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Information-geometrical characterization of statistical models which are statistically equivalent to probability simplexes

Hiroshi Nagaoka

TL;DR

This paper gives a characterization of statistical models on finite sets which are statistically equivalent to probability simplexes in terms of α-families including exponential families and mixture families.

Abstract

The probability simplex is the set of all probability distributions on a finite set and is the most fundamental object in the finite probability theory. In this paper we give a characterization of statistical models on finite sets which are statistically equivalent to probability simplexes in terms of $α$-families including exponential families and mixture families. The subject has a close relation to some fundamental aspects of information geometry such as $α$-connections and autoparallelity.

Information-geometrical characterization of statistical models which are statistically equivalent to probability simplexes

TL;DR

This paper gives a characterization of statistical models on finite sets which are statistically equivalent to probability simplexes in terms of α-families including exponential families and mixture families.

Abstract

The probability simplex is the set of all probability distributions on a finite set and is the most fundamental object in the finite probability theory. In this paper we give a characterization of statistical models on finite sets which are statistically equivalent to probability simplexes in terms of -families including exponential families and mixture families. The subject has a close relation to some fundamental aspects of information geometry such as -connections and autoparallelity.

Paper Structure

This paper contains 8 sections, 53 equations.

Table of Contents

  1. An introductory example
  2. Statement of the main result
  3. Some facts about condition (i)
  4. $\alpha$-family, e-family and m-family
  5. Proof of (i) $\Rightarrow$ (iv)
  6. Equivalence of (ii), (iii) and (iv)
  7. Proof of (ii) $\Rightarrow$ (i)
  8. Conclusion

Theorems & Definitions (3)

  • proof
  • proof
  • proof