Download Reasoning under Uncertainty, covering Section 6.1

CSCE 580
Artificial Intelligence
Ch.6 [P]: Reasoning Under Uncertainty
Section 6.1: Probability
Fall 2009
Marco Valtorta
[email protected]
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.
Probability does not exist.
--Pierre Simon de Laplace, 1812
--Bruno de Finetti, 1970
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
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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.
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Random Variables
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Possible World Semantics
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Semantics of Probability: finite case
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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.
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Some properties of finite probabilities
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More properties
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Semantics of Probability: general case
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Probability Distributions
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Conditioning
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Examples of evidence
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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.
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Observations and conditioning
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Semantics of Conditional Probability
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Semantics of Conditional Probability: Details
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Conditional probability is not the
probability of implication
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Compositional Measures of Belief
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Chain Rule
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Bayes’ Theorem
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Applications of Bayes’ Theorem
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