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CSC 450 AIMA 3e Chapter 13: Quantifying Uncertainty OUTLINE 2 Overview 1. rationale for a new representational language what logical representations can't do 2. 3. 4. 5. 6. 7. utilities & decision theory possible worlds & propositions unconditional & conditional probabilities random variables probability distributions using the Joint Probability Distribution for inference by enumeration for unconditional & conditional probabilities Quantifying Uncertainty 3 Quantifying Uncertainty consider our approach so far we've handled limited observability &/or non-determinism using belief states that capture all possible world states but the representation can become large, as can corresponding contingent plans, and it's possible that no plan can be guaranteed to reach the goal, yet the agent must act agents should behave rationally this rationality depends both on the importance of goals and on the chances of & degree to which they'll be reached Quantifying Uncertainty 4 A Visit to the Dentist we'll use medical/dental diagnosis examples extensively our new prototype problem relates to whether a dental patient has a cavity or not the process of diagnosis always involves uncertainty & this leads to difficulty with logical representations (propositional logic examples) (1) toothache cavity (2) toothache cavity gumDisease ... (3) cavity toothache (1) is just wrong since other things cause toothaches (2) will need to list all possible causes (3) tries a causal rule but it's not always the case that cavities cause toothaches & fixing the rule requires making it logically exhaustive Quantifying Uncertainty 5 Representations for Diagnosis logic is not sufficient for medical diagnosis, due to our Laziness: it's too hard to list all possible antecedents or consequents to make the rule have no exceptions our Theoretical Ignorance: generally, there is no complete theory of the domain, no complete model our Practical Ignorance: even if the rules were complete, in any particular case it's impractical or impossible to do all the necessary tests, to have all relevant evidence the example relationship between toothache & cavities is not a logical consequence in either direction instead, knowledge of the domain provides a degree of belief in diagnostic sentences & the way to represent this is with probability theory next slide: recall our discussion of ontological & epistemological commitments from 352 Quantifying Uncertainty 6 Epistemological Commitment ontological commitment what a representational language assumes about the nature of reality - logic & probability theory agree in this, that facts do or do not hold epistemological commitment the possible states of knowledge for logic, sentences are true/false/unknown for probability theory, there's a numerical degree of belief in sentences, between 0 (certainly false) and 1 (certainly true) Quantifying Uncertainty The Qualification Problem 7 for a logical representation the success of a plan can't be inferred because of all the conditions that could interfere but can't be deduced not to happen (this is the qualification problem) probability is a way of dealing with the qualification problem by numerically summarizing the uncertainty that derives from laziness &/or ignorance returning to the toothache & cavity problem in the real world, the patient either does or does not have a cavity a probabilistic agent makes statements with respect to the knowledge state, & these may change as the state of knowledge changes for example, an agent initially may believe there's an 80% chance (probability 0.8) that the patient with the toothache has a cavity, but subsequently revises that as additional evidence is available 8 Rational Decisions making choices among plans/actions when the probabilities of their success differ this requires additional knowledge of preferences among outcomes this is the domain of utility theory: every state has a degree of utility/usefulness to the agent & the agent will prefer those with higher utility utilities are specific to an agent, to the extent that they can even encompass perverse or altruistic preferences Quantifying Uncertainty 9 Rational Decisions making choices among plans/actions when the probabilities of their success differ we can combine preferences (utilities) + probabilities to get a general theory of rational decisions: Decision Theory a rational agent chooses actions to yield the highest expected utility averaged over all possible outcomes of the action this is the Maximum Expected Utility (MEU) principle expected = average of the possible outcomes of an action weighted by their probabilities choice of action = the one with highest expected utility Quantifying Uncertainty Revising Belief States 10 belief states in addition to the possible world states that we included before, belief states now include probabilities the agent incorporates probabilistic predictions of action outcomes, selecting the one with the highest expected utility AIMA3e chapters 13 through 17 address various aspects of using probabilistic representations an algorithmic description of the Decision Theoretic Agent function DT-AGENT (percept) returns an action persistent: belief-state, probabilistic beliefs about the current state of the world action, the agent's action update belief-state based on action and percept calculate outcome probabilities for actions, given action descriptions and current belief state select action with the highest expected utility, given probabilities of outcomes and utility information return action 11 Notation & Basics we should interpret probabilities as describing possible worlds and their likelihoods the sample space is the set of all possible worlds note that possible worlds are mutually exclusive & exhaustive for example, a roll