UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering CSCE 580 Artificial Intelligence Ch.6 [P]: Reasoning Under Uncertainty Section 6.1: Probability Fall 2009 Marco Valtorta Probability does not exist. --Bruno de Finetti, 1970 It is remarkable that a science which began with the consideration of games of chance should become the most important object of human knowledge... The most important questions of life are, for the most part, really only problems of probability... The theory of probabilities is at bottom nothing but common sense reduced to calculus. --Pierre Simon de Laplace, 1812

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Acknowledgment The slides are based on the textbook [P] and other sources, including other fine textbooks –[AIMA-2] –David Poole, Alan Mackworth, and Randy Goebel. Computational Intelligence: A Logical Approach. Oxford, 1998 A second edition (by Poole and Mackworth) is under development. Dr. Poole allowed us to use a draft of it in this course –Ivan Bratko. Prolog Programming for Artificial Intelligence, Third Edition. Addison-Wesley, 2001 The fourth edition is under development –George F. Luger. Artificial Intelligence: Structures and Strategies for Complex Problem Solving, Sixth Edition. Addison-Welsey, 2009

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Using Uncertain Knowledge Agents don't have complete knowledge about the world. Agents need to make decisions based on their uncertainty. It isn't enough to assume what the world is like. Example: wearing a seat belt. An agent needs to reason about its uncertainty. When an agent makes an action under uncertainty, it is gambling => probability.

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Probability Probability is an agent's measure of belief in some proposition---subjective or Bayesian probability. Example: Your probability of a bird flying is your measure of belief in the flying ability of an individual based only on the knowledge that the individual is a bird. – Other agents may have different probabilities, as they may have had different experiences with birds or different knowledge about this particular bird. –An agent's belief in a bird's flying ability is affected by what the agent knows about that bird.

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Numerical Measures of Belief Belief in proposition, f, can be measured in terms of a number between 0 and 1---this is the probability of f. –The probability f is 0 means that f is believed to be definitely false. –The probability f is 1 means that f is believed to be definitely true. Using 0 and 1 is purely a convention. The fact that f has a probability between 0 and 1 doesn't mean that f is true to some degree, but it means you are ignorant of its truth value. Probability is a measure of your ignorance. We are assuming that the uncertainty is epistemological— pertaining to an agent’s knowledge of the world—rather than ontological—how the world is. We are assuming that an agent’s knowledge of the truth of propositions is uncertain, not that there are degrees of truth. For example, if you are told that someone is very tall, you know they have some height; you only have vague knowledge about the actual value of their height.

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Random Variables

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Possible World Semantics

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Semantics of Probability: finite case

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Axioms of Probability (Kolmogorov): finite case P is a probability function if: These axioms are meant to be intuitive properties that we would like of any reasonable measure of belief. If a measure of belief follows these intuitive axioms, it is covered by probability theory, whether or not the measure is derived from actual frequency counts. These axioms form a sound and complete axiomatization of the meaning of probability. Soundness means that probability, as defined by the possible worlds semantics, follows these axioms. Completeness means that any system of beliefs that obeys these axioms has a probabilistic semantics.

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Some properties of finite probabilities

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering More properties

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Semantics of Probability: general case

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Probability Distributions

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Conditioning

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Examples of evidence

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Background Knowledge vs. Observations Offline computation is the computation done by the agent before it has to act. It can include compilation and learning. Offline, the agent takes background knowledge and data and compiles them into a usable form called a knowledge base. Background knowledge can either be given at design time or offline. An observation is a piece of information received online from users, sensors or other knowledge sources. Observations are implicitly conjoined, so that a set of observations is a conjunction of atoms. Neither users nor sensors provide rules directly from observing the world. The background knowledge allows the agent to do something useful with these observations. In many reasoning frameworks, the observations are added to the background knowledge. But in other reasoning frameworks (e.g, in abduction, probabilistic reasoning and learning), observations are treated separately from background knowledge.

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Observations and conditioning

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Semantics of Conditional Probability

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Semantics of Conditional Probability: Details

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Conditional probability is not the probability of implication

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Compositional Measures of Belief

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Chain Rule

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Bayes’ Theorem

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Applications of Bayes’ Theorem