## Statistical Decision Rules and Optimal Inference |

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### Contents

1 Statistical problems and statistical decisions | 1 |

2 Probability measures and conditional distributions | 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 |

### Common terms and phrases

a-algebra affine algebra arbitrary atoms barycentric coordinates bounded canonical parametrization Caph Caph(fl chart coincide completes the proof conditional distribution congruent consider constructive continuous convergent convex function convex set corollary corresponding countable covariant decision rule defined definition density determined differentiable directional statistic domain equal equivalent equivariant estimator exists exponent family faithful canonical families of probability finite finite-dimensional follows formula geodesic family geometry Hence homomorphisms identically information deviation Int Gy integral interval invariant laws Lebesgue lemma is proved likelihood function loss function manifold Markov morphism matrix measurable function measurable space metric monotone natural parametrization neighborhood nonnegative norm normalizing divisor obtain one-to-one orthoprojection outcomes parameter partial derivatives partition priori probability distributions probability measure problem S-measurable sequence simplex statistical decision strictly convex subfamily subset sufficient statistic tangent vectors tensor Theorem theory tion transformation unique values vector field zero