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Definition and some Properties of Information Entropy

100%
Zhang B., Nakamura Y.
Formalized Mathematics
|
2007
|
tom 15
|
nr 3
111-119
EN
In this article we mainly define the information entropy [3], [11] and prove some its basic properties. First, we discuss some properties on four kinds of transformation functions between vector and matrix. The transformation functions are LineVec2Mx, ColVec2Mx, Vec2DiagMx and Mx2FinS. Mx2FinS is a horizontal concatenation operator for a given matrix, treating rows of the given matrix as finite sequences, yielding a new finite sequence by horizontally joining each row of the given matrix in order to index. Then we define each concept of information entropy for a probability sequence and two kinds of probability matrices, joint and conditional, that are defined in article [25]. Further, we discuss some properties of information entropy including Shannon's lemma, maximum property, additivity and super-additivity properties.
2
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The Definition of Finite Sequences and Matrices of Probability, and Addition of Matrices of Real Elements

100%
Zhang B., Nakamura Y.
Formalized Mathematics
|
2006
|
tom 14
|
nr 3
101-108
EN
In this article, we first define finite sequences of probability distribution and matrices of joint probability and conditional probability. We discuss also the concept of marginal probability. Further, we describe some theorems of matrices of real elements including quadratic form.
3
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The Relevance of Measure and Probability, and Definition of Completeness of Probability

81%
Zhang B., Yamazaki H., Nakamura Y.
Formalized Mathematics
|
2006
|
tom 14
|
nr 4
225-229
EN
In this article, we first discuss the relation between measure defined using extended real numbers and probability defined using real numbers. Further, we define completeness of probability, and its completion method, and also show that they coincide with those of measure.
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