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Uncertainty in Artificial Intelligence Ramtin Raji Kermani, Mehrdad Rashidi, Hadi Yaghoobian Overview Uncertainty Probability Syntax and Semantics Inference Independence and Bayes' Rule Uncertain Agent sensors ? ? environment agent ? actuators model 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: “A will get me there on time if there's no accident on the bridge Truth & Belief Ontological Commitment: What exists in the world — TRUTH Epistemological Commitment: What an agent believes about facts — BELIEF Types of Uncertainty Uncertainty in prior knowledge E.g., some causes of a disease are unknown and are not represented in the background knowledge of a medical-assistant agent Uncertainty in actions E.g., actions are represented with relatively short lists of preconditions, while these lists are in fact arbitrary long Uncertainty in perception E.g., sensors do not return exact or complete information about the world; a robot never knows exactly its position Questions How to represent uncertainty in knowledge? How to perform inferences with uncertain knowledge? Which action to choose under uncertainty? How do we represent Uncertainty? We need to answer several questions: What do we represent & how we represent it? What can we do with the representations? What language do we use to represent our uncertainty? What are the semantics of our representation? 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? Uncertainty? What we call uncertainty is a summary of all that is not explicitly taken into account in the agent’s KB Uncertainty? Sources of uncertainty: Ignorance Laziness (efficiency?) Methods for handling uncertainty Creed: Default nonmonotonic logic: Theorworld is fairly normal. Abnormalities are rare my car does not have flat tire So,anAssume agent assumes normality, untila there is evidence of the contrary Assume A WetGrass |→ 0.7 Rain works unless contradicted by evidence 25 E.g., if Just an agent sees a bird x, world it assumes thatby x can fly, unless Creed: the opposite! The is ruled Murphy’s Issues: What assumptions are reasonable? How to Law handle it contradiction? has evidence that x is a penguin, an ostrich, a dead bird, a bird with broken wings, … Uncertainty is defined by sets, e.g., the set possible outcomes of an Creed: world is not divided between “normal” and “abnormal”, action, The the set of possible positions of a robot is it adversarial. nor Rules with fudge Possible factors:situations have various likelihoods (probabilities) The agent assumes the worst case, and chooses the actions that A25 |→0.3 get there on time maximizes a utility function in this case agent has probabilistic beliefs – pieces of knowledge with The Example: Adversarial search Sprinkler |→ 0.99 WetGrass associated probabilities (strengths) – and chooses its actions to maximize the expected value of some utility function Issues: Problems with combination, e.g., Sprinkler causes 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 Making decisions under uncertainty Suppose I believe the following: P(A25 gets me there on time | …) P(A90 gets me there on time | …) P(A120 gets me there on time | …) P(A1440 gets me there on time | …) Which action to choose? = = = = 0.04 0.70 0.95 0.9999 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 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 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. Random variables A discrete random variable is 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 variable has the domain <true,false>, e.g., Cavity (special case of discrete random variable) Continuous random variable as the domain of real numbers, e.g., Temp 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 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 Cavity Cavity Cavity = false Toothache = false = false Toothache = true = true Toothache = false = true Toothache = true Atomic events are mutually exclusive and exhaustive Atomic Events, Events & the Universe The universe consists of atomic events An event is a set of atomic events P: event [0,1] AB Axioms of Probability P(true) = 1 = P(U) P(false) = 0 = P() P(A B) = P(A) + P(B) – P(A B) A B U 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 Joint probability distribution Probability assignment to all combinations of values of random variables Toothache Toothache Cavity 0.04 0.06 Cavity 0.01 0.89 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] Conditional Probability Toothache Toothache A Cavity 0.04 0.06 Cavity 0.01 0.89 B U AB 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 Conditional Probability (continued) Definition of Conditional Probability: Product rule gives an alternative formulation: A general version holds for whole distributions: Chain rule is derived by successive application of product rule: P(A | B) = P(A B)/P(B) P(A B) = P(A | B) P(B) = P(B | A) P(A) P(Weather,Cavity) = P(Weather | Cavity) P(Cavity) 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) =… n = P(X i 1 i| | X1, ..., X i 1 )