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2016 | OriginalPaper | Chapter

12. Reasoning with Imperfect Knowledge

Author: Mariusz Flasiński

Publisher: Springer International Publishing

Published in: Introduction to Artificial Intelligence

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Abstract

If we reason about propositions in AI systems which are based on classic logic, we use only two possible logic values, i.e., true and false. However, in the case of reasoning about the real (physical) world such a two-valued evaluation is inadequate, because of the aspect of uncertainty

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Models applied for solving a problem of vague notions are introduced in the next chapter.

Mathematical foundations of probabilistic reasoning are introduced in Appendix I.

Such an observation is represented as a fact in a knowledge base.

Of course, all examples are simplified.

Strictly speaking, \(h_{i}\) means making a diagnosis (hypothesis) that the patient has the disease entity denoted by an index i.

Let us notice that it is really an a priori probability in the case of diagnosing John Smith, because for such an evaluation we do not take into account any symptoms/factors concerning him.

Let us notice that in order to use P( e) in formula ( 12.1) we have to compute this probability with formula ( 12.2). Thus, a priori probabilities \(h_{1}, h_{2}, \ldots , h_{n}\) should be known for all disease entities. We should also know the probabilities that a patient having an illness denoted by an index \(i = 1,2, \ldots , n\) has a temperature which is more than \(39.5\,^{\circ }\mathrm{C}\), i.e., \(P(e|h_i)\). We are able to evaluate these probabilities if we have corresponding statistical data.

This means that \(P(e_{1}, e_{2}, \ldots , e_{m} | h_{i}) = P(e_{1} | h_{i}) \cdot P(e_{2} | h_{i}) \cdot \cdots \cdot P(e_{m} | h_{i})\).

In this chapter we consider discrete random variables. Formal definitions of a random variable, a random vector, and distributions are contained in Appendix B.1.

In the first column of the table elementary events are placed. For each elementary event the value which is taken by the random variable is also defined.

Let us assume that a random vector \((X_1, X_2)\) is given, where \(X_1\) takes values \(a_1, \ldots , a_{m1}\) and \(X_2\) takes values \(b_1, \ldots , b_{m2}\). If we are interested only in the distribution of one variable and the other variable can take any values, then we talk about the marginal distribution of the first variable. Then, for example the marginal distribution of the variable \(X_1\) is determined in the following way: \(P(X_1 = a_i) = P(X_1 = a_i , X_2 = b_1) + \cdots + P(X_1 = a_i , X_2 = b_{m2}), \ i = 1, \ldots , m_1\).

The marginal distribution for the second variable \(X_2\) is determined in an analogous way. For an n-dimensional random vector we can determine the marginal distribution for any subset of variables, assuming that the remaining variables take any values.

That is, there are no directed cycles in the graph.

In order to simplify our considerations, without loss of generality of principles.

In our examples they are denoted T-T, T-F, F-T, F-F, respectively.

We can compute it as follows: \(P(\lnot \) L \() = 1.0 - P(\) L \() = 1.0 - 0.001 = 0.999\).

The variable F is a logical variable. Therefore, it is sufficient to determine the probabilities when F equals True. Probabilities for the value False are complements of these probabilities.

Conditional independence of variables is defined formally by Definition I.10 in Appendix I.

Earlier in the sense of indexing nodes of the network.

We have denoted such predecessors by \(Predecessors(X_{i})\).

Arthur Pentland Dempster—a professor of statistics at Harvard University. John W. Tukey (the Cooley-Tukey algorithm for Fast Fourier Transforms) was an adviser of his Ph.D. thesis. His work concerns the theory introduced in this section, cluster analysis, and image processing (the EM algorithm).

Glenn Shafer—a professor of statistics at Rutgers University. Apart from the development of DST, he proposed a new approach to probability theory based on game theory (instead of measure theory).

In the context of reasoning with incomplete knowledge, evidence means information gained by an AI system at some moment which is used as a premise of inference.

We mean an intuitive interpretation of this notion, not in the sense of probability theory.

Basic definitions of Dempster-Shafer Theory are included in Appendix I.3.

Let us notice that a probability measure P has the following property: \(P(\lnot \) genuine) = 1 \(- P(\) genuine). This property does not hold for a belief function Bel.

The stronger evidences are the less a value of Pl( S) is.

Let us assume that I have only one car.

Raymond Reiter—a professor of computer science and logic at the University of Toronto. His work concerns non-monotonic reasoning, knowledge representation models, logic programming, and image analysis.

Standard rules means such rules as the ones introduced in Chap. 6.

Robert C. Moore—a researcher at Microsoft Research and NASA Ames Research Center, Ph.D. in computer science (MIT). His work concerns NLP, artificial intelligence, automatic theorem proving, and speech recognition.

Drew McDermott—a professor of computer science at Yale University. His work concerns AI, robotics, and pattern recognition.

Jon Doyle—a professor of computer science at the Massachusetts Institute of Technology, Stanford University, and Carnegie-Mellon University. His work concerns reasoning methods, philosophical foundations of Artificial Intelligence, and AI applications in economy and psychology.

Such a proposition does not need to be removed (physically) from the knowledge base. It is enough to mark that the proposition is invalid (currently). If, for example, the removed justification is restored, then the system needs only to change its status to valid.

Johan de Kleer—a director of Systems and Practices Laboratory, Palo Alto Research Center (PARC). His work concerns knowledge engineering, model-based reasoning, and AI applications in qualitative physics.

Patrick John Hayes—a British computer scientist and mathematician, a professor of prestigious universities (Rochester, Stanford, Essex, Geneva). His work concerns knowledge representation, automated inference, philosophical foundations of AI, and semantic networks.

Title

Reasoning with Imperfect Knowledge

Book

Introduction to Artificial Intelligence

Print ISBN: 978-3-319-40020-4

Electronic ISBN: 978-3-319-40022-8

https://doi.org/10.1007/978-3-319-40022-8

DOI

https://doi.org/10.1007/978-3-319-40022-8_12

Author:

Mariusz Flasiński

Publisher

Springer International Publishing

Sequence number

Chapter number

Chapter 12