of a pair of dice has 36 possible worlds we use the Greek letter omega to refer to possible worlds refers to the sample space, to its elements (particular possible worlds) a basic axiom for probability theory (13.1) 0 P() 1, P() 1 as an example, for the dice rolls, each possible world is a pair (1, 1), (1, 2), ..., (6, 6) each with a probability of 1/36, all summing to 1 Quantifying Uncertainty 12 Notation & Basics assertions & queries in probabilistic reasoning these are usually about sets of possible worlds these are termed events in probability theory for AI, the sets of possible worlds are described by propositions in a formal language the set of possible worlds corresponding to a proposition contains those in which the proposition holds the probability of the proposition is the sum over those possible worlds Quantifying Uncertainty Propositions propositions another axiom of probability theory, using the Greek letter phi () for proposition (13.2) P() P() so for a fair pair of dice P(total = 7) = P((1+6))+P((6+1))+P((2+5))+P((5+2))+P(((3+4))+P((4+3)) =1/36+1/36+1/36+1/36+1/36+1/36 = 1/6 asserting the probability of a proposition constrains the underlying probability model without fully determining it 13 Propositions 14 propositions: unconditional & conditional probabilities P(total = 7) from the previous slide & similar probabilities are called unconditional or prior probabilities, sometimes abbreviated as priors they indicate the degree of belief in propositions without any other information, though in most cases, we do have other information, or evidence when we have evidence, the probabilities are conditional or posterior, given the evidence Conditional Probability P(A | B) is the probability of A given B Assumes that B is the only info known. Defined by: True P( A B) P( A | B) P( B) A AB B 15 Random Variables & Values 16 propositions in our probability notation by convention, the values for random variables use lower case letters, for example Weather = rain each random variable has a domain, its set of possible values for a Boolean random variable the domain is {true, false} also by convention, A = true is written as simply a, A = false as ¬a domains also may be arbitrary sets of tokens, like the {red, green, blue} of the map coloring CSP or {juvenile, teen, adult} for Age when it's unambiguous, a value by itself may represent the proposition that a variable has that value for example, using just sunny for Weather = sunny 17 Distribution Notation bold is used as a notational coding for the probabilities of all possible values of a random variable we may list the propositions or we may abbreviate, given an ordering on the domain as in the ordering (sunny, rain, cloudy, snow) for Weather then P(Weather) = <0.6, 0.1, 0.29, 0.01>, where bold indicates there's a vector of values this defines a probability distribution for the random variable Weather we can use a similar shorthand for conditional distributions, for example: P(X|Y) lists the values for P(X=xi | Y=yj) for all i,j pairs Quantifying Uncertainty Continuous Variables distributions are the probabilities of all possible values of a random variable there's alternative notation for continuous variables where there cannot be an explicit list: instead, express the distribution as a parameterized function of value for example, P(NoonTemp=x) = Uniform[18C,26C] (x) specifies a probability density function (pdf) that defines density function values for intervals of the NoonTemp variable values 18 Distribution Notation 19 for distributions on multiple variables we use commas between the variables: so P(Weather, Cavity) denotes the probabilities of all combinations of values of the 2 variables for discrete random variables we can use a tabular representation, in this case yielding a 4x2 table of probabilities this gives the joint probability distribution of Weather & Cavity tabulates the probabilities for all combinations Full Joint Distribution 20 semantics of a proposition the probability model is determined by the joint distribution for all the random variables: the full joint probability distribution for the Cavity, Toothache, Weather domain, the notation is: P(Cavity, Toothache, Weather) this can be represented as a 2x2x4 table given the definition of the probability of a proposition as a sum over possible worlds, the full joint distribution allows calculating the probability of any proposition over its variables by summing entries in the FJD 21 Inference With Probability using the full joint distributions for inference here's the FJD for the Toothache, Cavity, Catch domain of 3 Boolean variables as required by the axioms, the probabilities sum to 1.0 when available, the FJD gives a direct means of calculating the probability of any proposition just sum the probabilities for all the possible worlds in which the proposition is true Quantifying Uncertainty Inference With Probability P(toothache)=.108+.012+.016+.064 = .20 or 20% 22 Inference With Probability P(toothachecavity) = .20 + ??.072 + .008 .28 23 Inference With Probability 24 25 Inference for Probability given the full joint distribution & 13.9 we can answer all probability queries for discrete variables are we left with any unresolved issues? well, given n variables, and d as an upper bound on the number of values then the full joint distribution table size & corresponding processing of it are O(dn), exponential in n since n might be 100 or more for real problems, this is often simply not practical as a result, the FJD is not the implementation of choice for real systems, but functions more as the theoretical reference point (analogous to role of truth tables for propositional logic) the next sections we look at are foundational for developing practical systems Quantifying Uncertainty 26 END OF LECTURE Quantifying Uncertainty