Uncertainty Russell and Norvig: Chapter 13 CMCS424 Fall 2005 Uncertain Agent sensors ? ? environment agent ? actuators model An Old Problem … 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 Types of Uncertainty For example, to drive my car in the morning: Uncertainty in prior knowledge • It must not have been stolen during the night E.g., some causes of a disease are unknown and are • It must not have in flat tires not represented the background knowledge of a • There must be gasagent in the tank medical-assistant • The battery must not be dead Uncertainty in actions E.g.,ignition actions must are represented with relatively short lists • The work preconditions, while lists are in fact arbitrary • Iofmust not have lost thethese car keys long • No truck should obstruct the driveway • I must not have suddenly become blind or paralytic Etc… Not only would it not be possible to list all of them, but would trying to do so be efficient? 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 areCourtesy in factR.arbitrary Chatila long Uncertainty in perception E.g., sensors do not return exact or complete information about the world; a robot never knows exactly its position Types of Uncertainty Uncertainty in prior knowledge E.g., some causes of a disease are unknown and are Sources ofbackground uncertainty: not represented in the knowledge of a medical-assistant agent 1. Ignorance Uncertainty in actions E.g., actions are represented with relatively short lists 2. Laziness (efficiency?) of preconditions, while these lists are in fact arbitrary long Uncertainty in perception E.g., sensors do not return exactisoracomplete What we call uncertainty summary information about the world; a robot never knows of all thatitsisposition not explicitly taken into account exactly in the agent’s KB Questions How to represent uncertainty in knowledge? How to perform inferences with uncertain knowledge? Which action to choose under uncertainty? How do we 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 Handling Uncertainty Approaches: 1. Default reasoning 2. Worst-case reasoning 3. Probabilistic reasoning Default Reasoning 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, … Representation in Logic BIRD(x) ABF(x) FLIES(x) Very active research field in the 80’s PENGUINS(x) AB (x) F Non-monotonic logics: defaults, circumscription, BROKEN-WINGS(x) AB (x) F closed-world assumptions BIRD(Tweety) Applications to databases … Default rule: Unless ABF(Tweety) can be proven True, assume it is False But what to do if several defaults are contradictory? Which ones to keep? Which one to reject? Worst-Case Reasoning 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 Probabilistic Reasoning 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 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? 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 Notion of Probability P(AvA) = P(A)+P(A)-P(A A) You drive on Rt 1 to UMD often, and you notice that 70% of the times there is a traffic slowdown at the intersection of P(True) = P(A)+P(A)-P(False) PaintBranch & Rt 1. The next time you plan to drive on Rt 1, you will believe that the proposition “there is a slowdown 1at= theP(A) intersection of PB & Rt 1” is + P(A) True with probability 0.7 The probabilitySo:of a proposition A is a real P(A) = 1 - P(A) number P(A) between 0 and 1 P(True) = 1 and P(False) = 0 P(AvB) = P(A) + P(B) - P(AB) Axioms of probability Frequency Interpretation Draw a ball from a urn containing n balls of the same size, r red and s yellow. The probability that the proposition A = “the ball is red” is true corresponds to the relative frequency with which we expect to draw a red ball P(A) = ? Subjective Interpretation There are many situations in which there is no objective frequency interpretation: On a windy day, just before paragliding from the top of El Capitan, you say “there is probability 0.05 that I am going to die” You have worked hard on your AI class and you believe that the probability that you will get an A is 0.9 Bayesian Viewpoint probability is "degree-of-belief", or "degree-ofuncertainty". To the Bayesian, probability lies subjectively in the mind, and can--with validity--be different for people with different information e.g., the probability that Wayne will get rich from selling his kidney. In contrast, to the frequentist, probability lies objectively in the external world. The Bayesian viewpoint has been gaining popularity in the past decade, largely due to the increase computational power that makes many of the calculations that were previously intractable, feasible. Random Variables A proposition that takes the value True with probability p and False with probability 1-p is a random variable with distribution (p,1-p) If a urn contains balls having 3 possible colors – red, yellow, and blue – the color of a ball picked at random from the bag is a random variable with 3 possible values The (probability) distribution of a random variable X with n values x1, x2, …, xn is: (p1, p2, …, pn) with P(X=xi) = pi and Si=1,…,n pi = 1 Expected Value Random variable X