P(false) = 0 = P() • P(A  B) = P(A) + P(B) – P(A  B)

21. Prior probability • Prior (unconditional) probability • corresponds to belief prior to arrival of any (new) evidence • P(sunny)=0.72, P(rain)=0.1, etc. • Probability distribution gives values for all possible assignments: • Vector notation: Weather is one of <0.72, 0.1, 0.08, 0.1> • P(Weather) = <0.72,0.1,0.08,0.1> • Sums to 1 over the domain

22. Joint probability distribution • Probability assignment to all combinations of values of random variables • The sum of the entries in this table has to be 1 • Every question about a domain can be answered by the joint distribution • Probability of a proposition is the sum of the probabilities of atomic events in which it holds • P(cavity) = 0.1 [add elements of cavity row] • P(toothache) = 0.05 [add elements of toothache column]

23. A B U A  B Conditional Probability • P(cavity)=0.1 and P(cavity  toothache)=0.04 are both prior (unconditional) probabilities • Once the agent has new evidence concerning a previously unknown random variable, e.g., toothache, we can specify a posterior(conditional) probability • e.g., P(cavity | toothache) • P(A | B) = P(A  B)/P(B) [prob of A w/ U limited to B] • P(cavity | toothache) = 0.04/0.05 = 0.8

24. Conditional Probability (continued) • Definition of Conditional Probability:P(A | B) = P(A  B)/P(B) • Product rule gives an alternative formulation: P(A  B) = P(A | B)  P(B) = P(B | A)  P(A) • A general version holds for whole distributions:P(Weather,Cavity) = P(Weather | Cavity)  P(Cavity) • Chain rule is derived by successive application of product rule: P(X1, …,Xn) = P(X1,...,Xn-1) P(Xn | X1,...,Xn-1) = P(X1,...,Xn-2) P(Xn-1 | X1,...,Xn-2) P(Xn | X1,...,Xn-1) = … =