Chapter 5 STOCHASTIC METHODS Contents The Elements

Chapter 5 STOCHASTIC METHODS Contents • The Elements of Counting • Elements of Probability Theory • Applications of the Stochastic Methodology • Bayes’ Theorem CSc 411 Artificial Intelligence 1

Application Areas Diagnostic Reasoning. In medical diagnosis, for example, there is not always an obvious cause/effect relationship between the set of symptoms presented by the patient and the causes of these symptoms. In fact, the same sets of symptoms often suggest multiple possible causes. Natural language understanding. If a computer is to understand use a human language, that computer must be able to characterize how humans themselves use that language. Words, expressions, and metaphors are learned, but also change and evolve as they are used over time. Planning and scheduling. When an agent forms a plan, for example, a vacation trip by automobile, it is often the case that no deterministic sequence of operations is guaranteed to succeed. What happens if the car breaks down, if the car ferry is cancelled on a specific day, if a hotel is fully booked, even though a reservation was made? Learning. The three previous areas mentioned for stochastic technology can also be seen as domains for automated learning. An important component of many stochastic systems is that they have the ability to sample situations and learn over time. CSc 411 Artificial Intelligence 2

Set Operations Let A and B are two sets, U universe – – – – CSc 411 Cardinality |A|: number of elements in A Complement Ā: all elements in U but not in A Subset: A B Empty set: Union: A B Intersection: A B Difference: A - B Artificial Intelligence 3

Addition Rules The Addition rule for combining two sets: The Addition rule for combining three sets: This Addition rule may be generalized to any finite number of sets CSc 411 Artificial Intelligence 4

Multiplication Rules • The Cartesian Product of two sets A and B • The multiplication principle of counting, for two sets CSc 411 Artificial Intelligence 5 5

Permutations and Combinations • The permutations of a set of n elements taken r at a time • The combinations of a set of n elements taken r at a time CSc 411 Artificial Intelligence 6

Events and Probability CSc 411 Artificial Intelligence 7

Probability Properties • The probability of any event E from the sample space S is: • The sum of the probabilities of all possible outcomes is 1 • The probability of the compliment of an event is • The probability of the contradictory or false outcome of an event CSc 411 Artificial Intelligence 8

Independent Events CSc 411 Artificial Intelligence 9

The Kolmogorov Axioms Three Kolmogorov Axioms: From these three Kolmogorov axioms, all of probability theory can be constructed. CSc 411 Artificial Intelligence 10

Traffic Example Problem description A driver realizes the gradual slowdown and searches for possible explanation by means of car-based download system – Road construction? – Accident? Three Boolean parameters – S: whether slowdown – A: whether accident – C: whether road construction Download data – next page CSc 411 Artificial Intelligence 11

• Download data: The joint probability distribution for the traffic slowdown, S, accident, A, and construction, C, variable of the example A Venn diagram representation of the probability distributions is traffic slowdown, A is accident, C is construction. CSc 411 Artificial Intelligence 12

Variables CSc 411 Artificial Intelligence 13

Expectation CSc 411 Artificial Intelligence 14

Prior and Posterior Probability CSc 411 Artificial Intelligence 15

Conditional Probability A Venn diagram illustrating the calculations of P(d|s) as a function of p(s|d). CSc 411 Artificial Intelligence 16

Chain Rules The chain rule for two sets: The generalization of the chain rule to multiple sets We make an inductive argument to prove the chain rule, consider the nth case: We apply the intersection of two sets of rules to get: And then reduce again, considering that: Until is reached, the base case, which we have already demonstrated. CSc 411 Artificial Intelligence 17

Independent Events CSc 411 Artificial Intelligence 18

Probabilistic FSM CSc 411 Artificial Intelligence 19

Probabilistic Finite State Acceptor A probabilistic finite state acceptor for the pronunciation of “tomato”. CSc 411 Artificial Intelligence 20

The ni words with their frequencies and probabilities from the Brown and Switchboard corpora of 2. 5 M words. The ni phone/word probabilities from the Brown and Switchboard corpora. CSc 411 Artificial Intelligence 21

Bayes’ Rules • Given a set of evidence E, and a set of hypotheses H ={hi} The conditional probability of hi given E is: p(hi|E) = (p(E|hi) h(hi))/p(E) • Maximum a posteriori hypothesis (most probable hypothesis), since p(E)is a constant for all hypotheses arg max(hi) p(E|hi)p(hi) • E is partitioned by all hypotheses, thus p(E) = i p(E|hi)p(hi) CSc 411 Artificial Intelligence 22

General Form of Bayes’ Theorem The general form of Bayes’ theorem where we assume the set of hypotheses H partition the evidence set E: CSc 411 Artificial Intelligence 23

Applications of Bayes’ Theorem • Used in PROSPECTOR • A simple example: suppose to purchase an automobile: Dealers Go to probability Purchase a 1 probability 1 d 1 = 0. 2 p 1 = 0. 2 2 d 2 = 0. 4 p 2 = 0. 4 3 d 3 = 0. 4 p 3 = 0. 3 • The application of Bayes’ rule to the car purchase problem: CSc 411 Artificial Intelligence 24

Bayes Classifier • Naïve Bayes, or the Bayes classifier, that uses the partition assumption, even when it is not justified: . • Assume all evidences are independent, given a particular hypothesis CSc 411 Artificial Intelligence 25

The Traffic Problem The Bayesian representation of the traffic problem with potential explanations. The joint probability distribution for the traffic and construction variables Given bad traffic, what is the probability of road construction? p(C|T)=p(C=t, T=t)/(p(C=t, T=t)+p(C=f, T=t))=. 3/(. 3+. 1)=. 75 CSc 411 Artificial Intelligence 26