Computing & Information Sciences Kansas State University Wednesday, 24 Oct 2007 CIS 530 / 730: Artificial Intelligence Lecture 26 of 42 Wednesday. 24 October 2007 William H. Hsu Department of Computing and Information Sciences, KSU KSOL course page: http://snipurl.com/v9v3 Course web site: http://www.kddresearch.org/Courses/Fall-2007/CIS730 Instructor home page: http://www.cis.ksu.edu/~bhsu Reading for Next Class: Section 12.5 – 12.8, Russell & Norvig 2 nd edition nditional, Continuous, and Multi-Agent Plan Discussion: Probability Refresher
Lecture 26 of 42. Conditional, Continuous, and Multi-Agent Planning Discussion: Probability Refresher. Wednesday. 24 October 2007 William H. Hsu Department of Computing and Information Sciences, KSU KSOL course page: http://snipurl.com/v9v3 - PowerPoint PPT Presentation
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Computing & Information SciencesKansas State University
Wednesday, 24 Oct 2007CIS 530 / 730: Artificial Intelligence
Lecture 26 of 42
Wednesday. 24 October 2007
William H. Hsu
Department of Computing and Information Sciences, KSU
KSOL course page: http://snipurl.com/v9v3
Course web site: http://www.kddresearch.org/Courses/Fall-2007/CIS730
Instructor home page: http://www.cis.ksu.edu/~bhsu
Reading for Next Class:
Section 12.5 – 12.8, Russell & Norvig 2nd edition
Conditional, Continuous, and Multi-Agent PlanningDiscussion: Probability Refresher
Computing & Information SciencesKansas State University
Wednesday, 24 Oct 2007CIS 530 / 730: Artificial Intelligence
Universal Quantifiers in Planning
Quantification within Operators p. 383 R&N
ExamplesShakey’s World
Blocks World
Grocery shopping
Others (from projects?)
Exercise for Next Tuesday: Blocks World
Computing & Information SciencesKansas State University
Wednesday, 24 Oct 2007CIS 530 / 730: Artificial Intelligence
Practical Planning
The Real World What can go wrong with classical planning?
What are possible solution approaches?
Conditional Planning
Monitoring and Replanning (Next Time)
Adapted from Russell and Norvig
Computing & Information SciencesKansas State University
Wednesday, 24 Oct 2007CIS 530 / 730: Artificial Intelligence
Review:How Things Go Wrong in Planning
Adapted from slides by S. Russell, UC Berkeley
Computing & Information SciencesKansas State University
Wednesday, 24 Oct 2007CIS 530 / 730: Artificial Intelligence
Review:Practical Planning Solutions
Adapted from slides by S. Russell, UC Berkeley
Computing & Information SciencesKansas State University
Wednesday, 24 Oct 2007CIS 530 / 730: Artificial Intelligence
Adapted from slides by S. Russell, UC Berkeley
Conditional Planning
Computing & Information SciencesKansas State University
Wednesday, 24 Oct 2007CIS 530 / 730: Artificial Intelligence
Monitoring and ReplanningMonitoring and Replanning
Computing & Information SciencesKansas State University
Wednesday, 24 Oct 2007CIS 530 / 730: Artificial Intelligence
Adapted from slides by S. Russell, UC Berkeley
Preconditions for Remaining Plan
Computing & Information SciencesKansas State University
Wednesday, 24 Oct 2007CIS 530 / 730: Artificial Intelligence
Adapted from slides by S. Russell, UC Berkeley
Replanning
Computing & Information SciencesKansas State University
Wednesday, 24 Oct 2007CIS 530 / 730: Artificial Intelligence
Making Decisions under Uncertainty
Adapted from slides by S. Russell, UC Berkeley
Computing & Information SciencesKansas State University
Wednesday, 24 Oct 2007CIS 530 / 730: Artificial Intelligence
Probability:Basic Definitions and Axioms
Sample Space (): Range of a Random Variable X Probability Measure Pr()
denotes a range of “events”; X: Probability Pr, or P, is a measure over 2
In a general sense, Pr(X = x ) is a measure of belief in X = xP(X = x) = 0 or P(X = x) = 1: plain (aka categorical) beliefs (can’t be revised)All other beliefs are subject to revision
Kolmogorov Axioms 1. x . 0 P(X = x) 1 2. P() x P(X = x) = 1
3.
Joint Probability: P(X1 X2) Probability of the Joint Event X1 X2
Independence: P(X1 X2) = P(X1) P(X2)
1ii
1ii
ji21
XPXP
.XXji,X,X
Computing & Information SciencesKansas State University
Wednesday, 24 Oct 2007CIS 530 / 730: Artificial Intelligence
Basic Formulas for Probabilities
Product Rule (Alternative Statement of Bayes’s Theorem)
Proof: requires axiomatic set theory, as does Bayes’s Theorem
Sum Rule
Sketch of proof (immediate from axiomatic set theory)Draw a Venn diagram of two sets denoting events A and B
Let A B denote the event corresponding to A B…
Theorem of Total Probability Suppose events A1, A2, …, An are mutually exclusive and exhaustive
Mutually exclusive: i j Ai Aj =
Exhaustive: P(Ai) = 1
Then
Proof: follows from product rule and 3rd Kolmogorov axiom