## Statistical Decision Rules and Optimal InferenceNone available in plain English. |

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### Contents

1 | |

13 | |

3 Smooth manifolds and their mappings | 37 |

4Category theory and geometry | 50 |

THE FORMAL DECISION PROBLEM | 65 |

6 Markov geometry of families of probability distributions | 76 |

7 Geometry of dominated families of probability distributions | 92 |

8 Invariant information characteristics | 113 |

16 Continuously differentiable families of probability distributions | 228 |

GEOMETRY OF EXPONENT FAMILIES OF PROBABILITY DISTRIBUTIONS In this chapter we develop the theory of exponent families of prob... | 245 |

18 Exponent families of probability distributions | 262 |

19 Natural parametrization of exponent families | 279 |

20 Conjugate parametrizations of an exponent family of probability distributions | 290 |

21 Distributions of values of the directional statistic of an exponent family and related families | 303 |

22 Nonsynunetric Pythagorean geometry of the information deviation | 319 |

23 Charts of an exponent family of probability distributions | 334 |

EQUIVARIANT DIFFERENTIAL GEOMETRY OF A COLLECTION OF PROBABILITY DISTRIBUTIONS | 127 |

10 Project ive geometry of a collection of probability distributions | 140 |

11 Invariant Riemannian metric on a manifold of probability distributions | 156 |

12 Geodesic mean of probability distributions | 165 |

SMOOTH FAMILIES OF PROBABILITY DISTRIBUTIONS AND THE INFORMATION INEQUALITY 13 Finitedimensional approximation of infi... | 185 |

14 Differentiate families of probability distributions | 199 |

15 The information inequality | 213 |

OPTIMAL DECISION RULES IN THE EQUIVARIANT PROBLEM OF POINT ESTIMATION | 355 |

25 Estimation of the unknown density of a probability distribution of observations | 370 |

26 Invariant loss functions in problems of mathematical statistics 1 In the last two sections we considered wellknown examples of statistical point esti... | 391 |

27 Optimal estimators for smooth families of probability distributions | 403 |

28 Quasihomogeneous families of probability distributions | 437 |

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algebra assertion assume atoms bounded canonical Caph(fl charge chart closed coincide collection complete condition connection consider constant constructive continuous convergent convex coordinates corollary corresponding decision rule defined definition density derivatives described determined differentiable directional domain equal equivalent estimator example exists exponent family fact field finite fixed follows formula function geodesic geometry given Hence identically independent inequality integral interval invariant laws Lebesgue Lemma limit linear linear connection lower manifold mapping Markov mean measurable metric monotone morphism natural neighborhood norm normal object observations obtain parameter partition positive probability distributions problem projection PROOF proved random relation relative respect satisfies sequence simplex smooth space statistic strictly subfamily subset sufficient tangent tangent vectors Theorem theory tion transformation transition translation true unique values vector zero

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Page 8 - One is that if someone mistreats you, repay him in the same way: an eye for an eye, a tooth for a tooth, a death for a death.