CSc411Artificial Intelligence1 Chapter 5 STOCHASTIC METHODS Contents The Elements of Counting Elements of Probability Theory Applications of the Stochastic Methodology Bayes’ Theorem

CSc411Artificial Intelligence2 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 and 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.

CSc411Artificial Intelligence3 Set Operations Let A and B are two sets, U universe –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

CSc411Artificial Intelligence4 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

CSc411Artificial Intelligence5 The Cartesian Product of two sets A and B The multiplication principle of counting, for two sets 5 Multiplication Rules

CSc411Artificial Intelligence6 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 Permutations and Combinations

CSc411Artificial Intelligence7 Events and Probability

CSc411Artificial Intelligence8 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 Probability Properties

CSc411Artificial Intelligence9 Independent Events

CSc411Artificial Intelligence10 The Kolmogorov Axioms Three Kolmogorov Axioms: From these three Kolmogorov axioms, all of probability theory can be constructed.

CSc411Artificial Intelligence11 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

CSc411Artificial Intelligence12 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. Download data:

CSc411Artificial Intelligence13 Variables

CSc411Artificial Intelligence14 Expectation

CSc411Artificial Intelligence15 Prior and Posterior Probability

CSc411Artificial Intelligence16 A Venn diagram illustrating the calculations of P(d|s) as a function of p(s|d). Conditional Probability

CSc411Artificial Intelligence17 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 n th 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. Chain Rules

CSc411Artificial Intelligence18 Independent Events

CSc411Artificial Intelligence19 Probabilistic FSM

CSc411Artificial Intelligence20 A probabilistic finite state acceptor for the pronunciation of “tomato”. Probabilistic Finite State Acceptor

CSc411Artificial Intelligence21 The ni words with their frequencies and probabilities from the Brown and Switchboard corpora of 2.5M words. The ni words The ni phone/word probabilities from the Brown and Switchboard corpora.

CSc411Artificial Intelligence22 Given a set of evidence E, and a set of hypotheses H ={h i } The conditional probability of h i given E is: p(h i |E) = (p(E|h i )  h(h i ))/p(E) Maximum a posteriori hypothesis (most probable hypothesis), since p(E)is a constant for all hypotheses arg max(h i ) p(E|h i )p(h i ) E is partitioned by all hypotheses, thus p(E) =  i p(E|h i )p(h i ) Bayes’ Rules

CSc411Artificial Intelligence23 The general form of Bayes’ theorem where we assume the set of hypotheses H partition the evidence set E : General Form of Bayes’ Theorem

CSc411Artificial Intelligence24 Used in PROSPECTOR A simple example: suppose to purchase an automobile: Applications of Bayes’ TheoremDealers Go to probability Purchase a1 probability 1 d1 = 0.2 p1 = d2 = 0.4 p2 = d3 = 0.4 p3 = 0.3 The application of Bayes’ rule to the car purchase problem:

CSc411Artificial Intelligence25 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 Bayes Classifier

CSc411Artificial Intelligence26 The Bayesian representation of the traffic problem with potential explanations. The joint probability distribution for the traffic and construction variables The Traffic Problem 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