with n values x1,…,xn and distribution (p1,…,pn) E.g.: X is the state reached after doing an action A under uncertainty Function U of X E.g., U is the utility of a state The expected value of U after doing A is E[U] = Si=1,…,n pi U(xi) Joint Distribution k random variables X1, …, Xk The joint distribution of these variables is a table in which each entry gives the probability of one combination of values of X1, …, Xk Example: Toothache Toothache 0.04 0.06 Cavity 0.01 0.89 Cavity P(CavityToothache) P(CavityToothache) Joint Distribution Says It All Toothache Toothache 0.04 0.06 Cavity 0.01 0.89 Cavity P(Toothache) = ?? P(Toothache v Cavity) = ?? Conditional Probability Definition: P(A|B) =P(AB) / P(B) Read P(A|B): probability of A given B can also write this as: P(AB) = P(A|B) P(B) called the product rule Generalization P(A B C) = P(A|B,C) P(B|C) P(C) Bayes’ Rule P(A B) = P(A|B) P(B) = P(B|A) P(A) P(A|B) P(B) P(B|A) = P(A) Example Given: Toothache Toothache 0.04 0.06 Cavity 0.01 0.89 Cavity P(Cavity)=0.1 P(Toothache)=0.05 P(Cavity|Toothache)=0.8 Bayes’ rule tells: P(Toothache|Cavity)=(0.8x0.05)/0.1 =0.4 Representing Probability Naïve representations of probability run into problems. Example: Patients in hospital are described by several attributes: Background: age, gender, history of diseases, … Symptoms: fever, blood pressure, headache, … Diseases: pneumonia, heart attack, … A probability distribution needs to assign a number to each combination of values of these attributes 20 attributes require 106 numbers Real examples usually involve hundreds of attributes Practical Representation Key idea -- exploit regularities Here we focus on exploiting (conditional) independence properties Example customer purchases: Bread, Bagels and Butter (R,A,U) Bread Bagels Butter p(r,a,u) 0 0 0 0.24 0 0 1 0.06 0 1 0 0.12 0 1 1 0.08 1 0 0 0.12 1 0 1 0.18 1 1 0 0.04 1 1 1 0.16 Independent Random Variables Two variables X and Y are independent if P(X = x|Y = y) = P(X = x) for all values x,y That is, learning the values of Y does not change prediction of X If X and Y are independent then P(X,Y) = P(X|Y)P(Y) = P(X)P(Y) In general, if X1,…,Xn are independent, then P(X1,…,Xn)= P(X1)...P(Xn) Requires O(n) parameters Example #1 Butter p(u) 0 0.52 1 0.48 Bread Bagels Butter p(r,a,u) 0 0 0 0.24 Bagels p(a) 0 0 1 0.06 0 0.6 0 1 0 0.12 1 0.4 0 1 1 0.08 Bread p(r) 1 0 0 0.12 0 1 0 1 0.18 1 1 1 0 0.04 1 1 0.16 Bagels Butter 0 1 p(a,u) Bread Bagels 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1 P(a,u)=P(a)P(u)? P(r,a)=P(r)P(a)? p(r,a) Example #1 Butter p(u) 0 0.52 1 0.48 Bread Bagels Butter p(r,a,u) 0 0 0 0.24 Bagels p(a) 0 0 1 0.06 0 0.6 0 1 0 0.12 1 0.4 0 1 1 0.08 Bread p(r) 1 0 0 0.12 0 0.5 1 0 1 0.18 1 0.5 1 1 0 0.04 1 1 0.16 Bagels Butter 0 0 0 1 p(a,u) Bread Bagels p(r,a) 0.36 0 0 0.3 1 0.24 0 1 0.2 1 0 0.16 1 0 0.3 1 1 0.24 1 1 0.2 P(a,u)=P(a)P(u)? P(r,a)=P(r)P(a)? Conditional Independence Unfortunately, random variables of interest are not independent of each other A more suitable notion is that of conditional independence Two variables X and Y are conditionally independent given Z if P(X = x|Y = y,Z=z) = P(X = x|Z=z) for all values x,y,z That is, learning the values of Y does not change prediction of X once we know the value of Z notation: I( X ; Y | Z ) Car Example Three propositions: Gas Battery Starts P(Battery|Gas) = P(Battery) Gas and Battery are independent P(Battery|Gas,Starts) ≠ P(Battery|Starts) Gas and Battery are not independent given Starts Example #2 Hotdogs Mustard Ketchup p(h,m,k) 0 0 0 0.576 0 0 1 0.144 0 1 0 0.064 0 1 1 0.016 1 0 0 0.004 1 0 1 0.036 1 1 0 0.016 1 1 1 0.144 Mustard Ketchup p(m,k) 0 0 0.58 0 1 0.18 1 0 0.08 1 1 0.16 P(m,k)=P(m)P(k)? Mustard p(m) 0 0.76 1 0.24 Ketchup p(k) 0 0.66 1 0.34 Example #2 H M K p(h,m,k) 0 0 0 0.576 0 0 1 0.144 0 1 0 0.064 0 1 1 1 0 1 Mustard Hotdogs p(m|h) 0 0 0.9 0 1 0.2 0.016 1 0 0.1 0 0.004 1 1 0.8 0 1 0.036 1 1 0 0.016 1 1 1 0.144 Ketchup Hotdogs p(k|h) 0 0 0.8 0 1 0.1 Mustard Ketchup Hotdogs p(m,k|h) 1 0 0.2 0 0 0 0.72 1 1 0.9 0 1 0 0.18 1 0 0 0.08 1 1 0 0.02 0 0 1 0.02 0 1 1 0.18 1 0 1 0.08 1 1 1 0.72 P(m,k|h)=P(m|h)P(k|h)? Example #1 Bread Bagels Butter p(r,a,u) 0 0 0 0.24 0 0 1 0.06 Bread Butter p(r|u) 0 1 0 0.12 0 0 0.69… 0 1 1 0.08 0 1 0.29… 1 0 0 0.12 1 0 0.30… 1 0 1 0.18 1 1 0.70… 1 1 0 0.04 1 1 1 0.16 Bagels Butter p(a|u) Bread Bagels Butter p(r,a|u) 0 0 0.69… 0 0 0 0.46… 0 1 0.5 0 1 0 0.23… 1 0 0.30… 1 0 0 0.23… 1 1 0.5 1 1 0 0.08… 0 0 1 0.12… 0 1 1 0.17... 1 0 1 0,38… 1 1 1 0.33… P(r,a|u)=P(r|u)P(a|u)? Summary Example 1: I(X,Y|) and not I(X,Y|Z) Example 2: I(X,Y|Z) and not I(X,Y|) conclusion: independence does not imply conditional independence! Example: Naïve Bayes Model A common model in early diagnosis: Symptoms are conditionally independent given the disease (or fault) Thus, if X1,…,Xn denote whether the symptoms exhibited by the patient (headache, high-fever, etc.) and H denotes the hypothesis about the patients health then, P(X1,…,Xn,H) = P(H)P(X1|H)…P(Xn|H), This naïve Bayesian model allows compact representation It does embody strong independence assumptions